Characteristics of Conservation
Laws for Finite Difference
Equations
Timothy J. Grant
Submitted for the degree of Doctor of Philosophy
September 2011
Supervisor: Prof. Peter Hydon
Department of Mathematics
Faculty of Engineering and Physical Sciences
c
Timothy
J. Grant 2011
Abstract
The main result of this thesis is to find a characteristic for conservation laws (CLaws) of explicit
difference equations and prove that there is a one-to-one correspondence between the equivalence
class of characteristics and conservation laws. This result enables us to prove the converse of
Noether’s theorem for explicit difference equations: that there is a one-to-one correspondence
between variational symmetries and conservation laws. Using the characteristic we show that
the infinite hierarchy of conservation laws generated by the Gardner method for dpKdV are
distinct.
We use the characteristic to form a new method for finding CLaws of quad-graph equations that
is similar to the existing method for searching for symmetries though more complicated. We
apply this method to three quad-graph equations to find new five-point CLaws.
The second focus of this thesis has been to develop a method for discretizing a scalar PDE, with
polynomial nonlinearities, so as to locally preserve as many conservation laws as possible. This
method is applied to the KdV equation resulting in new and known schemes that preserve the
first and second CLaws together and the first and third CLaws together. These methods are
then studied numerically and compared with a multisymplectic and a volume preserving scheme.
ii
Acknowledgements
I am very grateful to Peter Hydon, without whose guidance, help and especially patience this
thesis would never have been written; and to the Natural Environment Research Council for
funding this project.
I’d also like to thank my various office and house mates over the years for making my time in
Surrey an enjoyable, and on occasions, amusing experience. Special thanks must go to Neil, for
coffee, cricket and keeping me sane. Chris also deserves credit for, somehow, surviving living
with me for three years.
Thanks to Dave for allowing me to use Phoenix, even though it was unsuccessful.
I must also thank everyone at Guildford Park Church for welcoming me into their lives and
especially the Matthews family for letting me live with them as I finish writing this thesis.
Thanks to Sarah as she’s patiently waited for me to finish.
Ultimately, I thank the triune God, without whose help I can do nothing:
1 Unless the LORD builds the house, those who build it labour in vain.
Unless the LORD watches over the city, the watchman stays awake in vain.
2 It is in vain that you rise up early and go late to rest,
eating the bread of anxious toil;
for he gives to his beloved sleep.
Psalm 127:1-2.
iii
Contents
1 Introduction
1.1
1.2
1.3
1.4
1
Conservation Laws of Differential Equations . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Kovalevskaya form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Numerical Solution of Differential Equations . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
Traditional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.2
Geometric integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
CLaws of Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.1
Direct construction of CLaws . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3.2
CLaws of the dpKdV equation . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.3
Finding a characteristic? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2 Brute Force
17
2.1
Some Existing Explicit Mass and Momentum Conserving Schemes . . . . . . . .
17
2.2
The Brute-Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.1
Discretization approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.2
Groebner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.3
The recipe for finding schemes . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3
A Three-Parameter Family of Methods . . . . . . . . . . . . . . . . . . . . . . . .
27
2.4
Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.4.1
Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4.2
Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.5
Some Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3 Characterizing CLaws of Difference Equations
37
3.1
Scalar Ordinary Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
Partial Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
v
Contents
3.2.1
Kovalevskaya form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.2
The characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.2.3
A trivial CLaw implies a trivial characteristic . . . . . . . . . . . . . . . .
45
3.2.4
A trivial characteristic implies a trivial CLaw . . . . . . . . . . . . . . . .
45
3.2.5
Example: the characteristic on solutions for dpKdV . . . . . . . . . . . .
48
3.3
The Gardner CLaws for dpKdV . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4
Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4 A New Approach to Finding CLaws
4.1
53
Reconstruction of CLaws from Seeds . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.1.1
Example: reconstruction of CLaws of dpKdV . . . . . . . . . . . . . . . .
54
4.2
The Adjoint of the Linearized Symmetry Operator . . . . . . . . . . . . . . . . .
58
4.3
Finding CLaws for Quad-Graph Equations . . . . . . . . . . . . . . . . . . . . . .
60
4.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
5 Numerical Schemes
73
5.1
Discrete Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.2
Method for PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.3
Numerical Schemes for KdV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
5.3.1
Schemes that preserve the first and second CLaws . . . . . . . . . . . . .
83
5.3.2
Schemes that preserve the first and third Claws . . . . . . . . . . . . . . .
99
5.3.3
Preserving all three CLaws together? . . . . . . . . . . . . . . . . . . . . . 109
5.4
Direct construction of explicit schemes to preserve one CLaw . . . . . . . . . . . 111
5.4.1
Preserving the second CLaw of KdV . . . . . . . . . . . . . . . . . . . . . 112
5.4.2
Preserving the third CLaw of KdV . . . . . . . . . . . . . . . . . . . . . . 113
5.4.3
Preserving the fourth CLaw of KdV . . . . . . . . . . . . . . . . . . . . . 113
5.4.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Analysis and Numerical Experimentation
115
6.1
Linear Stability Analysis of Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2
A Nonlinear Stability Consideration . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3
Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4
Numerical Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Conclusions and Further Study
7.1
153
The Best-Performing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
vi
Contents
7.2
Summary and Some Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . 157
A Linearizations of the Schemes
159
B Maple Code for Solving the ALSC
163
C Maple Code for Finding Discretizations
169
vii
List of Figures
2.1
Points used in brute force method . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2
Dispersion relations for the linearized three-parameter scheme . . . . . . . . . . .
30
2.3
Schemes with the same linearization as the Zabusky-Kruskal scheme (θ = 0) . . .
32
2.4
Schemes with different linearizations . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.5
Different schemes with the same linearization: θ = − 32 . . . . . . . . . . . . . . .
34
3.1
A 2-dimensional P∆E in Kovalevskaya form . . . . . . . . . . . . . . . . . . . . .
41
3.2
Transformation of a quad-graph equation into Kovalevskaya form . . . . . . . . .
42
3.3
A graphical representation of the terms in the characteristic . . . . . . . . . . . .
46
3.4
A Gardner CLaw of dpKdV and its characteristic
. . . . . . . . . . . . . . . . .
50
4.1
Pictorial representation of the adjoint of the linearized symmetry condition . . .
61
4.2
Terms that depend on u−10 in the ALSC
. . . . . . . . . . . . . . . . . . . . . .
63
4.3
Terms that depend on u30 in the ALSC . . . . . . . . . . . . . . . . . . . . . . .
65
5.1
The linear terms in the characteristic and O∆E . . . . . . . . . . . . . . . . . . .
76
5.2
The Euler operator acting on a product of linear terms . . . . . . . . . . . . . . .
77
5.3
A term with an even number of points . . . . . . . . . . . . . . . . . . . . . . . .
77
5.4
Linear terms with two independent variables . . . . . . . . . . . . . . . . . . . .
78
5.5
A symmetric discretization of a quadratic term . . . . . . . . . . . . . . . . . . .
80
5.6
f1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The discretization of 12 u2 in F
84
Stencils for the eight-point discretizations of Q2 and uux . . . . . . . . . . . . . .
90
5.8
More stencils for the eight-point discretizations of Q2 and uux . . . . . . . . . . .
91
5.9
Stencils of the ten-point discretizations of Q2 and uux . . . . . . . . . . . . . . .
97
5.10 More stencils of the ten-point discretizations of Q2 and uux . . . . . . . . . . . .
98
5.7
5.11 Stencils of eight-point discretizations of Q3 and KdV . . . . . . . . . . . . . . . . 104
5.12 Stencils of ten-point discretizations of Q3 and KdV . . . . . . . . . . . . . . . . . 107
6.1
KdV (c = 8) with eight-point schemes with the best value of ψ . . . . . . . . . . 122
ix
List of Figures
6.2
KdV (δ = 0.0222 ) with eight-point schemes with the best value of ψ . . . . . . . 123
6.3
KdV (δ = 0.0222 ) with eight-point schemes with the best value of ψ and a refined
mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4
KdV (c = 8) with eight-point schemes that exactly preserve the structure of the
1st and 2nd CLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5
KdV (δ = 0.0222 ) with eight-point schemes that exactly preserve the structure of
the 1st and 2nd CLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6
KdV (c = 8) with eight-point schemes, skewed in different directions, that preserve
the 1st and 2nd CLaws in characteristic form . . . . . . . . . . . . . . . . . . . . 129
6.7
KdV (c = 8) with eight-point schemes with the parameters found to best preserve
the second CLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.8
KdV (c = 1) with eight-point schemes . . . . . . . . . . . . . . . . . . . . . . . . 131
6.9
KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 2nd CLaws
in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.10 KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 2nd CLaws
in characteristic form with the same linearization as the narrow box scheme . . . 133
6.11 KdV (c = 8) with the first ten-point scheme that preserve the 1st and 2nd CLaws
in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.12 KdV (c = 8) with ten-point schemes that preserve the 1st and 2nd CLaws in
characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.13 KdV (δ = 0.0222 ) with the first ten-point scheme that preserve the 1st and 2nd
CLaws in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.14 KdV (c = 8) with eight-point schemes that preserve the 1st and 3rd CLaws . . . 139
6.15 KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 3rd CLaws
140
6.16 KdV (c = 8) with ten-point schemes that preserve the 1st and 3rd CLaws . . . . 143
6.17 KdV (δ = 0.0222 ) with ten-point schemes that preserve the 1st and 3rd CLaws . 144
6.18 The three-step explicit scheme that preserves the 1st and 3rd CLaws . . . . . . . 145
6.19 The three-step explicit scheme that preserves the 1st and 3rd CLaws for δ = 0.0222 146
6.20 Dispersion relations of numerical schemes for the one-way wave equation . . . . . 148
6.21 Dispersion relation for eight-point schemes that preserve the 1st and 2nd CLaws
with different discretizations for ut . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.22 Dispersion relation for ten-point schemes that preserve the 1st and 2nd CLaws . 149
6.23 Dispersion relations for eight-point schemes that preserve the 1st and 3rd CLaws 150
6.24 Dispersion relations for ten-point schemes that preserve the 1st and 3rd CLaws . 151
6.25 Dispersion relation for the explicit three-step scheme . . . . . . . . . . . . . . . . 151
7.1
Most notable schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2
The most notable schemes with µ = ν . . . . . . . . . . . . . . . . . . . . . . . . 156
x
List of Tables
2.1
The three-parameter family and its densities . . . . . . . . . . . . . . . . . . . . .
28
3.1
Densities for the three and five point CLaws of transformed dpKdV . . . . . . .
43
3.2
Seeds of the transformed dpKdV . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.1
Solutions of the ALSC for the Hydon–Viallet equation . . . . . . . . . . . . . . .
67
4.2
Solutions of the ALSC for the potential Lotka–Volterra equation . . . . . . . . .
67
4.3
Solutions of the ALSC for the potential Hydon–Viallet equation . . . . . . . . . .
68
4.4
CLaws of the Hydon–Viallet equation . . . . . . . . . . . . . . . . . . . . . . . .
70
4.5
CLaws of potential Hydon–Viallet equation . . . . . . . . . . . . . . . . . . . . .
71
4.6
CLaws of the potential Lotka–Volterra equation . . . . . . . . . . . . . . . . . . .
72
5.1
Eight-point schemes with 180◦ antisymmetry that preserve the first and second
CLaws of KdV in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.2
◦
Ten-Point schemes with 180 antisymmetry that preserve the first and second
CLaws in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
5.3
Eight-Point schemes with 180◦ antisymmetry that preserve the first and third
CLaws in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4
Ten-Point schemes with 180◦ antisymmetry that preserve the First and Third
CLaws in characteristic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.1 Linearized eight-point schemes that preserve the first and second CLaws and their
dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
A.2 Selection of the linearized ten-point schemes that preserve the first and second
CLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.3 Linearized eight-point schemes that preserve the first and third CLaws and their
dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.4 Linearized ten-point schemes that preserve the first and third CLaws and their
dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
xi
Chapter 1
Introduction
This thesis is concerned with the study of conservation laws (CLaws) of difference equations
and in particular their characteristics, which enable us to identify when different CLaws are
equivalent. Therefore we begin this introduction with a brief summary of the relevant continuous theory. We then discuss the numerical solution of differential equations by finite difference
methods, and how it is desirable to preserve geometric features, such as CLaws, in the discretization. This motivates our discussion of difference equations and their CLaws. We then conclude
the introduction with an outline of the thesis.
1.1
Conservation Laws of Differential Equations
A conservation law (CLaw) of a system of differential equations, ∆ = 0, is a divergence expression that vanishes on solutions of the system,
DivF = 0
when ∆ = 0.
(1.1)
∆ ≡ ut + uux + uxxx = 0,
(1.2)
For example the KdV equation,
has an infinite number of CLaws. In particular it has the CLaws
1 2
u + uxx = ∆,
0 =Dt (u) + Dx
(1.3)
2
1 3
1
1 2
u + Dx
u + uuxx − u2x = u∆,
(1.4)
0 =Dt
2
3
2
1 3
1 4
0 =Dt
u − u2x + Dx
u + u2 uxx − 2ux uxxx + u2xx − 2u2x u = u2 − 2ux Dx ∆, (1.5)
3
4
where
Dx ≡
∂
∂
∂
∂
+ ux
+ uxx
+ uxt
+ ...
∂x
∂u
∂ux
∂ut
is the total x derivative and Dt is the total t derivative. Drazin and Johnson [20] state that,
when applied to the water wave problem, (1.3) describes the conservation of mass, (1.4) the
conservation of momentum and (1.5) the conservation of energy. A CLaw is trivial if either F
1
Chapter 1: Introduction
vanishes on solutions of the system or if DivF ≡ 0 (a null divergence - the divergence expression is
zero without needing to be on solutions of the differential equations). Two CLaws are equivalent
if they differ by a trivial CLaw. If the system is in Kovalevskaya form (see §1.1.1) then we can
integrate the CLaw by parts to find an equivalent CLaw,
DivF̃ = Q · ∆,
(1.6)
which is said to be in characteristic form. The multiplier Q is called a characteristic of the
CLaw. For instance, the characteristic form of (1.5) is
DivF̃ = u2 + 2uxx ∆,
so Q = u2 + 2uxx is the characteristic and (1.5) is equivalent to the CLaw
Dt
1 3
u − u2x
3
+ Dx
1 4
u + u2 uxx + 2ux ut + u2xx
4
= 0.
(1.7)
A characteristic is trivial if it vanishes on solutions of the system of PDEs. Two characteristics that differ by a trivial characteristic are said to be equivalent. Given a system of PDEs
in Kovalevskaya form, Alonso showed [37, 45], there is a one-to-one correspondence between
equivalence classes of characteristics and equivalence classes of conservation laws. Therefore
characteristics can be used to identify when two seemingly different CLaws are equivalent.
The equivalence between classes of CLaws and characteristics leads to various applications of
characteristics. Characteristics have their most celebrated application in Noether’s Theorem
[43, 45] where they are used to construct CLaws from variational symmetries. Anco and Bluman
[8, 9] present a direct construction method for CLaws of PDEs in Kovalevskaya form. The method
uses characteristics and does not require the PDE to have a variational structure. Their method
states that if a solution of the adjoint of the linearized symmetry condition satisfies additional
constraints then that solution is a characteristic for a CLaw. Their method is a consequence
of the fact that the kernel of the Euler operator consists of total divergences1 . Anco, Bluman
and Wolf [7] show that if a nonlinear system of PDEs has a characteristic that depends on the
arbitrary solution of some linear system of PDEs then, provided additional conditions are met,
the nonlinear system can be invertibly mapped into the adjoint of the linear system.
Having seen the usefulness of characteristics it seems desirable to develop an analogous theory
for CLaws of difference equations.
1.1.1
Kovalevskaya form
Since the results concerning characteristics require the PDE to be in Kovalevskaya form, a brief
discussion is presented here. A system of equations is in Cauchy-Kovalevskaya form [45, 9] if
∂ nα uα
= Γα (t, x, ũ(n) ),
∂tnα
α = 1 . . . , q,
(1.8)
where t, x are the independent variables, u are the dependent variables and ũ(n) denotes all
derivatives of each uα up to (but not including) order nα w.r.t. t and arbitrary order in x.
1 This
result is part of the fact that the Variational Complex is exact on totally star-shaped domains - see
Olver [45] for details
2
Chapter 1: Introduction
Theorem 1.1.1. The Cauchy-Kovalevskaya Theorem [45]
Given a system of equations in Cauchy-Kovalevskaya form (1.8); with Cauchy data given by
∂ kα uα
α = 1, . . . , q, kα = 1, . . . , nα − 1,
(1.9)
(x, t0 ) = hα
kα (x),
∂tkα
where the hα
kα are analytic functions on the hyperplane t = t0 for x in a neighbourhood of a
point x0 ; if the functions Γα are analytic in their arguments then there exists a unique analytic
solution u = f (x, t) for the Cauchy problem, (1.8) and (1.9), defined for some neighbourhood of
the point (x0 , t0 ).
1.2
1.2.1
Numerical Solution of Differential Equations
Traditional approach
One of the main motivations for the study of difference equations is to better understand and
improve the numerical integration of differential equations. A good introduction to the numerical
solution of differential equations is Iserles’ book [31]. He illustrates the finite difference approach
to numerical integration by the Euler method for ODEs. The aim is to find an approximate
solution of
dy(t)
= f (t, y(t)), t ≥ t0 , y(t0 ) = y0 .
dt
The solution is given by
Z t
y(t) − y(0) =
f (τ, y(τ ))dτ ≈ f (t0 , y(t0 ))(t − t0 ),
t0
where the integral has been approximated in the simplest way: by the rectangle rule. Assuming
that the region of integration is split into time steps of equal length, tn+1 − tn = ν, the result is
the (forward) Euler method
yn+1 = yn + νf (tn , yn ),
n = 0, 1, . . . ,
where yn is the numerical approximation of the exact solution y(tn ). Iserles states that all finitedifference methods for numerically solving ODEs, such as Runge-Kutta and multi-step schemes,
are a generalization of the above idea.
The local truncation error (LTE) of a numerical method is the error made in one step of the
method. It is found by replacing the numerical approximation with the exact solution and
expanding using Taylor series. For example the local truncation error for the Euler method is
given by
LTE =y(tn + ν) − y(tn ) − νf (tn , y(tn )),
=y(tn ) + νy 0 (tn ) +
=
ν 2 00
y (ξ),
2!
ν 2 00
y (ξ) − y(tn ) − νf (tn , y(tn )),
2!
for some ξ ∈ [tn , tn+1 ],
so the LTE of the Euler method is O(ν 2 ). In order for a numerical method to be used it must
be convergent: as the step size tends to zero, the numerical solution tends to the true solution.
So, when solving the ODE on the interval [t0 , t0 + t∗ ],
lim
max
ν→0+ n=0,1,...,bt∗ /νc
||yn − y(tn )|| = 0.
3
Chapter 1: Introduction
A numerical method is convergent if for every ODE with a Lipschitz function f the method
converges. This highlights the traditional approach, which is to come up with widely applicable
methods with good control of local errors.
There are various methods for numerically solving PDEs. However our interest is in difference
equations, so we will only discuss the finite difference method. For the finite difference method
the continuous independent variables are replaced by a set of discrete of points. For simplicity
we consider only PDEs with a single spatial variable, x, and time, t. Typically a uniform grid
is used so that xm = x0 + mµ and tn = t0 + nν, where µ and ν are constants. The dependent
variable is approximated by a grid-function umn so that umn ≈ u(xm , tn ) and derivatives are
approximated by finite differences. For example the heat equation,
∂u ∂ 2 u
= 0,
−
∂t
∂x2
can be solved using the Euler method
um(n+1) − umn
u(m−1)n − 2umn + u(m+1)n
= 0.
−
ν
µ2
The Local truncation error (LTE) of a finite difference scheme is the error between the PDE and
the scheme about the point (xm , tn ). To perform this calculation the grid-function is replaced
by a Taylor expansion of the exact solution i.e.
u(m+i)(n+j) 7→ u(xm + iµ, t + jν) = u(xm , tn ) + iµux (xm , tn ) + jνut (xm , tn ) + . . . .
For example, the LTE for the Euler discretization of the heat equation is given by
LTE =ut (xm , tn ) − uxx (xm , tn )−
u(xm , tn + ν) − u(xm , tn ) u(xm − µ, tn ) − 2u(xm , tn ) + u(xm + µ, tn )
−
+
,
ν
µ2
=O(µ2 , ν).
If the LTE tends to zero as the step sizes tend to zero then the method is consistent. In the
above example the time and space steps can tend to zero independently and the LTE will tend
to zero. More generally, for a method to be consistent, the step sizes may need to tend to zero
along a specific refinement path, ν = λµr where λ and r are both positive constants, in order
for the LTE to tend to zero [42]. If the discretization is consistent along any refinement path, so
that λ and r can take any positive values, then the discretization is unconditionally consistent.
The whole purpose of finite difference schemes is that they are convergent. The numerical
solution should tend to the exact solution as the step sizes tend to zero. For an evolution
equation, let un denote the points in the grid function at the nth time level, arranged in a
vector and, similarly, let u(tn ) denote the exact solution at the nth time level. In order to define
convergence, the space is equipped with the Euclidean norm

1/2
X
||g||µ = µ
|gj |2  ,
j
where the sum is over all the grid points in a time level. A method is convergent [31], along a
refinement path, if for every initial condition and for all t∗ > 0 it is true that
lim
max ∗ ||un − u(tn )||µ = 0.
µ→0
n=0,1,...,bt /νc
4
Chapter 1: Introduction
A numerical method is stable along a refinement path if for every t∗ there exists a constant c(t∗ )
such that
||unµ || ≤ c(t∗ ),
n = 0, 1, . . . , bt∗ /νc ,
µ → 0.
For linear equations the stability of the numerical method can be easily checked using Fourier
analysis. This is not the case for nonlinear problems. For well-posed linear evolutionary PDEs
the Lax-Richtmyer equivalence theorem states that difference schemes are convergent if and only
if they are stable and consistent. We have no such theory for nonlinear equations.
The intuition that the theorem [Lax-Richtmyer] gives for problems that fall outside the scope of Lax-Richtmyer, however, is faulty, since consistency and stability
are often insufficient for convergence, and convergence need not imply stability in
general.[1]
A major distinction between numerical schemes is whether they are explicit or implicit. An
explicit scheme depends on only one point on the (n + 1)th time level (all other points are at
earlier time levels) and the numerical scheme is uniquely solved for this point - thus the PDE
can be solved one point at a time. This is not the case for implicit schemes and so the whole
(n + 1)th row of points must be solved for at the same time. For explicit schemes for hyperbolic
equations to be stable, they must satisfy the Courant-Friedrichs-Lewy (CFL) condition. For
hyperbolic equations the initial conditions along an interval determine uniquely the solution of
the PDE for some set of points (x, t) called the domain of dependence. If we attempt to use an
explicit method such that the points being solved for stay outside the domain of dependence as
the step sizes tend to zero, then the numerical solution cannot converge there. Thus the CFL
condition states that, for hyperbolic equations, the ratio µν should be small enough so that the
new point fits into the domain of dependence.
1.2.2
Geometric integration
Geometric integration seeks numerical schemes that preserve geometric features of differential
equations rather than focusing on the control of local errors of generic methods. A good review
is provided by Budd and Piggot [17]. Geometric structures provide constraints for the behaviour
of the system and so it is desirable to have a numerical scheme that has analogues of the same
constraints. Thus the numerical scheme will replicate the desired qualitative behaviour and may
have improved stability and accuracy compared with generic methods applied to the differential
equations.
Numerical schemes have been developed to preserve various geometric structures. Below, we
provide a brief overview of methods that preserve symplecticity and first integrals for ODEs,
and multisymplecticity and conservation laws for PDEs, as these are the most relevant methods
to this thesis. However, these are not the only structures people have sought to preserve.
Symmetries of equations are another important feature that numerical methods can be built
to preserve. Physical problems commonly have scaling symmetries, as many physical laws hold
for small and large scales. Therefore numerical methods have been developed to preserve these
scaling symmetries and more generally Lie point symmetries; such methods generally require
a moving mesh. Particular attention has focused on differential equations with a variational
structure [36]. Schemes that preserve CLaws result (by a discrete analogue of Noether’s theorem)
5
Chapter 1: Introduction
by forming a discrete analogue of the Lagrangian which inherits the variational symmetries of
the continuous problem. However not all PDEs have a variational structure so this method
cannot always be applied.
1.2.2.1
Geometric integration of ODEs
The main focus for geometric integration has been on ODEs, as has been documented in the
monograph [24]. The lecture notes of McLachlan and Quispel [38] provide a good introduction.
The idea, as has already been stated, is that the numerical method shares some of the geometric
properties of the ODE system and this should result in good long term behavior. In particular we
would like the numerical integrator to share properties of the flow map in phase space. However
in order to define maps on the phase space we are limited to one-step methods:
Multi-step methods do not define a map on phase space, because more than one
initial condition is required. They can have geometric properties, but in a different (product) phase space which can alter the effects of the properties. This puts
geometric integration firmly into the “single step” camp. If a system is defined on
a sphere, one should stay on that sphere: anything else introduces spurious, nonphysical degrees of freedom. [38]
Perhaps the most celebrated example of geometric integration is the use of symplectic integrators
for Hamiltonian ODEs (see for example [34]). Some well known methods have been shown to
be symplectic integrators, for example the implicit midpoint rule, the Stömer-Verlet method
and the Gauss collocation methods. The following is taken from [17]: a system of ODEs is in
canonical Hamiltonian form if
du
= J −1 ∇H,
dt
(1.10)
where u = (p, q)T with p, q ∈ Rd , H = H(p, q) is the Hamiltonian function, ∇ is the operator
T
∂
∂
∂
∂
∂
∂
,
,...,
,
,
,...,
,
∂p1 ∂p2
∂pd ∂q1 ∂q2
∂qd
and J is the skew-symmetric matrix
J=
0
−Id
Id
0
!
,
where Id is the identity matrix of dimension d. The flow of (1.10) is a symplectic map of the
phase space ψ : R2d 7→ R2d i.e.
ψ 0T Jψ 0 = J,
where ψ 0 is the Jacobian of the flow map. In fact, a flow is symplectic if and only if the flow is
Hamiltonian [24].
Symplecticity has the following, important, geometrical interpretation. If M is
any 2-dimensional manifold in R2d , we can define Ω(M ) to be the integral of the sum
over the oriented areas, of its projections onto the (pi , qi ) plane (so if d = 1 this is
the area of M ). If ψ is a symplectic map, then Ω(M ) is conserved throughout the
evolution. [17]
6
Chapter 1: Introduction
In addition, for autonomous Hamiltonian equations the Hamiltonian is a conserved quantity. An
important fact is that the composition of two symplectic maps is a symplectic map. This leads
to the idea of a symplectic integrator
A numerical one-step method is called symplectic if the one-step map
y1 = Φh (y0 )
is symplectic whenever the method is applied to a smooth Hamiltonian system.[24]
Budd and Piggot state:
a symplectic discretization of a Hamiltonian problem is a Hamiltonian perturbation of the original. The importance of this observation cannot be over-emphasized.
It implies that we can use the theory of perturbed Hamiltonian systems (in particular
KAM theory) to analyze the resulting discretization and it places a VERY strong
constraint on the possible dynamics of the discrete system. [17].
However, Ge and Marsden [57] proved for non-integrable equations that a numerical method
with fixed time steps cannot be symplectic and exactly preserve energy at the same time.
The desire to preserve energy leads to methods that discretize the ODE so as to preserve first
integrals. McLachlan and Quispel [38], who we follow here, outline how this is done (more
details are in [40]). The method used is to rewrite the ODE as a skew gradient system which
is discretized using a discrete gradient. For example, given a system of ODEs dx
dt = f (x) which
has a first integral I(x),
dx
d
I=
· ∇I = f (x)T ∇I = 0,
dt
dt
if we choose
S=
f (x)(∇I)T − (∇I)f T (x)
,
|∇I|2
then S T = −S and the ODE can be rewritten as
dx
= S∇I.
dt
They note that the choice of S is not unique. The skew discrete-gradient system is then discretized as
xn+1 − xn
¯
= S̃ ∇I(x
n , xn+1 ),
ν
¯ is a discrete gradient which is defined by the axioms
where ∇
¯ · (xn+1 − xn ),
I(xn+1 ) − I(xn ) =(∇I)
¯
∇I(x
n , xn+1 ) =∇I(xn ) + O(xn+1 − xn ),
and S̃ is any consistent antisymmetric matrix, such as S̃(xn , xn+1 ) = S((xn + xn+1 )/2). The
discretization then has the same first integral:
¯ · (xn+1 − xn ),
I(xn+1 ) − I(xn ) =(∇I)
¯ T S̃(∇I)
¯ = 0.
=ν(∇I)
7
(1.11)
(1.12)
Chapter 1: Introduction
We note that ∇I is the characteristic of the first integral,
d
dx
I=
· ∇I = ∇I ·
dt
dt
dx
− S∇I
dt
T
+ ∇I · (S∇I) = 0,
as the last term vanishes by the skew-symmetry of S. Systems of equations with p first integrals
are treated in a similar manner [40]. Instead of a skew-symmetric matrix the system is written
as a skew-symmetric (p + 1)-tensor acting on the gradients of the first integrals. This is then
discretized, using discrete gradients, to give a method that preserves all the first integrals. Steeb
et al. [53] extend their work to time-dependent first integrals.
1.2.2.2
Geometric integration of PDEs
The geometric integration of PDEs has been far less studied. One of the aims of this thesis is to
construct finite difference schemes for KdV that locally preserve its physically relevant CLaws.
Therefore we outline below some of the progress made in preserving CLaws of PDEs.
Furihata [23] constructs finite difference schemes for equations of the form
∂u
=
∂t
∂
∂x
2n+1
δG
,
δu
n ∈ N,
that inherit the energy conservation property i.e.
Z
d
G(u, ux )dx = 0.
dt
He does this by discretizing the energy function, G, and then applying a discrete variational
derivative to construct the difference scheme. Using this method (the discrete variational derivative method - DVDM) he provides a scheme for KdV that preserves the energy (1.5). In [32]
Koide and Furihata generate schemes for the regularized long wave equation (BBM equation)
that preserve the mass and momentum and the mass and energy using the DVDM. For the
nonlinear momentum preserving scheme they show that if the step sizes satisfy a certain condition then solutions exist and are unique. The BBM equation, unlike KdV which has an infinite
number of CLaws, has only three, physically relevant, CLaws [44, 21]; therefore the DVDM is
applicable to equations that are not integrable.
The most famous finite difference scheme for solving KdV is Zabusky and Kruskal’s scheme (Z-K)
from their paper [56] in which they coined the phrase solitons. Their scheme is a two-step explicit
method that has finite difference analogs of the mass and momentum CLaws. Sanz-Serna [52]
showed that the Z-K scheme is subject to nonlinear instability, he then provided a scheme with
an adaptive time-step that preserves the mass and momentum exactly with periodic boundary
conditions; however the divergence form of the momentum CLaw is lost.
In addition to the above methods, which preserve the divergence expression, McLachlan [39] constructs spatial discretizations of PDEs so that the resulting ODE systems have as first integrals
the conserved quantities of the PDEs. The ODE system can then be integrated using a discrete
gradient method to preserve the conserved quantities. Unlike the other methods, this does not
preserve the local form of the CLaw. De Frutos and Sanz-Serna demonstrated the benefits of
preserving CLaws [19]; they showed that if a numerical method conserves the momentum for the
KdV equation then that numerical method can perform better than a non-conservative scheme
with more accurate LTEs. They showed that this was because, for the one-soliton solution,
8
Chapter 1: Introduction
the numerical error for a conservative scheme went into the phase rather than the amplitude of
the soliton. This highlights the desirability of creating finite difference methods that preserve
CLaws.
Multisymplectic PDEs
Much recent focus has been on multisymplectic PDEs. Bridges and Reich [14] provide an
overview of methods for Hamiltonian PDEs from which the following is taken. A Hamiltonian
PDE in one spatial direction is in multisymplectic form if it can be written as
Mzt + Kzx = ∇z S(z),
z ∈ Rd ,
d ≥ 3,
(1.13)
where M and K are constant skew-symmetric matrices, S is a smooth function of z and ∇z S(z)
is the classical gradient on Rd . KdV can be put in this form with d = 4, z = (φ, u, v, w),



M=

0
−1
0
0


1 0 0
0 0 0 

,
0 0 0 
0 0 0


K=

0
0
0
−1
0 0
0 −1
1 0
0 0
1
0
0
0



,

and S(z) = 21 v 2 − uw + 61 u3 . Equations of the form (1.13) have the property that symplecticity
is conserved2
ωt + κx = 0,
with ω :=
1
dz ∧ Mdz,
2
κ :=
1
dz ∧ Kdz.
2
(1.14)
If S does not depend explicitly on t and x then energy and momentum are conserved:
Et + Fx = 0,
It + Gx = 0,
1
E(z) = S(z) − zT Kzx ,
2
1 T
G(z) = S(z) − z Mzt ,
2
1 T
z Kzt ,
2
1
I(z) = zT Mzx .
2
F (z) =
A numerical scheme is a multisymplectic integrator if it preserves a discrete version of the
conservation of symplecticity (1.14). Just as symplectic integrators for ODEs should be onestep methods, multisymplectic methods should be compact (i.e. ‘one-step methods’ in space and
time).
In [12, 11] Ascher and McLachlan investigate multisymplectic schemes for KdV. They seek to
understand the smooth behaviour of the multisymplectic schemes by studying the numerical
dispersion of the linearized equations. From this, they suggest that box-schemes are better than
schemes that have a non-compact spatial discretization because the latter my introduce artificial
wiggles into the solution. Frank et al. [22] study the dispersion relations of linear multisymplectic
PDEs and their discretizations. They state that that the sign of the numerical group velocity
should be the same as the sign of the actual group velocity. In [13] Bridges and Reich prove
the remarkable result that abstract linear Hamiltonian PDEs in multisymplectic form
- discretized with the centered box scheme - conserve energy and momentum exactly;
moreover, it is the local energy and momentum conservation that is preserved by the
discretization.
However they state [14] that with a uniform discretization it is not possible in general to preserve
energy and momentum exactly along with the symplectic structure.
2 For
an introduction to differential forms see [54].
9
Chapter 1: Introduction
1.2.2.3
Our aim
CLaws are very important when modeling physical phenomena. The definition of a CLaw (1.1) as
a divergence expression is a local property, and if integrated over the spatial domain (assuming
vanishing boundary conditions) results in a quantity that is constant on solutions. Because
(1.1) is a local constraint, preserving it provides a greater a constraint on the behaviour than
conserving the quantity that results from the spatial integration.
Our aim is to be able to construct finite difference schemes for PDEs that have finite difference
analogues of the CLaws of the differential equations. We wish to be able to do this without
reference to any special structure such as multisymplecticity so that the method can be applied
to PDEs that don’t have these structures. Just as the variational complex allows the computation
of CLaws of PDEs with no variational structure, the same is true for difference equations. It
is hoped that developing our understanding of CLaws of difference equations will enable finite
difference schemes to be built that have analogues of desired CLaws, with no reference to a
variational structure. However in this thesis we will only attempt to discretize KdV to preserve
the three physically relevant CLaws.
1.3
CLaws of Difference Equations
A system of difference equations has independent variables on a lattice n = (n1 , . . . , np ). The
dependent variables are uα
n for α = 1, . . . , q and take values in R. The natural operators on the
lattice are the shift operators which are defined by
Si : nj 7→ nj + δij ,
α
Si : uα
n 7→ un+1i ,
and SJ = S1j1 . . . Spjp
where δji is the Kronecker delta and 1i is the p-tuple with 1 in the ith place and 0’s in every
other place. A system of difference equations ∆ = (∆1 , . . . , ∆q ) is thus written as
∆α (n, [u]) = 0,
α = 1, . . . , q,
where [u] denotes a finite number of shifts of the dependent variables. When investigating P∆Es
with two independent variables we use (m, n) in preference to (n1 , n2 ). So
Sm : (m, n) 7→ (m + 1, n),
Sn : (m, n) 7→ (m, n + 1).
A conservation law (CLaw) of a system of P∆Es is a divergence expression that vanishes on
solutions of the system:
DivF :=
p
X
i=1
(Si − I)F i = 0 when [∆] = 0,
(1.15)
where [∆] denotes any finite shifts of the system and I is the identity operator. The functions
F i are known as the densities of the CLaw and may have the independent variables and shifts
of the dependent variables as arguments. For ordinary difference equations (O∆Es), (1.15) is
replaced by (Sn − I)φ = 0 when [∆] = 0, so that φ is a first integral. A CLaw can be trivial in
two ways:
1. All of the densities vanish on solutions, i.e. F|[∆]=0 = 0.
10
Chapter 1: Introduction
2. DivF ≡ 0, without reference to the equation [∆] = 0. For instance, this occurs if F is the
difference analogue of a total curl (see [45]).
For brevity we refer to the densities of a trivial CLaw as trivial densities.
An important result, due to Kuperschmidt [33], is that an expression is a total divergence if and
only if it is in the kernel of the discrete Euler operator3 . The component of the discrete Euler
operator that corresponds to the dependent variable uα is given by
X
∂f (n, [u])
Eα (f ) =
S−J
.
∂uα
n+J
J
1.3.1
Direct construction of CLaws
Hydon presented a method for constructing first integrals of O∆Es [25] and a direct construction
method for CLaws of second order scalar P∆Es [26], that are in the analogue to Kovalevskaya
form:
u(m+2)(n+p) = ω(m, n, um , um+1 ),
(1.16)
where um denotes all variables of the form um(n+j) . Therefore, given initial conditions z =
(m, n, um , um+1 ), all values u(m+k)n for k ≥ 2 can be found using shifts of (1.16).
The simplest non-trivial CLaw for (1.16) is an expression of the form
(Sm − I)F (m, n, um , u(m+1)(n+p) ) + (Sn − I)G(m, n, um , um+1 ) = 0,
when (1.16) holds. Hence
(Sn − I)G(m, n, um , um+1 ) = F (m, n, um , u(m+1)(n+p) ) − F (m + 1, n, um+1 , ω).
(1.17)
This does not involve the shift operator in the m direction, Sm . Thus the independent variable m
can now be regarded as a parameter. Therefore (1.17) can be regarded as a functional difference
equation involving one independent variable, n, and two dependent variables, unm and unm+1 .
The LHS of (1.17) is a total difference and is thus in the kernel of the Euler operator [33, 29],
which has two components
X
∂
,
Em =
(Sn )−j
∂um(n+j)
j
Em+1 =
X
(Sn )−j
j
∂
∂u(m+1)(n+j)
.
So by applying the Euler Operator to (1.17) the following equations are obtained
Em F (m, n, um , u(m+1)(n+p) ) − F (m + 1, n, um+1 , ω) = 0,
Em+1 F (m, n, um , u(m+1)(n+p) ) − F (m + 1, n, um+1 , ω) = 0.
(1.18)
(1.19)
By specifying which points F depends on, (1.19) can be solved using invariant differentiation.
For P∆Es with no singular points, if an expression is in the kernel of the Euler operator, it must
be a total divergence. Hence, having found an F that satisfies (1.19), a suitable G must exist.
This can be found using the homotopy operator [29]; however, it is often easier to use inspection.
Hydon [26] describes how the method can be generalized to CLaws which depend on shifts of ω
in the n direction and systems rather than scalar equations.
3 Just
as for differential equations, this forms part of the fact that the Variational Complex for difference
equations is exact on trivial domains [29].
11
Chapter 1: Introduction
1.3.2
CLaws of the dpKdV equation
A quad-graph equation is a scalar P∆E of the form
∆ ≡ u11 − ω(m, n, u00 , u10 , u01 ) = 0,
(1.20)
where m, n ∈ Z are the independent variables and the dependent variable has the values uij :=
u(m + i, n + j) at the four points indicated. We add the usual constraint that (1.20) can be
solved uniquely for each uij . Given initial conditions, z = (m, n, ui0 , u0j ) for i ≥ 0 and j > 0,
(1.20) can be used to determine any uij in the upper right quadrant relative to the point (m, n).
Rasin and Hydon present a modified method for constructing CLaws of quad-graph equations
[47] which they apply to the dpKdV equation
(u11 − u00 )(u10 − u01 ) = 1.
(1.21)
This can be rewritten as
u11 = ω,
ω=
1
+ u00 ,
u10 − u01
(1.22)
u10 = Ω,
Ω=
1
+ u01 .
u11 − u00
(1.23)
and
1.3.2.1
Three-point CLaws
The simplest non-trivial CLaws of quad-graphs are those that have densities that depend only
on the points u00 , u01 , u10 and so have the form
(Sm − I)F (m, n, u00 , u01 ) + (Sn − I)G(m, n, u00 , u10 ) = 0,
(1.24)
on solutions of dpKdV. So
F (m + 1, n, u10 , ω) − F (m, n, u00 , u01 ) + G(m, n + 1, u01 , ω) − G(m, n, u00 , u10 ),
(1.25)
which is a functional equation. Instead of using the Euler operator to eliminate G, terms that
depend on ω are eliminated using the commuting differential operators
L1 =
where ω,u01 =
∂ω
∂u01 ,
∂
ω,u01 ∂
−
,
∂u01
ω,u00 ∂u00
L2 =
∂
ω,u10 ∂
−
∂u10
ω,u00 ∂u00
etc. Applying L2 L1 to (1.25) results in the condition
∂2F
∂2G
+
− (u10 − u01 )2
2
∂u00
∂u200
∂2F
∂2G
−
∂u00 ∂u01
∂u00 ∂u10
− 2(u10 − u01 )
∂F
∂G
+
∂u00
∂u00
= 0,
(1.26)
where F = F (m, n, u00 , u01 ) and G = G(m, n, u00 , u10 ). Differentiating (1.26) w.r.t. u01 three
times eliminates G, giving the condition on F that
∂5F
∂5F
∂4F
− (u10 − u01 )2
+ 4(u10 − u01 )
= 0.
2
3
4
∂u00 ∂u01
∂u00 ∂u01
∂u00 ∂u301
12
(1.27)
Chapter 1: Introduction
As F is independent of u10 , (1.27) can be split into an overdetermined system of PDEs for F by
comparing coefficients of u10 .
∂5F
= 0,
∂u00 ∂u401
∂4F
∂5F
+4
= 0,
2u01
4
∂u00 ∂u01
∂u00 ∂u301
∂4F
∂5F
∂5F
− u201
− 4u01
= 0.
2
3
4
∂u00 ∂u01
∂u00 ∂u01
∂u00 ∂u301
u210 :
u110 :
u010 :
(1.28)
(1.29)
(1.30)
Instead of replacing u11 with ω in the divergence expression (1.24), u10 can be replaced with Ω
to give
F (m + 1, n, Ω, u11 ) − F (m, n, u00 , u01 ) + G(m, n + 1, u01 , u11 ) − G(m, n, u00 , Ω),
(1.31)
As before, operators can be defined which eliminate Ω,
L̂1 =
∂
Ω,u00 ∂
−
,
∂u00
Ω,u01 ∂u01
L̂2 =
∂
Ω,u11 ∂
−
.
∂u11
Ω,u01 ∂u01
Applying these to (1.31) yields
∂ 2 G̃
∂2F
−
− (u11 − u00 )2
∂u201
∂u201
∂2F
∂ 2 G̃
+
∂u00 ∂u01
∂u01 ∂u11
!
− 2(u11 − u00 )
∂F
∂G
−
∂u01
∂u01
= 0,
(1.32)
where G̃ = G(m, n + 1, u01 , u11 ). Differentiating (1.32) w.r.t. u00 three times eliminates G̃ giving
the further condition on F
∂5F
∂4F
∂5F
2
−
(u
−
u
)
+
4(u
−
u
)
11
00
11
00
∂u300 ∂u201
∂u400 ∂u01
∂u300 ∂u01
(1.33)
which, by comparing coefficients of u11 , yields more equations that F must satisfy:
u211 :
u111 :
u011 :
∂5F
=0
∂u400 ∂u01
∂4F
∂5F
4 3
+2 4
u00 = 0
∂u00 ∂u01
∂u00 ∂u01
∂4F
∂5F
∂5F
−4 3
u00 +
u2 = 0
−
3
2
∂u00 ∂u01
∂u00 ∂u01
∂u400 ∂u01 00
(1.34)
(1.35)
(1.36)
Solving the overdetermined system of equations, (1.28) to (1.30) and (1.34) to (1.36), gives
F =C1 (m, n)u200 u201 + C2 (m, n)u200 u01 + C3 (m, n)u00 u201 +
+ C4 (m, n)u00 u01 + f1 (m, n, u01 ) + g1 (m, n, u00 ).
(1.37)
The arbitrary function g1 can be removed by adding the trivial density
FT = (Sn − I)g1 (m, n, u00 ),
GT = −(Sm − I)g1 (m, n, u00 ),
(1.38)
to F and G respectively.
The form of F has been determined by differentiating the divergence expression on solutions
five times. This has created a hierarchy of equations that must all be satisfied in order for the
divergence expression to be a CLaw. By going up this hierarchy, conditions can be found for
13
Chapter 1: Introduction
the unknown functions f1 , Ci and G. By solving these conditions, and moving up the hierarchy,
eventually the most general three-point CLaw is found and the arbitrary constants determine
the individual CLaws. This process results in four non-trivial CLaws:
1.
F =u00 u201 − u200 u01 + u00 − u01 ,
G =u200 u10 − u00 u210 ,
2.
F =(−1)m+n+1 (u00 u201 + u200 u01 − u00 − u01 ),
G =(−1)m+n (u200 u10 + u00 u210 ),
1
m+n+1
2 2
,
3. F =(−1)
u00 u01 − 2u00 u01 +
2
G =(−1)m+n u200 u210 ,
4.
1.3.2.2
1
F =(−1)m+n+1 (u00 u01 − ),
2
G =(−1)m+n (u00 u10 ).
Five-point CLaws
Rasin and Hydon also find all the five-point CLaws of dpKdV. These are CLaws which have
densities depending on u10 , u20 , u11 , u01 , u02 . Shifts of dpKdV (1.22) are used to eliminate the
dependence on u21 and u12 that occur in the shifted densities. The five point CLaws found, that
are not trivially related to the three-point Claws, are
F = ln(u11 − u02 ),
1
G = ln
− u01 + u10 ,
u20 − u11
1
− u11 + u02 ,
6. F = ln
u10 − u01
5.
G = ln (u01 − u10 ) ,
7. F =n ln u11 − u02 +
1
− (m − 1) ln(u02 − u11 ),
u01 − u10
1
G =n ln(u01 − u10 ) − m ln
− u01 + u10 .
u20 − u11
Rasin and Hydon have also used their method to find CLaws for the best-known classes of
quad-graph equations [48, 49].
1.3.3
Finding a characteristic?
A characteristic for CLaws of difference equations cannot be obtained in the same way as for
PDEs. For PDEs, the chain rule ensures that each CLaw is linear in the highest-order derivatives
through which the dependence on ∆ occurs. Integration by parts is then used to find the
characteristic. The analogue of integration by parts for difference equations is summation by
parts. Typically, CLaws depend nonlinearly on [∆], as is shown above, so it is not possible to
construct a characteristic merely using summation by parts. Therefore a new method for putting
a CLaw of a difference equation in characteristic form is needed. In particular the proof that
there is a one-to-one correspondence between equivalence classes of CLaws and characteristics
will not be analogous to the PDE case.
14
Chapter 1: Introduction
In [41] Mikhailov et al. define a cosymmetry of a difference equation as being a member of the
kernel of the adjoint of the linearized symmetry condition (ALSC). In a parenthetical comment,
they claim that a cosymmetry is a characteristic of a CLaw. However they give no justification
for this, and without a proof of one-to-one correspondence, cosymmetries cannot identify CLaws.
In §4.2 we show that characteristics are indeed in the kernel of the ALSC however we provide
a solution of the ALSC for the potential Lotka–Volterra equation that does not correspond to a
CLaw.
1.4
Thesis Outline
In Chapter 2 we present a brute force method to find discretizations of KdV that preserve
the mass and momentum. Using this method we obtain a three parameter family of explicit
schemes that contains the famous Zabusky-Kruskal scheme [56]. This motivates the need for a
characteristic for difference CLaws.
The continuous theory for characteristics cannot be carried over to difference equations, so in
Chapter 3 we define a characteristic and show there is a one-to-one correspondence between
equivalence classes of CLaws and characteristics. A corollary of this is that there is a one-to-one
correspondence between variational symmetries and CLaws.
We then show in Chapter 4 that, analogously to the continuous theory, the characteristic lies
in the kernel of the adjoint of the linearized symmetry operator. This provides a new way to
construct CLaws which we explore with the Hydon-Viallet equation, potential Lotka-Volterra
equation and the potential Hydon-Viallet equations.
Having defined a characteristic, in Chapter 5, we provide a method for searching for finite
difference schemes for KdV that locally preserve as many of its first three CLaws as possible.
We find novel schemes that preserve the mass and momentum, and we find the norm-preserving
scheme used in [11]. When searching for schemes that preserve the mass and energy we find a
novel explicit and novel eight-point schemes as well as the ten-point scheme of Furihata [23].
Finally in Chapter 6 we analyze the schemes we have found.
15
Chapter 2
Brute Force
In this chapter we outline a brute-force method for discretizing a PDE, ∆ = 0, in a way that
preserves some of its CLaws. We restrict attention to a PDE with two independent variables x
and t, that has CLaws of the form
Dt (Gi ) + Dx (Fi ) = 0.
(2.1)
The terms Gi and Fi will be referred to as the density and flux respectively. The method
requires that the difference scheme can be uniquely solved for one of its points, so we consider
only explicit schemes. The method finds schemes with uniform steps that have finite difference
analogues of the PDE’s CLaws. By this we mean that the discretization has CLaws
(Sn − I) f (Sm − I) e
Gi +
Fi = 0,
ν
µ
e =0
when ∆
(2.2)
where µ is the spatial step and ν is the time step, and tildes represent discretizations of the
corresponding continuous terms. We use this method to find a three-parameter family of schemes
for KdV that preserve its first (1.3) and second (1.4) CLaws together (which we shall refer to as
the mass and momentum conservation laws respectively).
As has already been stated, in this thesis we only study discretizations of the KdV equation,
∆ ≡ ut + uux + uxxx . However the methods presented do not rely on any special structure of
KdV so they should be applicable to other equations. The advantage of using KdV is that the
exact solution of the initial value problem is known for soliton solutions and there are lots of
discretizations to compare any resulting schemes with. Having used the method to find schemes
that preserve the first two CLaws for KdV, we then compare the schemes by studying the stability
and dispersion relation of the linearized schemes and some basic numerics.
2.1
Some Existing Explicit Mass and Momentum Conserving Schemes
The most famous finite difference scheme for KdV is the Zabusky-Kruskal scheme (Z-K) [56]
u01 =u0−1 −
ν
3µ (u10
+ u00 + u−10 )(u10 − u−10 ) −
17
ν
µ3 (u20
− 2u10 + 2u−10 − u−20 ).
(2.3)
Chapter 2: Brute Force
The scheme preserves the structure of the first two CLaws with discrete densities given by
f1 = 1 u00 + 1 u0−1 = u(xm , tn ) + O(ν),
G
2
2
1
f1 = (u−10 u00 + u200 + u2−10 ) + 1 (u−20 − u−10 − u00 + u10 )
F
6
2µ2
= 12 u(xm , tn )2 + uxx (xm , tn ) + O(µ),
f2 = 1 u00 u0−1 = 1 u(xm , tn )2 + O(ν).
G
2
2
f2 = 1 u2−10 u00 + u−10 u200 + 1 (u−20 u00 − u−10 u00 + u−10 u10 )
F
2
2µ2
= 13 u(xm , tn )3 + u(xm , tn )uxx (xm , tn ) − 21 ux (xm , tn )2 + O(µ).
Most schemes do not seek to conserve the CLaws in the above form but rather seek to preserve
the conserved quantities, which are found by integrating the CLaw over the spatial domain with
periodic boundary conditions. The first two CLaws of KdV (1.3) and (1.4) yield
Z
0 =Dt (u) dx
Z
1 2
dx,
0 =Dt
2u
which we will refer to as the conservation of mass and momentum respectively. We shall say
that a scheme preserves mass or momentum if
X
fi = 0,
(Sn − I)µ
G
(2.4)
m
fi is a finite difference approximation to Gi . A finite difference scheme that preserves the
where G
structure of a CLaw (2.2) will approximately preserve the conserved quantity but in general it
will not be exact. However, preserving the conserved quantity does not imply the local structure
of the CLaw is preserved.
Applying periodic boundary conditions to the Z-K scheme and summing over the spatial domain
implies that
X
X
u00 =
u0−1 ;
m
m
thus the mass is conserved exactly. The momentum is conserved approximately, because
X
X
1
(u01 )2 − 12
(u0−1 )2 = O(ν 3 ), as ν → 0.
2
m
m
The fact that the momentum is not conserved exactly means that solutions are not necessarily
bounded. Sanz-Serna [52] showed that the Z-K scheme is subject to nonlinear instability. He
√
also derived a linearized stability condition, 3 3ν ≤ 2µ3 .
Sanz-Serna used the Z-K scheme to create a method with an adaptive time-step that exactly
preserves the mass and momentum. First note that the Z-K scheme is of the form
u01 − u0−1 = 2νKm (un ),
(2.5)
where
Km = −(Sm − I)
18
f1
G
.
µ
(2.6)
Chapter 2: Brute Force
Then, because of the periodic boundary conditions, the preservation of the second CLaw with
the Z-K scheme implies
X
m
u00 Km (un ) = −
X (Sm − I)
m
µ
The time step, for the adaptive scheme, is given by
t0 = 0,
t1 = ν,
tn+1 = 2τn + tn−1 ,
f2 = 0.
F
(2.7)
n = 1, 2, 3, . . . ,
where
τn =
P
n) −
m (um (tP
um (tn−1 ))Km (u(tn ))
.
2
K
m (u(tn ))
m
(2.8)
Sanz-Serna’s scheme is then given by
um (tn+1 ) = um (tn−1 ) + 2τn Km (u(tn )).
(2.9)
Having summed over the spatial domain and applied the boundary conditions, τn is a function
of n only thus the scheme is in the form of a difference CLaw
f1 (u(tn ))) = 0.
(Sn − I)(um (tn ) + um (tn−1 )) + (Sm − I)(2τn G
Summing over the spatial domain shows that the mass is still exactly conserved. To show the
exact conservation of momentum, (2.9) is squared and then summed over the spatial domain,
X
X
um (tn+1 )2 =
(um (tn−1 ) + 2τ n Km (u(tn )))2 ,
(2.10)
m
m
=
X
um (tn−1 )2 + 4τn
m
=
X
X
um (tn−1 )Km (u(tn )) + 4(τn )2
m
2
um (tn−1 ) + 4τn
m
=
X
X
X
Km (u(tn ))2 ,
m
um (tn )Km (un )
using (2.8),
(2.11)
m
um (tn−1 )2
using (2.7)
(2.12)
m
However this preservation of the momentum is not due to a local CLaw. From (2.11)
X (Sn − I) um (tn )2 + um (tn−1 )2 X
0=
−
um (tn )Km (un )
τn
4
m
m
X (Sn − I) um (tn )2 + um (tn−1 )2 X (Sm − I)
=
+
F2 .
τn
4
µ
m
m
but once the summation is removed,
(Sn − I) um (tn )2 + um (tn−1 )2
(Sm − I)
C2 =
+
F2 ,
τn
4
µ
(2.13)
the divergence expression does not vanish on solutions of (2.9) so the CLaw is not preserved.
The reason the momentum is conserved is the application of (2.8) which involves summing over
the spatial domain; hence this is not a local effect. If the time-step in (2.13) is replaced by a
constant step ν then
u01 + u0−1 u01 − u00
u01 + u0−1 u01 − u00
C2 =
− u00 Km = u00
− Km ,
(2.14)
2
2ν
2u00
2ν
19
Chapter 2: Brute Force
so the term in brackets is an implicit scheme that preserves the second CLaw exactly but doesn’t
conserve the first CLaw (this is shown by applying the discrete Euler operator to the scheme).
However this scheme involves dividing by u00 which may become zero and the scheme is nonlinear
in u01 .
By preserving the momentum exactly, stability of the scheme is ensured. However this only
applies when the boundary conditions are periodic. Preserving the divergence form of the CLaw
should provide a constraint on the behaviour of solutions regardless of the boundary conditions.
Therefore, by preserving the divergence form of the CLaws, we aim to find better methods.
2.2
The Brute-Force Method
The brute-force method begins with the observation that the first CLaw (1.3) is KdV. Therefore it
is trivial to preserve the first CLaw: discretize the density and flux and then apply their respective
difference operators; this gives a discretization of KdV that is itself a difference CLaw. The
density and flux of the second CLaw are then discretized and the respective difference operators
are applied. This gives a discretization of the second CLaw. By evaluating the discretization
of the second CLaw on solutions of the discretization of KdV an expression that is required to
be zero is obtained. From this expression conditions are found on the discretizations of KdV.
Satisfying these conditions results in a numerical scheme for KdV that preserves the first two
CLaws.
2.2.1
Discretization approach
The method is based on forming the most general discretization of terms in the PDE or densities
and fluxes with a given set of points. The discretizations are based on Taylor series expansions
of the grid function about the point (xm , tn ),
uij ≈ u(xm + iµ, tn + jν),
= u + iµux + jνut +
(iµ)2
(jν)2
.
uxx +
utt + iµjνuxt + . . .
2!
2!
(xm ,tn )
It is assumed that ν = λµr for some fixed r > 0 and λ > 0, and the discretization must be
consistent. For example using the points uij for i = A, . . . , B, j = C, . . . , D the discretizations
of the linear terms in KdV are
ug
xxx =
B
D
1 XX
αij uij ,
µ3
B
D
1XX
βij uij ,
ν
uet =
i=A j=C
(2.15)
i=A j=C
where the necessary conditions on the coefficients are
0=
B X
D
X
αij , 0 =
i=A j=C
B X
D
X
iαij , 0 =
i=A j=C
0=
B X
D
X
i=A j=C
B X
D
X
i2 αij , 3! =
i=A j=C
βij ,
1=
B X
D
X
jβij .
B X
D
X
i3 αij ,
(2.16)
i=A j=C
(2.17)
i=A j=C
However, for the scheme to be consistent, there are additional conditions that need to be satisfied.
For ug
xxx → uxxx as µ → 0 the additional conditions are found from the Taylor series expansions
20
Chapter 2: Brute Force
and are either restrictions on the values that r can take or are extra constraints on the coefficients.
In general, for k, l ∈ N, (excluding the case k = 3, l = 0, otherwise the scheme will not converge
to uxxx , see (2.16)) it is necessary that
P k l
k l
X
ij i j αij µ ν
l
=
lim
λ
ik j l αij µlr+k−3 = 0;
lim
µ→0
µ→0
µ3
ij
hence
lr > 3 − k or
X
ik j l αij = 0.
ij
The conditions (2.16) ensure these are satisfied for l = 0. For n = k + l ≤ 3 the remaining
conditions are
P
X
X
ij jαij ν
lim
= lim λ
jαij µr−3 = 0, so r > 3 or
jαij = 0,
(2.18)
3
µ→0
µ→0
µ
ij
ij
P
X
X
ij ijαij µν
r−2
=
lim
λ
ijα
µ
=
0,
so
r
>
2
or
ijαij = 0,
(2.19)
lim
ij
µ→0
µ→0
µ3
ij
ij
P 2
2
X
X
ij j αij ν
j 2 αij µ2r−3 = 0, so r > 32 or
lim
= lim λ2
j 2 αij = 0,
(2.20)
3
µ→0
µ→0
µ
ij
ij
P 3
3
X
X
ij j αij ν
3
3
3r−3
=
lim
λ
j
α
µ
=
0,
so
r
>
1
or
j 3 αij = 0,
(2.21)
lim
ij
µ→0
µ→0
µ3
ij
ij
P
2
2
X
X
ij ij αij µν
lim
= lim λ2
ij 2 αij µ2r−2 = 0, so r > 1 or
ij 2 αij = 0,
(2.22)
3
µ→0
µ→0
µ
ij
ij
P 2
2
X
X
ij i jαij µ ν
lim
= lim λ
i2 jαij µr−1 = 0, so r > 1 or
i2 jαij = 0.
(2.23)
3
µ→0
µ→0
µ
ij
ij
Then, to avoid extra constraints on the coefficients, r must satisfy
3
,
n
3
2
n ≥ 2, k = 0, 1 : r > , r >
,
n
n−1
2
1
3
n ≥ 3, k = 0, 1, 2 : r > , r >
,r >
n
n−1
n−2
n ≥ 1,
k=0:
r>
note
3
2
1
≥
≥
.
n
n−1
n−2
Thus as n increases the restriction on r decreases. So fixing r > 3 will avoid any additional
constraints beyond (2.16).
In order to have uet → ut as µ → 0 it is necessary that (excluding l = 1, k = 0)
P k l
k l
X
ij i j βij µ ν
0 = lim
= lim λl−1
ik j l βij µk+(l−1)r = 0;
µ→0
µ→0
ν
ij
therefore
k + (l − 1)r > 0 or
X
ik j l βij = 0.
ij
The constraint on r is immediately satisfied for l ≥ 1. Thus the only remaining cases have l = 0,
in which case n = k > r is needed to avoid any extra conditions on the coefficients. For n ≤ 3
21
Chapter 2: Brute Force
the extra conditions are
P
ij
lim
µ→0
lim
µ→0
lim
µ→0
P
ij
P
ij
X
1X
iβij µ1−r = 0, so r < 1 or
iβij = 0,
µ→0 λ
ij
ij
iβij µ
= lim
ν
i2 βij µ2
ν
i3 βij µ3
ν
X
1X 2
i βij µ2−r = 0, so r < 2 or
i2 βij = 0,
µ→0 λ
ij
ij
= lim
X
1X 3
i βij µ3−r = 0, so r < 3 or
i3 βij = 0.
µ→0 λ
ij
ij
= lim
(2.24)
(2.25)
(2.26)
Thus if (2.16) and (2.17) are the only conditions imposed on the discretizations then for consistency r > 3 and r < 1 which is not possible. Therefore extra conditions need to be imposed on
the αij and βij so that there is a set of values of r where both uet → ut and ug
xxx → uxxx . There
is not a unique way of doing this. One such choice is that (2.18), (2.19) and (2.24) are imposed
then 32 < r < 2.
KdV also contains the quadratic term uux so, to discretize this term, products of Taylor series
need to be examined
uij ukl =u2 + µ(i + k)uux + ν(j + l)uut + µ2 iku2x + µν(jk + il)ux ut + ν 2 jlu2t +
µ2 (i2 + k 2 )
ν 2 (j 2 + l2 )
.
uuxx + µν(ij + kl)uuxt +
uutt + H.O.T.
2!
2!
(2.27)
(xm ,tn )
Therefore to discretize uux


D
B X
B
B X
D
B
X
X
X
X
1

uu
gx =
γijkj uij ukj +
γijkl uij ukl  ,
µ
j=C
i=A k=i
(2.28)
i=A l=j+1 k=A
with the necessary conditions
0=
P
X
γijkl ,
1=
X
(i + k)γijkl ,
(2.29)
where
is short hand for the summation used in (2.28) (alternatively u2 could be discretized
and then the difference operator in the m direction applied to it). As for the linear terms, extra
conditions may be required on the coefficients to ensure that the method is consistent. The only
terms in (2.27) that can cause problems when discretizing uux are those that are purely of the
form ν n . The order one and two terms are
X γijkl (j + l)ν
X
lim
= lim λ
γijkl (j + l)µr−1 = 0,
µ→0
µ→0
µ
X
so r > 1 or
γijkl (j + l) = 0,
2
2
2
X γijkl (j + 2jl + l )ν
X
lim
= lim λ2
γijkl (j 2 + 2jl + l2 )µ2r−1 = 0,
µ→0
µ→0
µ
X
so r > 21 or
γijkl (j 2 + 2jl + l2 ) = 0,
and as n increases the restriction on r decreases. With 23 < r < 3 it is clear that no extra
conditions are required on the γijkl other than (2.29) to have a consistent discretization of KdV.
2.2.2
Groebner bases
The method we present for finding discretizations will result in large systems of polynomial
equations to solve. These polynomials will be in terms of the undetermined coefficients in the
22
Chapter 2: Brute Force
discretizations. To solve this large system of equations we will use the method of Groebner
bases. We outline the basic idea and refer the reader to [15, 16, 18, 35] for further information.
Definitions and notation has been taken from [18] and [35].
α2
αn
1
A monomial in x1 , . . . , xn is a product of the form xα
1 · x2 · . . . · xn , where all the exponents
α1 , . . . αn are non-negative integers. To simplify notation let α = (α1 , . . . , αn ), which is termed
α2
αn
1
a multi-index, then xα := xα
1 · x2 · . . . · xn .
A polynomial f , over a field F, is a finite linear combination of monomials (with coefficients in
F). So
X
f=
aα xα , where aα ∈ F,
α
and the sum is over a finite number of the n-tuples α. The set of all polynomials in x1 , . . . , xn
with coefficients in F is denoted k[x1 , . . . , xn ].
An affine variety is the set of all solutions of a system of polynomial equations i.e. given polynomials f1 , . . . , fs over a field the affine variety is
V(f1 , f2 , . . . , f2 ) := {(a1 , . . . , an ) ∈ Fn : fi (a1 , . . . , an ) = 0 for all 1 ≤ i ≤ s}.
The polynomial ideal generated by the polynomials f1 , . . . , fs ∈ k[x1 , . . . , xn ] is
( s
)
X
hf1 , . . . , fs i :=
hi fi : h1 , . . . hs ∈ k[x1 , . . . , xn ] .
i=1
In order to perform operations on the set of polynomials we need to define a monomial ordering.
An ordering on the monomials must be compatible so that
xα > xβ =⇒ xγ · xα > xγ · xβ ,
where α, β and γ are all multi-indicies. The monomial x0 = 1 is the least monomial in any polynomial ring. The two most important orderings for the work in this thesis are the lexicographic
ordering (lex) and total degree ordering (tdeg, also known as the graded reverse lexicographic
ordering). For the lexicographic order xα > xβ if α1 = β1 , α2 = β2 , . . . , αi > βi . For the total
degree ordering xα > xβ if |α| > |β| where |α| = α1 + α2 + . . . + αn . If |α| = |β| then there
exists an i such that αn = βn , αn−1 = βn−1 , . . . , αi ≤ βi . The leading monomial of a polynomial with respect to an ordering is the greatest monomial occurring in the polynomial. The
leading term of a polynomial is the leading monomial multiplied by its coefficient. For example,
if f = 4xy 2 z + 4z 2 − 5x3 + 7x2 z 2 ∈ k[x, y, z], if we order its terms w.r.t. the lex ordering with
x > y > z then
f = −5x3 + 7x2 z 2 + 4xy 2 z + 4z 2 ,
and its leading monomial is LM (f ) = x3 and its leading term is LT (f ) = −5x3 . If we order the
terms using the tdeg ordering with x > y > z then
f = 4xy 2 z + 7x2 z 2 − 5x3 + 4z 2
and its leading monomial is LM (f ) = xy 2 z and its leading term is LT (f ) = 4xy 2 z.
We are now in a position to give an example of multivariate polynomial devision. If we have two
polynomials f1 = xy−1 and f2 = y 2 −1 then we can divide another polynomial, f = x2 y+xy 2 +y 2
23
Chapter 2: Brute Force
with respect to these two polynomials as follows. The leading term of f1 divides LT (f ) w.r.t.
the lex order with x > y, so
f = x(xy − 1) + xy 2 + x + y 2 .
Now LT (f1 ) also divides the leading term xy 2 in the remainder, so
f = (x + y)(xy − 1) + x + y 2 + y.
Neither the LT (f1 ) or LT (f2 ) divide the leading monomial of the remainder, so we move on to
the next monomial y 2 . The LT (f1 ) cannot divide this term, so we divide by f2
f = (x + y)(xy − 1) + x + (y 2 − 1) + y + 1.
None of the monomials in the remainder can be divided by LT (f1 ) or LT (f2 ) so the algorithm
terminates. Thus
f = (x + y)f1 + (y 2 − 1)f2 + x + y + 1.
Clearly if the remainder was zero f ∈ hf1 , f2 i however we now note that the remainder changes
if the devision is performed w.r.t. f2 first and then f1 ; we obtain
f = (x + 1)f2 + xf1 + 2x + 1,
so the remainder is not unique! The remainder depends on the order of the divisions. Thus we
cannot conclude f 6∈ hf1 , f2 i.
The solution to this problem is given by Groebner bases. They have the property that whatever
the order of the polynomial devisions the remainder is unique.
Fix a monomial order. A finite subset G = g1 , . . . , gt of an Ideal I is said to be a
Groebner basis if
hLT (g1 ), . . . , LT (gt )i = hLT (I)i.
Equivalently, but more informally, set {g1 , . . . , gt } ⊂ I is a Groebner basis of I if and
only if the leading term of every element of I is divisible by one of the LT (gi ).[18]
A Groebner basis can be calculated from a basis for a polynomial idea using Buchberger’s
algorithm or its variants. Suppose we have the polynomial ideal generated by
f1 := x2 + y + z − 1,
f2 := x + y 2 + z − 1
f3 := x + y + z 2 − 1,
I = hf1 , f2 , f3 i. We want to represent this ideal with a Groebner basis I = hgi i using the lex
ordering and x > y > z. The resulting basis is
g1 =x + y + z 2 − 1,
g2 =y 2 − y − z 2 + z,
g3 =2yz 2 + z 4 − z 2 ,
g4 =z 6 − 4z 4 + 4z 3 − z 2 = z 2 (z − 1)2 (z 2 + 2z − 1).
We note here that the first polynomial depends on all three variables, the second two depend on
y and z only, and the final polynomial depends on z only. This occurs because during the process
24
Chapter 2: Brute Force
of finding the Groebner basis, polynomials are added to the basis with lower leading terms in
the ordering. Thus by using the lex ordering we will obtain polynomials in the basis that do not
depend on the highest variables in the ordering. By the ideal-variety correspondence theorem
[18], the variety described by the original polynomials is the same as the variety described by
the polynomials that form the Groebner basis. Thus we can now solve the original system of
equations fi = 0 by solving g4 (z) = 0, then working back up the system of equations to find the
corresponding y values (from g3 = 0 and g4 = 0) and finally the corresponding x values from
g1 = 0.
This suggests that all we need to do is calculate the Groebner basis for our system of polynomial
equations using the lex ordering and we can then solve the resulting system of equations. However, calculating the Groebner basis is exceedingly expensive in memory and time. In practice
calculating a Groebner basis w.r.t. the tdeg is cheaper than the lex order. This enables us to
check the consistency of a system of equations. If the resulting Groebner basis (whatever the
chosen ordering) is h1i then the original system of equations is inconsistent, they have no solution
because they imply 1 = 0 (the Weak Nullstellensatz). Therefore we can calculate the Groebner
basis for our system of equations using the tdeg ordering and see if resulting basis does not equal
h1i. If this is true, then the system of equations has some solution in an algebraically closed field
(for our equations this is C). We can then use the Groebner Walk algorithm to change basis to
one with lex ordering. We can then solve the system as outlined above. However it still may not
have solutions in R which we require.
Calculating a Groebner basis is an expensive operation whatever the monomial ordering is.
For a set of polynomials, in n variables, with total degree not exceeding d then degree of the
n−1
polynomials in the Groebner basis is bounded by 2( 12 d2 + d)2 . It can be shown that for
sufficiently large n there exists a constant c and a set of polynomials such that every Groebner
cn
Basis of the set contains an element that exceeds 22 [16]. However, the efficiency of the
algorithm for a given problem, is affected by the ordering of the polynomials and the ordering of
the variables, so changing these can affect whether a problem is tractable. Due to the expense
of finding Groebner bases, the method we outline in this thesis will be limited by our ability to
calculate the Groebner basis.
2.2.3
The recipe for finding schemes
The details of the brute-force method are now outlined. This method was only used to search
for discretizations of KdV that preserve the first two CLaws using the points shown in Figure
2.1. (It is superseded by the method in Chapter 5.)
1. Choose points for the discretization of KdV to depend on. For our example, the points
u−20 , u−10 , . . . , u20 , u0−1 , u01 were chosen. Thus the flux for the first CLaw must be of the
f1 = F
f1 (u20 , u−10 , u00 , u10 ) and the density must be of the form G
f1 = G
f1 (u0−1 , u00 ),
form F
see Figure 2.1. (With this choice of points the resulting discretization for KdV will be
consistent provided r > 3 because uet only uses variables of the form u0j .)
f1 be the most general Taylor series approximation of 1 u2 + uxx depending on the
2. Let F
2
f1 be the most general Taylor series approximating u. It
chosen points. Similarly let G
is probably best not to impose extra constraints to ensure consistency at this stage as
the choice of constraints is not unique. Note that not all the points need to be used to
25
Chapter 2: Brute Force
00
Fi
Gi
Figure 2.1: Points used to discretize KdV; the boxes enclose points that the fluxes, Fei , and
fi , depend on
densities, G
approximate each term in a density. For instance, fewer points might be chosen for the
nonlinear term.
f1 and G
f1 into (2.2) and expand out. This gives a family of discretizations
3. Substitute F
of KdV that preserve the first CLaw. Note that for a PDE that is not itself a CLaw,
or if it is not desirable to preserve the first CLaw, then the previous step is replaced by
forming the most general discretization of the PDE with the chosen points. In order to be
able to evaluate the second CLaw on solutions of the discretization it is necessary that the
scheme can be solved uniquely for one point. Thus it is most natural to form an explicit
f1 is linear in u−20 or u10 ,
discretization of the PDE. In the example it is necessary that F
f
f
or G1 is linear in u00 or u0−1 . G1 has been constructed to be linear in u00 therefore the
scheme is solved uniquely for u01 . Hence the schemes found in the example will be explicit.
4. Choose points for the terms in the density and flux of the second CLaw to depend
f2 =
on. Use these points to form the most general discretizations. In the example, F
1 3
1 2
f
F2 (u20 , u−10 , u00 , u10 ) is the most general discretization for 3 u + uuxx − 2 ux , with the
f2 = G
f2 (u0−1 , u00 ) is the most general discretization for 1 u2 .
chosen points. Similarly G
2
f2 and F
f2 into (2.2) and expand out to give the second CLaw.
5. Substitute G
6. Substitute the finite difference scheme for the PDE (KdV) into the second CLaw so that
the second CLaw is evaluated on solutions of the previously found scheme. In general it
may be necessary to substitute shifts of the numerical scheme into the second CLaw; for
an explicit scheme this is equivalent to pulling back the CLaw to the initial conditions.
7. The second CLaw is required to vanish on solutions of the numerical scheme. Thus comparing the coefficients of the uij and νs and µs splits the second CLaw into an overdetermined
system of quadratic equations in the coefficients of the Taylor approximations.
8. Solve the system (using a computer algebra system to calculate a Groebner basis).
9. If a solution exists then a scheme has been found that locally preserves the structure of
the first two CLaws.
10. Check that the discretization of the PDE is consistent. If it is not, and there are still undetermined coefficients, impose extra conditions from §2.2.1 to make the method consistent.
26
Chapter 2: Brute Force
Note: if we have n points and a pth order term, up , then the number of terms needed to
approximate it is given by a p-multicombination
!
n−1+p
(n + p − 1)!
1
=
= n(n + 1) . . . (n + p − 1).
(2.30)
(n − 1)!p!
p!
p
!
5
1 2
For example, the first CLaw contains 2 u , therefore
= 10 terms are needed to approxi2
mate it, i.e. c1 u−20 2 +c2 u−20 u−10 +. . .+c10 u10 2 , with the condition c1 +. . .+c10 = 12 . The third
CLaw (1.5) has a quartic term 14 u4 and a third derivative −2ux uxxx in its densities, therefore the
scheme will need to include more points, so the number of terms greatly increases. The number
of terms can be reduced by restricting which points the higher powers of u depend on, but it is
not clear how this should be done.
2.3
A Three-Parameter Family of Methods
The method, with the specified dependency, produced a three-parameter (α, β, γ) family of finite
difference methods for KdV (shown in Table 2.1) that all preserve the first two CLaws.
The first thing to note is that all these methods are two-step explicit methods. They all can
be written as (2.5), so Sanz-Serna’s adaptive time-step alteration can be applied to obtain
exact preservation of the momentum. The parameters only affect how the non-linear term is
approximated. In order to have a more compact discretization any nonlinear terms containing
u20 can be eliminated by satisfying the conditions:
u20 2 :
u10 u20 :
u00 u20 :
2
2
2
1
α + β + γ − = 0,
3
3
3
3
2
− γ = 0,
3
2
− α = 0.
3
Therefore α = 0, β = 12 and γ = 0. Similarly, to remove the other outlying nonlinear points that
contain u−20 , the conditions
u−20 2 :
u00 u−20 :
u−10 u−20 :
2
α = 0,
3
2
2
2
1
− α − β − γ + = 0,
3
3
3
3
4
2
1
− α − β + = 0,
3
3
3
need to be satisfied. This also gives α = 0, β = 21 and γ = 0. So these parameter choices result
in the most compact scheme, with the chosen points, that preserves the first two CLaws. This
scheme is the Z-K scheme.
Another sensible approach to choosing parameter values is to attempt to minimize the local
truncation error (LTE) of
uu
gx = uux + 65 − 23 α− 43 β − 23 γ (uxxx u + 2uxx ux )µ2 + 12 −2α−β −γ (uxxx ux + u2xx )µ3 + O(µ4 ).
So to make this term a fourth-order approximation in space,
α = α,
β=
3
4
+ α,
27
γ = − 41 − 3α,
Chapter 2: Brute Force
and one such choice would be α = γ = −1/16, β = 11/16. Despite the free parameter, it is not
possible to make the discretization for this term a fifth-order approximation. We note that the
Z-K scheme is only a second-order scheme, however it should be noted that the ug
xxx term is a
second-order approximation so there may not be any advantage in increasing the accuracy of
the nonlinear term beyond this.
Table 2.1: The three-parameter family and its densities
ν
µ3
ν
+ 3µ
+ 2αν
3µ
+ 2βν
3µ
+ 2γν
3µ
u01 = u0−1 +
(u−20 − u20 − 2u−10 + 2u10 ) +
u−20 u00 + u10 2 − u10 u−10 + u−20 u−10 − u−10 u00 − u20 2 +
2u−10 u00 − u00 u20 − 2u−20 u−10 − 2u10 2 + u20 2 + u−20 2 − u−20 u00 + 2u−10 u10 +
2u−10 u00 − u−20 u−10 + u−10 u10 − 2u10 2 − u−20 u00 + u20 2 − u00 u10 + u−10 2 +
u−10 u00 − u10 u20 + u−10 u10 − u10 2 − u−20 u00 + u20 2
f1 = 1 u00 + 1 u0−1
G
2
2
f1 = u−20 −u−102−u00 +u10 +
F
+ α3
+ β3
+ γ3
1
u10 2 + u−20 u00 + u−20 u−10 +
6
2µ
u−20 2 − 2u−20 u−10 − u−20 u00 + u−10 2 + u−10 u10 − u10 2 +
u−10 2 − u10 2 − u−20 u00 − u−20 u−10 + u−10 u00 + u00 2 +
u−10 u00 − u10 2 − u−20 u00 + u00 u10
f2 = 1 u00 u0−1
G
2
f2 = u−20 u00 −2u−102u00 +u−10 u10 +
F
2.4
u00 2 +
u−20 u−10 u00 + u−10 u10 2 + u−20 u00 2 − u−10 u00 2 +
u−10 2 u10 − 2u−20 u−10 u00 + u−20 2 u00 + 2u−10 u00 2 − u−10 u10 2 − u−20 u00 2 +
u−10 2 u00 − u−10 u10 2 − u−20 u00 2 + 2u−10 u00 2 − u−20 u−10 u00 +
u−10 u00 u10 − u−10 u10 2 − u−20 u00 2 + u−10 u00 2 .
2µ
+ α3
+ β3
+ γ3
1
6
Linear Analysis
We now linearize the difference scheme about the constant solution u = ρ. To linearize the
scheme we take the Gâteaux derivative (see §4.2 for a definition) of the scheme acting on vmn
and then substitute u00 = ρ to give
ρν
ν
v(m−2)n − v(m+2)n − 2vm−1n + 2v(m+1)n −
v(m+1)n − v(m−1)n +
3
µ
µ
2ρθν
+
v(m+2)n − 2v(m+1)n + 2v(m−1)n − v(m−2)n ,
(2.31)
3µ
vm(n+1) = vm(n−1) +
where θ = α + 2β + γ − 1 so that the Z-K scheme’s linearization has θ = 0. This is now a
discretization for
ut + ρux + uxxx = 0.
(2.32)
Also note that, in order for the uu
gx term to be a third-order approximation to uux , we require
that θ = 1/4.
28
Chapter 2: Brute Force
2.4.1
Stability analysis
In order to check the linear stability we follow Vliegenthart’s analysis for the Z-K scheme [55]
using Fourier analysis. Assume that the solution is of the form vmn = Aξ n eiωm ; then the scheme
is stable if |ξ| ≤ 1 for all values of ω and ρ. Substituting this solution into (2.31) leads to the
condition that
2iν
2θρ
1
ξ2 +
ξ − 1 = 0.
sin(ω) ρ + 2(1 − cos(ω))
− 2
µ
3
µ
This equation is of the form ξ 2 + 2iλξ − 1 = 0 where λ ∈ R; its roots satisfy
ξ1 + ξ2 = 2iλ,
ξ1 ξ2 = −1,
√
thus ξ1 = a + iλ, ξ2 = −a + iλ and |ξi | = a2 + λ2 = 1. Therefore if |λ| ≤ 1 a suitable a ∈ R
exists and |ξi | = 1. Thus for linear stability we require
4θρ
2
ν ≤ 1.
ρ
sin(ω)
+
(sin(ω)
−
sin(ω)
cos(ω))
−
2
µ
3
µ
The parameter θ is fixed for any particular scheme and we assume that |ρ| ≤ umax where
|u(x, t)| ≤ umax . Then applying the triangle equality twice gives
√ !
4θρ
3 3
ν
2
|ρ| + − 2 ≤ 1,
µ
3
µ
4
√ !
√
3 3
2
ν µ umax (1 + 3 |θ|) +
≤ µ3 .
(2.33)
2
This suggests that increasing |θ| requires a slightly more severe step-size restriction. However its
contribution is part of the µ2 term so there is very little difference between the linear stability
of the schemes provided |θ| is not large.
2.4.2
Dispersion relation
The dispersion relation for wave solutions (u(x, t) = ei(ωt+kx) ) of (2.32) is
ω = k 3 − ρk.
The phase velocity is
ω
k
= k 2 − ρ and the group velocity is
dω
dk
= 3k 2 − ρ.
The numerical dispersion relation is calculated by using the trial solution vmn = ei(kµm+ωνn) in
the linearized schemes [34]. The trial solution is 2π-periodic so we only need to consider
−π ≤ µk ≤ π,
−π ≤ νω ≤ π.
The resulting dispersion relation is
sin(ων) =
2ν sin(kµ)
(1 − cos(kµ)) − ρµ2
µ3
2θ
3 (1
− cos(kµ)) +
1
2
.
(2.34)
This relation has three parameters: the step size µ, λ = µν3 and ρ. It is clear that with ρ = 0 all
the schemes are identical. In figures 2.2(a), 2.2(b), 2.2(c) the dispersion relation is plotted for
2
ρ = 4.5, ρ = 2.25 and ρ = 1 with µ = 15
and λ = 31 (which are the values chosen in the numerical
29
Chapter 2: Brute Force
experiments in §2.5). From Figure 2.2 all the schemes allow spurious high speed waves in space
and time. This is due to the non-compact stencils in both space (four rather than three steps)
and time (two steps rather than one). However, locally about k = 0 and for small |ω|, for most
values of θ the dispersion relations have the correct shape. However, from these figures we can
see that there exists a bifurcation point for θ where the dispersion relation no longer locally has
the correct shape; instead of ω increasing with k, ω decreases so the numerical phase and group
velocities will have the wrong signs. In fig.2.2(b) this occurs for θ ≈ 38, whereas in fig.2.2(a)
this occurs at a lower value of θ. From (2.34) the term in brackets on the RHS becomes negative
3
3
2
for all values of k for θ > 2ρµ
2 − 8 = 37.125 for ρ = 2.25. We also note that the θ = − 3 scheme
behaves very similarly to the Z-K scheme (θ = 0).
(a) ρ = 4.5
(b) ρ = 2.25
(c) ρ = 1
Figure 2.2: Dispersion relations for the linearized three-parameter scheme with different values
of ρ (µ = 2/15, λ = 1/3)
2.5
Some Numerics
We conducted some numerical experiments modelling a single soliton solution. The time and
spatial steps used were ν = µ3 /3 and µ = 40/M where M is the number of points in the x
direction. The initial condition used was u(x, 0) = 12sech2 (x) which is depicted by a dashed
black line. The actual solution profile is shown by the black line. The error in the first, second
and third CLaws was found by calculating
µ
X
m
um,n ,
µ
X
m
u2m,n ,
µ
X
1 3
3 um,n
m
+
1
µ2 um,n
(um−1,n − 2um,n + um+1,n ) ,
at each time level and seeing how much these quantities change.
Figure 2.3 shows four schemes each with the same linearization (θ = 0) as the Z-K scheme. All
the schemes behave very similarly, with the soliton gaining amplitude and speed compared with
the true solution, however their sensitivity to the spatial discretization varies with decreasing
M, the schemes becoming unstable at different values. The (2, 1, −3) scheme is unstable with
M = 300 and shows sawtooth waves overriding the soliton solution, at M = 200 the (1, 0, 0)
and the (0, 0, 1) are exhibiting the same sawtooth behaviour and by M = 200 the Z-K scheme
(0, 12 , 0) has clearly developed sawtooth waves and is unstable with M = 150. This suggests that
the more compact Z-K scheme is coping best with the coarsest discretization.
30
Chapter 2: Brute Force
Comparing schemes with different values of θ resulted in Figure 2.4. From this figure it seems
that as θ decreases the amplitude and speed of the soliton decrease. For θ = − 32 the profile
of the numerical soliton appears to be remarkably close to the actual solution and the third
CLaw is being approximated better. This could be because the third CLaw is being locally
preserved or it could be because the soliton is traveling at roughly the correct speed; hence
the scheme is just more accurate. All the solutions have developed a wave-train following the
soliton however the (0, 0, 0) scheme has a very small amplitude and it is necessary to zoom in to
discern the (0, 1/6, 0) scheme’s wave-train. By reducing the number of spatial steps we can see
how the different schemes cope with coarse discretizations. At M = 250 the (0, 2/3, 0) scheme
has developed sawtooth waves, at M = 200 the Z-K scheme has clear sawtooth waves and the
(−1/16, 11/16, −1/16) scheme resembles noise, at M = 100 the (0, 0, 0) scheme has become a
lower slower soliton with a larger wave-train following, however at M = 50 the (0, 1/6, 0) scheme
still looks like a soliton - it is coping with the very coarse discretization incredibly well; far better
than the Z-K scheme even though it is not compact. In Figure 2.5 different schemes all with
θ = − 32 are examined. All these schemes show similar results, they all are very close to the
actual soliton and have a very small wave train following the numerical soliton, the smallest of
which belongs to the (0, 61 , 0) scheme. They all preserve the third CLaw better than the Z-K
scheme though there is a clear difference between the schemes. However, overall it suggests that
schemes with the same linearization behave very similarly.
The remarkable behaviour of the θ = − 32 schemes could be a result of the specific choice of
step sizes or initial condition. However, by experimenting with different speed solitons and
with different step sizes the same superior behaviour is observed. The dispersion relation and
linearized stability condition do not appear to account for the superior behaviour of schemes
with θ = − 23 , which is a two-parameter family of discretizations given by setting α = 31 − 2β − γ.
The good preservation of the third CLaw could be because it is locally preserved; however, using
the method of Chapter 5 no discrete third CLaw has so far been found. The good preservation
of the third CLaw may be a consequence of preserving the phase speed better than the other
schemes. However in Figure 2.4 we can see that the (−1, 1, − 32 ) scheme is closest to the actual
position of the soliton but the (0, 16 , 0) scheme preserves the third CLaw significantly better than
it. This, to our knowledge, new two-parameter family seems to outperform the Zabusky-Kruskal
scheme which has the most compact discretization for the nonlinear term.
31
Chapter 2: Brute Force
32
Figure 2.3: Different (α, β, γ) schemes with the same linearization as the Zabusky-Kruskal scheme (θ = 0). The green and blue schemes seem to be
behaving almost identically.
Chapter 2: Brute Force
33
Figure 2.4: Some (α, β, γ) schemes with different linearizations. The blue scheme has θ = 0, the red scheme has θ = 1/4, the green scheme has θ = − 32
the magenta scheme has θ = 13 and the cyan scheme has θ = −1.
Chapter 2: Brute Force
34
Figure 2.5: Different (α, β, γ) schemes with the same linearization: θ = − 23
Chapter 2: Brute Force
2.6
Summary
In this section we have developed a brute-force method for discretizing KdV to produce explicit
schemes that preserve the first two CLaws. Using this method we found a three-parameter
family of discretizations (that includes the famous Zabusky-Kruskal scheme) that preserve the
first two CLaws. From some basic numerics it appears that this family contains a two-parameter
family which outperforms the Zabusky-Kruskal scheme, but it is not currently understood why
it performs so well.
The brute-force method developed in this chapter is limited to schemes that can be solved
uniquely for a given point only. The method may require substituting in various shifts of the
numerical scheme to evaluate the desired CLaw on solutions of the numerical scheme, but it
seems neater to look for the CLaw in characteristic form where this won’t be necessary. Another
problem is that the densities and fluxes contain high order polynomial terms that require discretizing. Working with characteristics of CLaws, rather than densities and fluxes, will require
discretization of fewer and lower order terms. In addition it is desirable to look for general
implicit schemes where it is not possible to uniquely solve the discretization for a given point.
All this suggests that we should search for CLaws in characteristic form, which we do in Chapter
5.
The desire to search for numerical discretizations of KdV that preserve its CLaws in characteristic
form motivates the need to develop a theory of characteristics for difference equation CLaws,
which we do in the next chapter.
35
Chapter 3
Characterizing CLaws of
Difference Equations
We showed in §1.3.3 that a characteristic for difference equations cannot be obtained in the
same way as for PDEs because the CLaws may depend nonlinearly on the difference equation.
Therefore a new method for putting a CLaw of a difference equation into characteristic form
is needed. As a consequence, the proof that there is a one-to-one correspondence between
equivalence classes of CLaws and characteristics will not be analogous to the PDE case.
In this chapter we will start by defining a characteristic for first integrals of O∆Es in §3.1.
Having defined a characteristic, we show that a characteristic is trivial if and only if the first
integral is trivial. In §3.2.2 we provide a definition for a characteristic of CLaws of P∆Es which
subsumes the definition for O∆Es. Then in §3.2.3 and §3.2.4 we prove the equivalence relation
between classes of CLaws and characteristics giving us a method for characterizing difference
CLaws. We then look at some applications. The most important consequence is that we provide
the first proof that there is a one-to-one correspondence between variational symmetries and
CLaws of Euler–Lagrange equations in Kovalevskaya form. In addition we use the characteristic
to show that the CLaws in the infinite hierarchy of CLaws for dpKdV, that are generated by
the Gardner transformation in [51], are distinct (without needing to take a continuum limit).
Before proceeding, it is helpful to recall, from §1.3, the definition of a CLaw for a difference
equation and the two ways in which it can be trivial. A conservation law (CLaw) of a system
of P∆Es, ∆ = 0, is a divergence expression that vanishes on solutions of the system:
DivF :=
p
X
i=1
(Si − I)F i = 0 when [∆] = 0,
(3.1)
where [∆] denotes any finite shifts of the system. (For scalar P∆Es, we use ∆ in place of ∆).
The functions F i are known as the densities of the CLaw and may have the independent variables
and shifts of the dependent variables as arguments. For ordinary difference equations (O∆Es),
(1.15) is replaced by (Sn − I)φ = 0 when [∆] = 0, so that φ is a first integral. A CLaw can be
trivial in two ways:
1. All of the densities vanish on solutions, i.e. F|[∆]=0 = 0.
2. DivF ≡ 0, without reference to the equation [∆] = 0. For instance, this occurs if F is the
difference analogue of a total curl (see [45]).
37
Chapter 3: Characterizing CLaws of Difference Equations
For brevity we refer to the densities of a trivial CLaw as trivial densities. In addition we say
that a CLaw (3.1) is in characteristic form if DivF = Q · ∆, in which case, Q is a characteristic
for the CLaw.
3.1
Scalar Ordinary Difference Equations
Although the main focus of this chapter is on P∆Es, it is instructive to look at O∆Es first. A
k th -order scalar O∆E is of the form
∆ ≡ uk − γ(n, u0 , . . . , uk−1 ) = 0,
∂γ
6= 0,
∂u0
(3.2)
where n ∈ Z is the independent variable. The dependent variable has the values ui := u(n + i).
Let z = {n, u0 , . . . , uk−1 } denote the set of variables from which all ui for i ≥ k can be found.
We refer to this set as the ‘initial conditions’. A first integral of (3.2) is a nonconstant function,
φ(n, u0 , . . . , uk−1 ), that is constant on solutions. So
Sn φ − φ = 0
when ∆ = 0.
(3.3)
It is useful to refer to constant solutions of (3.3) as trivial first integrals, by analogy with trivial
CLaws of the second kind.
Equation (3.2) can be used to express uk in terms of z and ∆. A nontrivial first integral must
depend on uk−1 , otherwise (3.3) does not depend on ∆ so the only way for φ to be a first integral
is to be identically constant. Therefore the CLaw is
C(z, ∆) := φ(n + 1, u1 , . . . , uk−1 , ∆ + γ(n, u0 , . . . , uk−1 )) − φ(n, u0 , . . . , uk−1 ),
where C(z, 0) = 0. We now use the fundamental theorem of calculus to write the CLaw in the
form
Z 1
Z 1
d
C(z, λ∆) dλ = ∆
C(z, ∆) =
φ,k (m + 1, u1 , . . . , uk−1 , λ∆ + γ(m, u0 , . . . , uk−1 )) dλ.
λ=0
λ=0 dλ
(Throughout this chapter, the partial derivative of a function, f , w.r.t. its ith continuous argument is denoted by f,i ). Thus we define the characteristic to be
Z 1
∂C(z, λ∆)
Q(z, ∆) :=
dλ.
(3.4)
∂λ∆
λ=0
As with differential equations, a trivial characteristic is one that vanishes on solutions, so that
Q(z, 0) = 0.
A first integral is trivial if and only if it is identically constant, because the first kind of triviality
cannot occur as [∆] does not occur in the unshifted φ. In this case C(z, ∆) ≡ 0, so the proof
that a trivial first integral has a trivial characteristic is immediate.
To show that a trivial characteristic implies a trivial first integral, observe that
Z 1
Q(z, 0) =
φ,k (n + 1, u1 , . . . , uk−1 , γ(n, u0 , . . . , uk−1 )) dλ
λ=0
=φ,k (n + 1, u1 , . . . , uk−1 , γ(n, u0 , . . . , uk−1 )).
(3.5)
Thus if the characteristic for a first integral is trivial then
0 = φ,k (n + 1, u1 , . . . , uk−1 , γ(n, u0 , . . . , uk−1 )).
38
(3.6)
Chapter 3: Characterizing CLaws of Difference Equations
As γ,1 6= 0, γ can be used as an independent variable instead of u0 , so
∂
φ(n + 1, u1 , . . . , uk−1 , γ) = 0.
∂γ
Hence φ is a trivial first integral.
Having dealt with the question of triviality, we now show how to reconstruct the first integral
from the characteristic. For simplicity, we use a second-order example, but the same procedure
applies more generally.
The O∆E
∆ ≡ u2 − γ(n, u0 , u1 ) = 0,
γ=
n
1
u0 + ,
n+1
u1
(3.7)
has a first integral
1
φ(n, u0 , u1 ) = nu0 u1 − n(n + 1).
2
(3.8)
1
1
(Sn − I)φ =(n + 1)u1 u2 − (n + 1)(n + 2) − nu0 u1 + n(n + 1).
2
2
(3.9)
So
Using (3.5), the characteristic evaluated on solutions is
φ,2 (n + 1, u1 , γ) = (n + 1)u1 .
(3.10)
Using (3.7) to replace u2 in (3.9) and simplifying produces
(Sn − I)φ =(n + 1)u1 ∆,
which is in characteristic form.
To reconstruct the first integral from the characteristic, we have
∂
φ(n + 1, u1 , γ) = (n + 1)u1 .
∂γ
Solving this and then changing variables gives
φ(n, u0 , u1 ) = nu0 u1 + f (n, u0 ).
This is then used as an ansatz in the direct construction method for first integrals [25]. If φ is a
first integral of (3.7) then
1
n
(n + 1)u1
u0 +
+ f (n + 1, u1 ) = nu0 u1 + f (n, u0 ),
n+1
u1
and so
f (n + 1, u1 ) − f (n, u0 ) = −n − 1.
(3.11)
Differentiating (3.11) w.r.t. u0 shows that f (n, u0 ) = g(n). Substituting this back into (3.11)
results in the condition that
(Sn − I)g(n) = −n − 1.
Thus g(n) = − 21 n(n + 1), and so the first integral (3.8) has been reconstructed.
39
Chapter 3: Characterizing CLaws of Difference Equations
In general, reconstructing the first integral for scalar O∆Es is simple. Given a k th -order O∆E
(3.2), the characteristic for a first integral evaluated on solutions is
Q̄(z) := Q(z, 0) = φ,k (n + 1, u1 , . . . , uk−1 , γ(z)).
(3.12)
Because γ depends on u0 , it can be treated as an independent variable in place of u0 , so
Q̄(n, u1 , . . . , uk−1 , γ) =
Therefore
φ(n + 1, u1 , . . . , uk−1 , γ) =
Z
∂
φ(n + 1, u1 , . . . , uk−1 , γ).
∂γ
Q̄(n, u1 , . . . , uk−1 , γ)dγ + f (n + 1, u1 , . . . , uk−1 ),
where f is an arbitrary function. Therefore
Z
φ(n, u0 , . . . , uk ) = Q̄(n − 1, u0 , . . . , uk−1 )duk−1 + f (n, u0 , . . . , uk−2 ).
All that remains is to find f by the direct construction method. Substituting φ into C and
evaluating the result on solutions gives the condition that f must satisfy. Because Sn f does not
depend on u0 , differentiating with respect to u0 results in a differential equation for f . This
equation can be integrated to give an expression for f that depends on an arbitrary function
depending on n, u1 , . . . , uk−2 . Substituting this back into the CLaw and differentiating w.r.t. u1
gives a differential equation that can be solved to find the new arbitrary function. This process
of differentiating and then integrating is continued until an arbitrary function depending on n
only is obtained. This function must satisfy a linear difference equation that can hopefully be
solved. Thus the CLaw is reconstructed.
3.2
3.2.1
Partial Difference Equations
Kovalevskaya form
A crucial step in the above working was to replace uk by ∆+γ. We can do something similar for a
scalar difference equation with p independent variables, n = (n1 , . . . , np ), if it is in Kovalevskaya
form,
un1 +k,n̂ = ω(n, u0 , u1 , . . . , uk−1 ).
(3.13)
Here n̂ = (n2 , . . . , np ) and ui denotes all variables of the form un1 +i,n̂+m̂ (with m̂ ∈ Zp−1 ) on
which the function depends. For the P∆E to be k th -order w.r.t. n1 there must exist some m̂
6= 0. When not used inside a function, ui represents the (p − 1)-dimensional
such that ∂u ∂ω
n1 ,n̂+m̂
hyperplane of points of the form un1 +i,n̂+m̂ . A schematic example of a 2-D scalar P∆E in
Kovalevskaya form is shown in Figure 3.1.
Let z = {n, u0 , u1 , . . . , uk−1 } be the minimal set of initial conditions from which shifts of (3.13)
can be used to find any point of the form un1 +j,n̂+m̂ for j ≥ k. The function ω may not depend
on all of these points, (see Figure 3.1). For ease of notation let u(n + m) = um = um1 m̂ where
m̂ = (m2 , . . . , mp ); therefore ui is the set of all uim̂ . The P∆E is now denoted by ∆ = 0, where
∆ ≡ uk0 − ω,
40
ω = ω(z).
Chapter 3: Characterizing CLaws of Difference Equations
n1
un1 +3,n2
ω
n
u2
2
u1
u0
Figure 3.1: A 2-dimensional P∆E in Kovalevskaya form, un1 +3,n2 = ω. The box encloses points
that ω depends on, which are represented by crosses. Dashed lines represent the initial conditions.
Let [∆] represent a finite number of shifts of the P∆E that may occur in an expression; also let
ωJ ≡ SJ ω. Therefore on solutions of (3.13), [∆] = 0.
An explicit scalar P∆E is one that can be transformed into Kovalevskaya form by a change of
independent variables from n to
ñ = An + b
for A ∈ GL(p, Z), det(A) = ±1, and b ∈ Zp .
(3.14)
The value of the dependent variable is unaffected by the above transformation of independent
variables if we define
ũñ = un(ñ) = un .
For example, quad-graph equations, which can be written in the form
um+1,n+1 = ω(m, n, um,n , um+1,n , um,n+1 ),
where m = n1 and n = n2 , are transformed to Kovalevskaya form by the shear
!
!
1 1
0
A=
, b=
.
0 1
0
(3.15)
For instance, the dpKdV equation (H1 in the ABS classification [6]) is
β−α
,
um+1,n − um,n+1
(3.16)
β−α
.
ũm̃+1,ñ − ũm̃+1,ñ+1
(3.17)
um+1,n+1 = um,n +
and it transforms to
ũm̃+2,ñ+1 = ũm̃,ñ +
As a consequence of the following lemma, transforming an explicit P∆E into Kovalevskaya form
by (3.14) will preserve its CLaws.
Lemma 3.2.1. If an explicit scalar P∆E is transformed by a linear transformation (3.14) then
there is a one-to-one correspondence between equivalence classes of the CLaws of the original
P∆E and the CLaws of the transformed P∆E.
Proof. The transformation for the 2-D case is of the form
!
!
!
m̃
a b
m
=
+
ñ
c d
n
41
e
f
!
.
Chapter 3: Characterizing CLaws of Difference Equations
um,n+1
ũm̃+1,ñ+1
um+1,n+1
ω
ω
um,n
ũm̃+2,ñ+1
um+1,n
ũm̃,ñ
ũm̃+1,ñ
Figure 3.2: Transformation of a quad-graph equation into Kovalevskaya form. The dashed lines
show the initial conditions required to sweep out half the plane.
The effect of the shift operators on the transformed independent variables is
Sm m̃ = a(m + 1) + bn + e = m̃ + a,
Sm ñ = c(m + 1) + dn + f = ñ + c,
Sn m̃ = am + b(n + 1) + e = m̃ + b,
Sn ñ = cm + d(n + 1) + f = ñ + d.
Thus
a c
Sm = Sm̃
Sñ ,
b d
Sn = Sm̃
Sñ .
Therefore given a CLaw with densities F and G,
a c
b d
(Sm − I)F + (Sn − I)G = (Sm̃
Sñ − I)F̃ + (Sm̃
Sñ − I)G̃


|a|
|b|
sgn(a)−1
sgn(b)−1
X
X
a−sgn(a)i+
b−sgn(b)i+
2
2
= (Sm̃ − I) sgn(a)
Sm̃
Sñc F̃ + sgn(b)
Sm̃
G̃+

(Sñ − I) sgn(c)
i=1
|c|
X
i=1
sgn(c)−1
c−sgn(c)i+
2
Sñ
F̃
+ sgn(d)
i=1
|d|
X
i=1
= (Sm̃ − I)F̂ + (Sñ − I)Ĝ,

sgn(d)−1
b d−sgn(b)i+
2
Sm̃
Sñ
G̃
(3.18)
where F̃ and G̃ denote the densities in terms of the transformed variables and the second line is
obtained by repeatedly summing by parts. From (3.18), it is clear that divergence expressions
are transformed to divergence expressions. If the original CLaw is trivial of the second kind then
it vanishes identically so the transformed CLaw must also vanish identically. If F |[∆]=0 = 0 and
G|[∆]=0 = 0 then F̃ |[∆]=0
= 0 and G̃|[∆]=0
= 0 so the new densities F̂ and Ĝ are trivial. Because
˜
˜
the transformation is invertible the converse is also true. The same reasoning applies to higher
dimensional P∆Es; thus the lemma is true.
For example, when the quad-graph equations are transformed by (3.15), the transformed equation has CLaws with the densities
F̂ = F̃ + G̃,
To transform back to the quad-graph, use
!
m
=
n
1
0
Ĝ = Sm̃ G̃.
−1
1
42
!
m̃
ñ
(3.19)
!
.
(3.20)
Chapter 3: Characterizing CLaws of Difference Equations
Then the densities for the CLaws are
−1
F = F̂ − Sm
Ĝ,
−1
G = Sm
Ĝ.
(3.21)
It should be noted that in both cases, (3.19) and (3.21), the new densities will not be on the
initial conditions, but the difference equations can be used to pull them back. As a concrete
example, we show the three-point and five-point CLaws of the transformed dpKdV equation
(3.17) in Table 3.1 (where the tildes have been dropped and um+i,n+j ≡ uij ).
Table 3.1: Densities for the three and five point CLaws of transformed dpKdV
m+1
F̂1 = − (−1) m
Ĝ1 = (−1)
(2 u00 u11 − β − 2 u00u10 +α)
2 u10 u0−1 +
β−α
u1−1 −u10
−α
F̂2 = −
+ (u00 − u10 ) (u00 u10 −
α) (u00 − u11 ) (u00 u11 − β)
β−α
β−α
u10 u0−1 + u1−1 −u10 − α
Ĝ2 = u10 − u0−1 − u1−1 −u10
m+1
F̂3 = − (−1) m
Ĝ3 = (−1)
u00 2 u11 − u00 β + u112 u00− u11 β − u00 2 u10+ u00α − u00 u10 2 + α u10
β−α
u1−1 −u10
u10 + u0−1 +
u10 u0−1 +
β−α
u1−1 −u10
−α
2
u11 2 + 4 u11 u00 β − β 2 + 2 u00 2 u10 2 − 4 u00 u10 α + α2 −2 u00
2
m
β−α
β−α
2
Ĝ4 = (−1)
2 u10 2 u0(−1) + u1−1
−
4
α
u
u
+
10
0−1
−u10
u1(−1) −u10 + α
F̂4 = (−1)
m+1
β−α
F̂5 = − ln u10β−α
−
u
+
ln
u
+
00
0−1
−u
u
−u
11
1−1
10
β−α
Ĝ5 = ln u1−1 + (β − α) u0−2 + u1−2 −u1−1 − u0−1 −
β−α
u1−1 −u1,0
β−α
+
ln
−
u
F̂6 = − ln u00 + u10β−α
0(−1)
−u
u
−u
11
1−1
10
−1 β−α
β−α
Ĝ6 = ln (β − α) u0−2 + u1−2 −u1−1 − u0−1 − u1−1 −u10
−1
− u10
F̂7 = (m − n)F̂5 + nF̂6
Ĝ7 = (m − n + 1)Ĝ5 + nĜ6
A system of difference equations is in Kovalevskaya form if it is in the form
uin1 +ki ,n̂ = ω i (z)
for i = 1, . . . , q;
(3.22)
here
z = {n, u1k1 , u1k1 +1 , . . . , u1k1 −1 , . . . , uqkq , uqkq +1 , . . . , uqkq −1 }
0
0
0
0
where k0i < k i . The set uiki is the set of the ith dependent variable with the least shift in the
0
n1 direction that occurs in any of ω i s. In addition the system has a non-degeneracy condition:
43
Chapter 3: Characterizing CLaws of Difference Equations
there exist points uiki L̂ ∈ uiki such that
0
i
0
∂ω 1
∂u1 1
∂ω 1
∂uq q
...
k0 L̂q
k0 L̂1
..
.
..
.
q
∂ω
∂u1 1
q
∂ω
∂uq q
...
k0 L̂q
k0 L̂1
6= 0.
(3.23)
Given the initial conditions z, shifts of (3.22) can be used to find any point of the form uij m̂ for
j ≥ k i , j ∈ N.
Let
∆i ≡ u k i 0 − ω i ,
where ω i = ω i (z)
and let [∆i ] represent a finite number of shifts of the ith P∆E that occur in an expression.
Similarly let ∆ = (∆1 , . . . , ∆q )T and let [∆] represent all possible shifts of the equations, so on
solutions of (3.22), [∆] = 0.
3.2.2
The characteristic
The characteristic of a given CLaw is defined in the same manner as for O∆Es. Given the system
of equations
uiki 0 = ∆i + ω i ,
(3.24)
the shift operators are used to write any um in terms of the initial conditions, z, and [∆]. Let
C(z, [∆]) be an expression such that C(z, [0]) = 0. Then
Z
1
d
C(z, [λ∆1 ], [λ∆2 ], . . . , [λ∆q ]) dλ
dλ
λ=0
Z 1 X
∂C(z, [λ∆1 ], [λ∆2 ], . . . , [λ∆q ])
=
(SJ ∆i )
dλ.
∂SJ (λ∆i )
λ=0
C(z, [∆]) =
i,J
Now repeatedly summing by parts gives
C(z, [∆]) =
q
X
i=1
∆i
Z
1
λ=0
E∆i (C(z, [∆]))|∆7→λ∆ dλ + DivF,
for some F, where
E∆i (C(z, [∆])) :=
X
J
S−J
∂C(z, [∆1 ], [∆2 ], . . . , [∆q ])
∂SJ (∆i )
and F|[∆]=0 = 0.
(3.25)
Hence F is a trivial density of the first kind. If C(z, [∆]) is a CLaw (i.e. a divergence expression),
then by subtracting DivF from both sides of (3.25), the equivalent CLaw
C̃(z, [∆]) = C(z, [∆]) − DivF = Q(z, [∆]) · ∆
is obtained, where
Qi (z, [∆]) :=
Z
1
λ=0
(E∆i (C(z, [∆])))|∆7→λ∆ dλ.
(3.26)
The q-tuple Q is a multiplier of the system of equations, so being consistent with the continuous
case and [30], it is defined to be the characteristic of the CLaw.
44
Chapter 3: Characterizing CLaws of Difference Equations
A trivial characteristic is one that vanishes on solutions, so Q(z, [0]) = 0. For PDEs, the
characteristic of a CLaw is trivial if and only if the CLaw is trivial. We will show that, with our
definition of the characteristic (3.26), the same is true for P∆Es. Observe that
i
i
Q̄ (z) := Q (z, [0]) = lim
µ→0
Z
1
λ=0
Eµ∆i (C(z, [µ∆]))∆7→λ∆ dλ =
=E∆i (C(z, [∆]))|[∆]=0 ,
Z
1
λ=0
E∆i (C(z, [∆]))|[∆]=0 dλ
(3.27)
so we need to prove that a CLaw is trivial if and only if E∆ (C)|[∆]=0 = 0. We define Q̄(z), i.e.
the characteristic evaluated on solutions, to be the seed of the CLaw.
3.2.3
A trivial CLaw implies a trivial characteristic
A trivial CLaw of the second kind (DivF ≡ 0) vanishes identically, so the proof is immediate.
However the first kind of triviality takes the form F(z, [0]) = 0. To deal with this we use the
identity
E∆l (DivF(z, [∆])) =E∆l (Div (F(z, [∆]) − F(z, [0])))


X
X ∂Si zj
∂
=
S−J 
· Si
F i (z, [∆]) − F i (z, [0])  +
l
∂S
∆
∂z
J
j
i,j
J


i
i
X
X ∂Si SI ∆l
X
∂F (z, [∆]) 
∂F (z, [∆])
+
S−J 
· Si
−
.
∂SJ ∆l
∂SI ∆l
∂SJ ∆l
i
J
(3.28)
i,I
Using the fact that
X
J,i,I
S−J
∂Si SI ∆l
∂F i (z, [∆])
· Si
l
∂SJ ∆
∂SI ∆l
=
X
I,i
Si−1 S−I Si
∂F i (z, [∆])
,
∂SI ∆l
equation (3.28) simplifies to
E∆l (DivF(z, [∆])) =
X
J,i,j
S−J
∂
∂Si zj
· Si
F i (z, [∆]) − F i (z, [0]) .
∂SJ ∆
∂zj
From this, it immediately follows that
(E∆l (DivF(z, [∆])))|[∆]=0 = 0.
Thus Q|[∆]=0 = 0.
3.2.4
A trivial characteristic implies a trivial CLaw
Having proved that the characteristic of a trivial CLaw is a trivial characteristic, trivial densities
can be added to the densities of any CLaw, C(z, [∆]), without affecting the value of Q̄. In
particular, by noting that F(z, [∆]) − F(z, [0]) is a trivial density of the first kind, any explicit
dependence on [∆] in the densities can be removed. Therefore it can be assumed w.l.o.g. that
the densities of the CLaws do not depend explicitly on [∆]. Hence the only kind of triviality
that can occur is if the divergence expression vanishes identically.
45
Chapter 3: Characterizing CLaws of Difference Equations
Sn−R Sm F ω0−R
ω00
ω0R
Sm F
u2
u1
m
u0
n
Figure 3.3: A graphical representation of the terms in the characteristic for a 2-dimensonal
scalar P∆E. The solid black box encloses points on which Sm F depends and the blue box
encloses points on which Sn−R Sm F depends. The dashed boxes enclose points on which ω0−R ,
ω00 and ω0R depend.
3.2.4.1
The 2-dimensional scalar case
We begin by illustrating the method of proof with a scalar 2-dimensional P∆E, ∆ = uk0 − ω(z);
the CLaw is assumed to depend on ∆ to SnR ∆. This CLaw must have densities of the form
F = F (m, n, u0 , . . . , uk−1 ),
and G = G(m, n, u0 , . . . , uk−1 ),
where we use F and G in preference to F 1 and F 2 respectively. Now Sn G cannot depend on
any ∆ terms because Sn uik = ui(k+1) ∈ ui ⊂ z for i = 0, . . . , k − 1. Therefore the seed is
R
X
∂
−j
j
Q̄ =
Sn
F (m + 1, n, u1 , . . . , uk−1 , [Sn ∆ + ω0j ])
.
j
∂Sn ∆
j=0
[∆]=0
The shifts of Sm F on which Q̄ depends are shown in Fig.3.3. Let u0L ∈ u0 be the leftmost point
in u0 on which ω00 depends. Therefore
∂ω0j
6 0,
=
∂u0(L+j)
∂ω0i
= 0,
∂u0(L+j)
∀j ∈ Z,
i>j
i, j ∈ Z.
So, when evaluated on solutions, each u0(L+j) ∈ u0 point (the square points in Figure 3.3) is
replaced by ω0j (the discs in Figure 3.3) as an independent variable in Q̄, because the determinant
∂(ω0−R ,...,ω0R )
of ∂(u0(L−R)
is non-zero. Thus
,...,u0(L+R) )
Q̄ =
R
X
j=0
Sn−j
∂
F (m + 1, n, u1 , . . . , uk−1 , [ω0j ]).
∂ω0j
This is effectively the discrete Euler operator for ω, treating m and the ui terms as parameters.
The kernel of this operator (see Lemma 3.2.2) is made up of total divergences, (Sn − I)h, and
functions of n and u only. Thus if the characteristic is zero on solutions of the P∆E then
F (m+1, n, u1 , . . . , uk−1 , [ω0j ]) =(Sn − I)h(m+1, n, u1 , . . . , uk−1 , [ω0j ])+
+ f (m+1, n, u1 , . . . , uk−1 ),
for some f , and so
F (z) = (Sn − I)h(m, n, u0 , . . . , uk−2 , [u(k−1)j ]) + f (m, n, u0 , . . . , uk−2 ).
Adding the trivial density
FT = − (Sn − I)h(m, n, u0 , . . . , uk−2 , [u(k−1)j ]),
GT =(Sm − I)h(m, n, u0 , . . . , uk−2 , [u(k−1)j ]),
46
Chapter 3: Characterizing CLaws of Difference Equations
to the original densities gives the equivalent densities
F̃ = f (m, n, u0 , . . . , uk−2 ),
G̃ = G + (Sm − I)h.
The G̃ density may contain ∆ terms but these can be removed by adding a trivial density. Thus
the divergence expression for these densities cannot contain any ∆ terms, so in order for it to
be a CLaw it must vanish identically and is thus trivial.
3.2.4.2
The general case
For a system of q equations in Kovalevskaya form the seed of a CLaw is
1 X
X
∂S
F
∂
1
=
S−J
.
Q̄α (z) =
S−J
(Si F i − F i )
α ∂SJ ∆α
∂S
∆
J
J
J,i
[∆=0]
(3.29)
[∆=0]
The simplification is because the unshifted densities only have arguments in z so the Si F i terms
must only have arguments in z for i = 2, . . . , p. The only way for shifts of the equation to appear
in the CLaw is by shifting in the n1 direction. Thus the p-tuple J can be assumed to have the
form (0, j2 , j3 , . . . , jp ). Because only derivatives of S1 F 1 appear in (3.29), no terms in uα
for
kα
0
α = 0, . . . , q occur explicitly in Q̄. Therefore, because the difference equation is in Kovalevskaya
form, and in particular the non-degeneracy condition (3.23) holds, the ωJ s can be treated as
independent variables. Thus (3.29) becomes
Q̄α (n, u, [ω]) =
X
J
S−J
∂S1 F 1
=: Eωα (S1 F 1 );
∂ωJα
(3.30)
here u represents arguments from {uα
|1 ≤ i ≤ k − 1, α = 1, . . . , q}. The operator Eω is formed
kα
i
by having Eωα as each component. This is the Euler-operator for the functions ω which treats
n1 and the u terms as parameters.
Lemma 3.2.2. The kernel of Eω consists of sums of functions that are independent of any ωs,
together with total divergences in the directions i = 2, . . . , p.
Proof. If f = f (n, u) then it is immediate that f ∈ ker(Eω ). Therefore assume that f =
f (n, u, [ω]). Let
ωλα := λω0α + (1 − λ)g α (n, u),
where g α is a function of our choice (which can usually be the zero function). Then observe that
by summing by parts, if there are p independent variables,
X ∂f
d
f (n, u, [ωλ ]) =
SJ (ω0α − g α (n, u))
dλ
∂SJ ωλα
α,J
X
p
X
X
∂f
α
α
+
(Si − I)hi (n, u, [ω], λ)
=
(ω0 −g )
S−J
α
∂S
ω
J
λ
α
i=2
J
p
X
X
=
(ω0α −g α )Eωλα (f (n, u, [ωλ ])) +
(Si − I)hi (n, u, [ω], λ).
α
i=2
47
(3.31)
Chapter 3: Characterizing CLaws of Difference Equations
The second summation in (3.31) does not contain i = 1 because the ωs are only shifted in the
n2 , . . . , np directions. So integrating (3.31) w.r.t. λ, f satisfies
Z 1
X
α
α
f (n, u, [ω]) − f (n, u, [g]) =
(ω0 −g )
Eωλα (f (n, u, [ωλ ])) dλ+
λ=0
α
+
p
X
i=2
Therefore if Eω (f ) = 0 then f (n, u, [ω]) =
functions H i , where
1
(Si − I)
Pp
i=2 (Si
i
H (n, u) = f (n, u, [g]),
Z
H (n, u, [ω]) =
Z
1
hi (n, u, [ω], λ) dλ.
λ=0
− I)H i (n, u, [ω]) + H 1 (n, u) for some
1
hi (n, u, [ω]), λ) dλ, for i = 2, . . . , p.
λ=0
Hence the characteristic is zero on solutions if and only if
S1 F 1 =
p
X
i=2
and so
(Si − I)H i (n + 11 , u, [ω]) + f (n + 11 , u)
F1 =
p
X
(Si − I)H i (z) + f (n, S1−1 u).
i=2
Therefore adding the trivial density
FT1 = −
p
X
i=2
(Si − I)H i (z),
FTi = (S1 − I)H i (n, z) i = 2, . . . , p,
(3.32)
to the densities of the CLaw, gives an equivalent CLaw which is trivial, because shifts of the
densities must stay on the initial conditions. Hence the divergence expression can only vanish if
it does so identically.
3.2.5
Example: the characteristic on solutions for dpKdV
To demonstrate the use of the characteristic, we calculate the seeds (using maple) of the CLaws
in Table 3.1 of the transformed dpKdV equation (3.17). The seeds are displayed in Table 3.2,
and are expressed in terms of ωs because treating these as independent variables leads to more
compact expressions than pulling back to write each Q̄i in terms of z.
Now the transformed dpKdV equation also has the following CLaw
β−α
β−α
m+1
m+1
F = (−1)
2 u11 u01 +
− β + (−1)
α − 2u11 u00 +
,
u11 − u12
u10 − u11
m+1 (−1)
β−α
β−α
G=
−2 u00 +
u10 u0(−1) +
+
y
u10 − u11
u1−1 − u10
2
!
(−1)m+1
β−α
β−α
+
2u10 u00 +
− 2 u00 +
β
y
u10 − u11
u10 − u11
α (−1)m+1
β−α
β−α
+
u00 +
+ u0(−1) +
,
y
u10 − u11
u1−1 − u10
β−α
β−α
y = − u0−1 −
+ u00 +
.
u1−1 − u10
u10 − u11
48
Chapter 3: Characterizing CLaws of Difference Equations
Table 3.2: Seeds of the transformed dpKdV
Q̄1 = 2 (−1)
m+1
m+1
u11 − 2 (−1)
u10
Q̄2 = −2u11 ω00 + α + u11 2 + 2u10 ω00 − β − u10 2
m
Q̄3 = (−1)
Q̄4 = 4 (−1)
m
−2u11 ω00 + α − u11 2 + 2u10 ω00 − β + u10 2
−u11 2 ω00 + αu11 + u10 2 ω00 − β u10
−1
− u12
+
−1
−1
−2
−1
β−α
−
u
+ (ω00 − ω1 ) + (β − α) (ω0−1 − ω00 )
u10 + ω0−1
− (ω−1 − ω00 )
11
−ω00
−2
Q̄5 = − (β − α) (ω00 − ω01 )
u11 +
β−α
ω00 −ω01
−1
u11 + ω00β−α
+
−ω01 − u10
−1
−1
−2
β−α
−
u
+ (ω0−1 − ω00 ) − (β − α) (ω0−1 − ω00 )
u10 + ω0−1
1−1
−ω00
Q̄6 = − (ω00 − ω01 )
−1
+ (β − α) (ω00 − ω01 )
−2
−1
−2
2
Q̄7 = (β − α) (ω0 − ω01 )
u11 + ω00β−α
−
u
− ω00 −ω
+ (m − n) Q̄5 + ω0−1−1
12
−ω01
−ω00 +
01
−1
−2
−1
β−α
+nQ̄6 + (β − α) (ω00 − ω01 )
u11 + ω00
+ (ω0−1 − ω00 )
−ω1 − u10
However, this CLaw’s seed is 2 (−1)m (−u10 + u11 ) = Q̄1 . Thus it is equivalent to the first CLaw
in Table 3.1. In fact it was constructed by shifting the densities of the first CLaw forwards in
both directions and then pulling back to the initial conditions. This illustrates the use of the
seeds in giving a complete characterization of equivalent CLaws.
3.3
The Gardner CLaws for dpKdV
In [51] Rasin and Schiff used a discrete version of the Gardner transformation to construct an
infinite number of CLaws for the dpKdV equation (3.16). They showed that these CLaws were
distinct by taking a continuum limit and showing that the resulting CLaws for the continuous
equation are distinct. By using the characteristic we show that the CLaws they found are distinct
without having to take a continuum limit.
In this section, we consider the dpKdV equation on the quad-graph rather than in Kovalevskaya
form. The dpKdV equation can be solved for any of the points on the quad-graph; hence we
choose the initial conditions z = {m, n, ui0 , u−j1 , u0k |i, j, ∈ N, k ∈ Z} which are shown by dashed
lines in Figure 3.4.
The densities for the CLaws they generated are the Fi ’s and Gi ’s in the expansion of
!
∞
∞
X
X
1
(i) i
F = − ln(u10 − u01 ) − ln 1 +
v00 =
Fi i ,
u10 − u01 i=1
i=0
!
∞
∞
X
X
1
(i+1)
G = ln − ln(u00 − u20 ) + ln 1 + (1)
v00 i = ln +
Gi i ,
v00 i=1
i=0
49
(3.33)
(3.34)
Chapter 3: Characterizing CLaws of Difference Equations
F3
u1,1
S1−4 S2 G3
u5,1
S2 G3
G3
S 1 F3
Figure 3.4: The figure on the left shows the densities for the third CLaw in the hierarchy; on
the right the extreme shifts of the densities in the characteristic are shown.
in powers of . Where
(1)
v00 =
1
,
u00 − u20
(i)
v00 =
i−1
X
1
(j) (i−j)
v v
,
u00 − u20 j=1 00 10
(i)
and v00 is referred to as the ith order v term.
To prove that the CLaws are distinct the following lemma is used.
Lemma 3.3.1.
∂
(α)
v
6 0
=
∂u(α+1)0 00
∂
(α)
v
=0
∂u(α+j)0 00
∀α ∈ N,
(3.35)
j > 1, j ∈ N.
(3.36)
Proof. The basic case, α = 1, is immediate. Therefore assume it is true for α = k − 1, where
k ≥ 2. For each i ∈ N,


k−1
X
∂
∂
1
∂
(k)
(j)
(k−j)
(j)
(k−j) 

v =
v
v10 +v00
S1 v00
.
∂u(k+i)0 00
u00 − u20 j=1 ∂u(k+i)0 00
∂u(k+i)0
The first term in the summation is zero for all j using (3.36), because the highest value j can
take is k − 1 but the lowest value of i is 1. Similarly for all j ≥ 2 the second term vanishes, so
only one term is left
2 ∂
∂
(k)
(1)
(k−1)
v = v00
S1
v
.
(3.37)
∂u(k+i)0 00
∂u(k+i−1)0 00
Thus, using the inductive assumption, if i = 1 then (3.37) is non-zero and if i > 1, (3.37) is
zero.
Thus for α ≥ 1
∂
∂u(α+2)0
(α+1)
v00
(1)
v00
!
=
1
∂
(1)
v00 ∂u(α+2)0
(α+1)
v00
(1)
= v00 S1
Also, as the second factor in (3.38) doesn’t depend on u00 ,
!
(α+1)
∂
∂
v00
6= 0.
(1)
∂u00 ∂u(α+2)0
v
00
50
∂
∂u(α+1)0
(α)
v00
6= 0.
(3.38)
(3.39)
Chapter 3: Characterizing CLaws of Difference Equations
(α+1)
v00
for α ≥ 1 thus,
(α+1)
v
is
using Lemma 3.3.1, this is the only term in Gα to depend on u(α+2)0 . Therefore S2 00(1)
Expanding out (3.34) shows that the highest order v term in Gα is
(1)
v00
v00
the only term in the CLaw to depend on u(α+2)1 and so depend on S1α+1 ∆, because S1 F depends
only on points in z except for u11 (see Fig.3.4). Thus the characteristic on solutions is
!
α
∂S1 Fα X −i ∂S2 Gα +
E∆ (S1 Fα + S2 Gα )|[∆]=0 =
+
S1
∂∆
∂S1i ∆ i=0
[∆]=0
!!
(α+1)
∂
v00
−(α+1)
+ S1
S2
.
(1)
∂u(α+2)0
v00
[∆]=0
The final term is the only term to depend on u−(α+1),1 , as the other terms are not shifted as far
back, i.e. using (3.39)
!!
(α+1)
∂
v00
∂2
−(α+1)
E∆ (S1 F α + S2 Gα )|[∆]=0 = S1
6= 0,
S2
(1)
∂u−(α+1)1
∂u00 ∂u(α+2)0
v
00
therefore this term cannot be a linear combination of the other terms. Therefore the characteristic does not vanish on solutions of dpKdV, so the CLaw is non-trivial. Furthermore
the lower-order CLaws (those with densities F1 , . . . , Fα−1 and G1 , . . . , Gα−1 ) do not depend on
u−(α+1)1 , so the α CLaw cannot be a linear combination of these CLaws. Thus the CLaws
generated by the Gardner transformation are distinct.
Recently Rasin [46] has used the Gardner method to generate an infinite hierarchy of CLaws
for all the equations in the ABS classification and one asymmetric equation, however he has not
shown that these CLaws are distinct from one another. Therefore it would be interesting to see
if the above method for the dpKdV can be applied to these hierarchies of CLaws to show that
they are distinct, or, if this is not possible, that the seed can be used in a different manner to
show the same result.
3.4
Noether’s Theorem
The most important application for characteristics of CLaws is in proving the converse of
Noether’s theorem for difference equations; therefore providing the difference equation analogue
of Alonso’s result [37]. The version of Noether’s theorem presented here is due to Hydon [27].
Given a Lagrangian, L, the Euler-Lagrange equations are given by
X
∂
, α = 1, . . . , q.
0 = E(L), where E = (E1 , . . . , Eq ) and Eα =
S−J
∂uα
n+J
J
An infinitesimal symmetry generator
X=
q
X
Qα
α=1
∂
,
∂uα
n
(3.40)
where Q = (Q1 , . . . , Qq )T is the characteristic of the symmetry, is a variational symmetry if
E(XL) ≡ 0.
51
Chapter 3: Characterizing CLaws of Difference Equations
Therefore XL must be a total divergence
X
p
(SJ Qα )
J,α
X
∂
L=
(Si − I)f i .
α
∂un+J
i=1
(3.41)
Summation by parts is then used to rewrite (3.41) as
Q · E(L) =
X
α
Qα Eα (L) ≡
X
J,α
Qα S−J
∂
L
∂uα
n+J
=
p
X
(Si − I)F i .
(3.42)
i=1
Thus if Q is the characteristic for a variational symmetry it is also the characteristic for a CLaw
of the Euler-Lagrange equations (∆ = E(L)). Two variational symmetries are equivalent if they
differ by a symmetry whose characteristic vanishes on solutions of the Euler-Lagrange equations
- i.e. a trivial symmetry. Therefore, if the Euler-Lagrange equations can be put in Kovalevskaya
form, in which case the CLaw is trivial only if the characteristic vanishes on solutions, there is
a one-to-one correspondence between equivalence classes of variational symmetries and CLaws.
3.5
Summary
In this chapter we have provided a definition for characteristics for CLaws of explicit difference
equations and shown there is a one-to-one correspondence between equivalence classes of CLaws
and characteristics. Thus the characteristics provide a systematic method to determine whether
two CLaws are equivalent. Using the characteristics we have then shown there is a one-toone correspondence between variational symmetries and CLaws of Euler-Lagrange equations in
Kovalevskaya form.
The primary use of the characteristic is to identify when CLaws are equivalent. We have demonstrated this by using the characteristic to show that the infinite hierarchy of CLaws generated
by the Gardner transformation for the dpKdV equation are distinct.
52
Chapter 4
A New Approach to Finding
CLaws
In the previous chapter we showed how characteristics evaluated on solutions of the difference
equations (which we term seeds) identify equivalence classes of CLaws. In this chapter we show
the seeds are in the kernel of the adjoint of the linearized symmetry condition (ALSC). This
yields a new method for constructing CLaws of difference equations, similar to the methods
used to find symmetries. However to use this method, we need to be able to reconstruct the
CLaws from the seeds. This is more complex than the equivalent problem for PDEs because the
seed may not itself be a characteristic and so we simply cannot use a homotopy operator on the
product of the seed and the difference equation. Therefore we begin this chapter by showing
how to reconstruct CLaws from seeds. We then show that the seeds are in the kernel of the
ALSC. Finally we show how to construct CLaws for various quad-graph equations, generating
new five-point CLaws.
4.1
Reconstruction of CLaws from Seeds
A characteristic for a CLaw in general depends on shifts of the difference equations. If the
characteristic is known then the densities for the CLaw can, in principle, be reconstructed using
homotopy operators [29]. However, it is the seed, i.e. the characteristic evaluated on solutions,
that identifies equivalence classes of CLaws. In §4.2, we show that the seed is in the kernel of
the adjoint of the linearized symmetry condition evaluated on solutions (4.8). Therefore it is
desirable to be able to reconstruct the CLaws from the seed, which may not be a characteristic,
in which case homotopy operators cannot be used immediately.
The key to the reconstruction of CLaws from Q̄ is Lemma 3.2.2. By choosing suitable g α s in
the functions ωλα = λω0 + (1 − λ)g α (n, u),
Z 1
X
F 1 (n + 11 , u, [ω]) =
(ω0α − g α )
Q̄α (n, u, [ωλ ]) dλ+
λ=0
α
+
m
X
i=2
(Si − I)H i (n, u, [ω]) + f (n + 11 , u).
The H i terms can be set to zero w.l.o.g. by adding a trivial CLaw (3.32); their contribution to
53
Chapter 4: A New Approach to Finding CLaws
the CLaw is placed on the other densities, which have not yet been determined. Thus
1
F (n + 11 , u, [ω]) =
X
(ω0α
α
α
−g )
Z
1
Q̄α (n, u, [ωλ ]) dλ + f (n + 11 , u).
(4.1)
λ=0
In general, Q̄, and as a result (4.1), contains negative shifts of ω. A more compact expression
for (4.1) is obtained by adding expressions of the form (3.32) to remove these terms. Shifting
a CLaw gives an equivalent CLaw, so we can assume that S1 F 1 has only positive shifts of ω.
In this case the negative shifts of ω occurring in Q̄ are the result of negatively shifting various
derivatives of S1 F 1 (see (3.30) and Figure 3.3). Equation (4.1) is a sum of terms; the term with
the most negative shift of ω in the ith direction will not contain the most positive shift of ω.
Therefore adding (Si − I) of this term to (4.1) will give an equivalent expression that is more
compact. Continuing this process gives a reconstruction for S1 F 1 that has no negative shifts of
ω.
Equation (4.1) (or its simplified version as discussed above) provides all the ω dependence of
the CLaw. Having found the ω dependence, a characteristic for the CLaw can be calculated by
replacing ωJα in (4.1) with S J ∆α + ωJα and applying (3.26), so
Qα (z, [∆]) =
Z
1
λ=0
E∆α (F 1 (n + 11 , u, [∆ + ω0 ]))|∆7→λ∆ dλ.
(4.2)
Thus the CLaw can be written as
C = Q(z, [∆]) · ∆ =
m
X
α=1
α
Qα (z, [uiki 0 − ω i (z)])(uα
kα 0 − ω (z)).
Homotopy operators [29] can then be used to find the densities. Alternatively, once the ω
dependence has been found, the arbitrary function f and the other densities can be found using
a shortened version of the direct construction method [26].
4.1.1
Example: reconstruction of CLaws of dpKdV
Using maple we can reconstruct the CLaws from the characteristics on solutions of dpKdV
given in §3.2.5. For these examples we can choose g(n, u) = 0, so ωλ = λω00 (in practice we
start with this choice and only change it if we cannot evaluate the integral). The fourth CLaw
in Table 3.1 has the fourth seed given in Table 3.2. Thus
F (m + 1, n, u10 , u11 , ω00 ) = ω00
Z
1
−4 (−1)
λ=0
m
−λ ω00 u11 2 + αu11 +u10 2 λ ω00 −βu10 dλ+ (4.3)
+ f (m+1, n, u10 , u11 )
= ω00 −2 (−1)
m
m
−ω00 u11 2+u10 2 ω00 −4 (−1) (αu11 −βu10 ) +
+ f (m + 1, n, u10 , u11 ) .
From §4.1, a characteristic is
(4.4)
∂
m
m
(∆+ω00 ) −2 (−1) −(∆+ω00 )u11 2+u10 2 (∆+ω00 ) −4(−1) (αu11−βu10 ) dλ,
λ=0 ∂∆
∆7→λ∆
m
= − 2 (−1) −∆u11 2 + u10 2 ∆ − 2ω00 u11 2 + 2u10 2 ω00 + 2α u11 − 2β u10 .
Z
Q(z, ∆) =
1
54
Chapter 4: A New Approach to Finding CLaws
Thus
C =Q∆ = 2 (−1)
m
m+1
(u21 − ω00 ) −u11 2 u21 − ω00 u11 2 + u10 2 u21 + u10 2 ω00 + 2α u11 − 2βu10 ,
=2 (−1) (β +α− u21 u10 −u00 u10 −u21 u11 −u00 u11 ) (u21 u10 −u21 u11 −u00 u10 +u00 u11 −β +α) .
(4.5)
This is a total divergence, so applying the homotopy operator [29] gives the required densities
1
m
(−1) 4u10 u0−1 β + 20u00 β u11 + 4u−11 αu01 + 2u−10 u01 β − 10u00 2 u11 2 +
6
1
m
− (−1) u00 2 u−1(−1) 2 − 2u00 βu−1(−1) − 4u00 u−10 α − 10u00 u10 α + 2u−10 2 u00 2 +
6
1
m
− (−1) 7u01 2 u11 2 − 14αu11 u01 − u−10 2 u01 2 − 2u10 2 u0−1 2 − 2u01 2 u−11 2 + 5u00 2 u10 2 ,
6
1
m
G = (−1) −10 u10 u20 α + 2u−10 2 u00 2 − u10 2 u0−1 2 − 5u00 2 u10 2 +
6
1
m
+ (−1) −2u00 βu−1(−1) + 10u00 u10 α + u00 2 u−1(−1) 2 + 2u10 u0−1 β
6
1
m
+ (−1) 4u20 u1−1 β +5u10 2 u20 2−2u20 2 u1−1 2−4u00 u−10 α−12nβ 2 + 12nα2 .
6
F =−
Note that these densities contain points of the form u−1j and u2j . Therefore to write these
densities on z they need to be shifted forwards and then the dpKdV equation (3.17) used to pull
all terms back onto the initial conditions, leading to even longer expressions. Finally, setting
[∆] terms to zero gives an equivalent CLaw with densities on z. Once in this form, (3.30) can
be used to verify the seed. It is not necessary to pull the density back to the initial conditions
to verify the seed: by treating u3i etc. as functions of z and [∆], (3.27) can be used to calculate
the seed, however due to the functional nature of the expressions care must be taken.
In practice it is much easier to use the direct construction method, which leads to more compact
expressions for the densities. Starting from (4.4), we can use the direct construction method.
The densities are
m−1
m−1
F =u11 −2 (−1)
−u11 u01 2+u00 2 u11 − 4(−1)
(αu01 −βu00 ) +f (m, n, u00 , u01 ),
G =G(m, n, u00 , u10 ).
Thus substituting these into the CLaw and evaluating on solutions gives
β −α
β −α
β−α
m
2
2
−2 (−1) − u00 +
u11 +u10 u00 +
+
C = u00 +
u10−u11
u10 −u11
u10 −u11
m
m−1
m−1
−4(−1) (αu11 −βu10 ))−u11 −2 (−1)
−u11 u01 2 +u00 2 u11 −4 (−1)
(α u01 −βu00 ) +
+ f (m + 1, n, u10 , u11 ) − f (m, n, u00 , u01 ) + G (m, n + 1, u01 , u11 ) − G (m, n, u00 , u10 )
which must be zero. Now by differentiating we obtain
0=
∂2f
∂2C
=
,
∂u00 ∂u01
∂u00 ∂u01
Thus f = f (m, n, u00 ) + h(m, n, u01 ); however, h can be set to zero by adding a trivial CLaw,
so that it is absorbed by G. Thus
0=
∂C
∂
m
m+1
= 4 (−1) u11 2 u01 + 4 (−1)
u11 α +
G (m, n + 1, u01 , u11 ) ,
∂u01
∂u01
55
Chapter 4: A New Approach to Finding CLaws
and therefore
m
m
G (m, n, u00 , u10 ) = −2 (−1) u10 2 u00 2 + 4 (−1) u10 α u00 + g (m, n, u10 ) .
Now
0=
∂C
∂f
=−
,
∂u00
∂u00
and so f = f (m, n). The final differentiation is
0=
∂C
∂g
=−
,
∂u10
∂u10
so g = g(m, n). The CLaw now simplifies to
m
m
(Sm − I)f (m, n) + (Sn − I)g(m, n) = 2 (−1) α2 − 2 (−1) β 2 ,
a solution of which is f = 0 and g = 2n(α2 − β 2 )(−1)m . So the reconstructed densities are
m
F =2u11 (−1) −u11 u01 2 + u00 2 u11 + 2α u01 − 2β u00 ,
m
G =2 (−1) −u10 2 u00 2 + 2 u10 α u00 + α2 n − β 2 n ,
which are equivalent to the densities found by the homotopy method, as they have the same
seed.
For a more complicated example, consider the densities of the sixth Claw of dpKdV shown in
Table 3.1. Its characteristic on solutions is found in Table 3.2; it depends on ω0−1 , ω00 and ω01 .
The characteristic is constructed so that terms depending on ω0−1 do not depend on ω01 . The
term that depends on ω0−1 is
−1
β −α
−1
−2
h1 (m, n, u1 , ω0−1 , ω00 ) := (ω0−1 −ω00 ) −(β −α)(ω0−1 −ω00 )
u10 +
−u1−1 .
ω0−1 −ω00
Then let
h2 (m, n, u1 , ω00 , ω01 ) :=ω00 Q̄6 (m, n, u1 , [λω]) + (Sn − I) (ω00 h1 (n, u1 , [λω]))
ω00 u10 − ω01 u10 − ω00 u11 + ω01 u11
,
=−
−λ ω00 u11 + λ ω00 u10 − λ ω01 u10 − β + λ ω01 u11 + α
which is a function that only depends on ω00 and ω01 , which is integrated w.r.t. λ to obtain an
expression for Sm F ,
Z
Sm F |[∆]=0 = h2 d λ + f (m + 1, n, u10 , u11 , u12 )
−ω00 u11 + ω00 u10 − ω01 u10 − β + ω01 u11 + α
= − ln
+ f (m + 1, n, u10 , u11 , u12 ).
−β + α
(4.6)
In principle, the characteristic can be calculated from (4.6) and the homotopy operator used to
find densities for the CLaw. However, as was demonstrated in reconstructing the fourth CLaw,
the direct construction method is preferable.
By relabeling (4.6) and choosing G to depend on appropriate values of u, the densities have the
form
−u11 u01 + u11 u00 − u12 u00 − β + u12 u01 + α
F = −ln
+f (m, n, u00 , u01 , u02 ),
−β + α
G =G (m, n, u00 , u10 , u01 , u11 ) .
56
Chapter 4: A New Approach to Finding CLaws
The dependence on the dpKdV equation is already determined by (4.6), so all that needs to
be determined is the arbitrary function f and the density in the n direction: G. These form a
CLaw so they must satisfy
(u10 −u11 ) (−u11 u01 +u11 u00 −u12 u00 −β +u12 u01 +α)
C = −ln
+f (m + 1, n, u10 , u11 , u12 ) +
(−β +α) (u11 −u12 )
−u11 u01 + u11 u00 − u12 u00 − β + u12 u01 + α
+ ln
− f (m, n, u00 , u01 , u02 ) +
−β + α
+ G (m, n + 1, u01 , u11 , u02 , u12 ) − G (m, n, u00 , u10 , u01 , u11 ) = 0
By differentiating C w.r.t. the continuous variables, linear differential equations are obtained for
the unknown equations. By differentiating w.r.t. to variables on the bottom row, we eliminate
Sm f and by choosing the most extreme point in the bottom row on which Sn G depends, u02 , G
is also eliminated. Differentiating again gives
0=
∂2f
∂2C
=
,
∂u00 ∂u02
∂u00 ∂u02
so f = f1 (m, n, u00 , u01 ) + fˆ2 (m, n, u01 , u02 ). However f2 can be set to zero w.l.o.g. by the
addition of a trivial density which is absorbed by f1 and G. As a result
0=
∂C
∂S2 G
=
,
∂u02
∂u02
hence G = G(m, n, u00 , u10 , u11 ). Now
0=
∂2C
∂ 2 f1
=
,
∂u00 ∂u01
∂u00 ∂u01
so f1 = f2 (m, n, u00 ), where a trivial CLaw has been added to remove the other arbitrary
function. As a result
∂C
∂S2 G
0=
=
,
∂u01
∂u01
so G1 = G(m, n, u10 , u11 ). Finally (on the bottom row) differentiate w.r.t. u00 to obtain
0=
∂C
∂f2
=
,
∂u00
∂u00
so f2 = f3 (m, n). Having found all the dependence of f on the continuous variables, when
differentiating w.r.t. variables on the next row only the G functions will occur; by choosing an
extreme point in the n direction only one of these functions can occur. So
0=
∂G
∂C
−1
= − (u10 − u11 ) −
.
∂u10
∂u10
Therefore G = − ln(u10 − u11 ) + g(m, n, u11 ). The final differential equation is
0=
∂g
∂C
=
,
∂u11
∂u11
so g = g(m, n). Thus, to complete the reconstruction, the following difference equation needs to
be satisfied
f3 (m + 1, n) − f3 (m, n) + g (m, n + 1) − g (m, n) = 0.
By setting f3 = g = 0 this is satisfied. So the reconstructed CLaw has densities
−u11 u01 + u11 u00 − u12 u00 − β + u12 u01 + α
F = − ln
,
−β + α
G = − ln (u10 − u11 ) .
57
Chapter 4: A New Approach to Finding CLaws
4.2
The Adjoint of the Linearized Symmetry Operator
The Gâteaux derivative of an r-tuple P ∈ Ar is the operator Dp : Aq 7→ Ar defined in [29]1 by
P [u + Q[u]] − P [u]
DP (Q) = lim
→0
d
.
= P [u + Q[u]]
d
=0
Explicitly, the Gâteaux derivative of P is the r × q matrix shift operator with entries
(DP )iα =
X ∂P i
Sm ,
∂uα
n+m
m
i = 1, . . . , r,
α = 1, . . . , q.
Therefore its adjoint with respect to the `2 inner product is
X
∂P i
(D∗P )αi =
S−m α
S−m , i = 1, . . . , r,
∂un+m
m
α = 1, . . . , q.
Consequently the Euler operator, E, is defined by
E(P [u]) = D∗P (1).
So using the Leibniz rule, for two r-tuples P and Q,
∗
E(P · Q) = DP∗ ·Q (1) = DP∗ (Q) + DQ
(P ).
A vector field X =
P
α
(4.7)
Qα ∂u∂α acting on an r-tuple P can now be written as
n
pr X(P ) = DP (Q).
Thus, given a system of l difference equations, ∆ = 0, the linearized symmetry condition is
0 = D∆ (Q)|[∆]=0 .
The Euler operator acting on a expression is zero if and only if that expression is a total divergence
[29]. Therefore an l-tuple Q is a characteristic of a CLaw if and only if
0 = E(Q · ∆) = D∗∆ (Q) + D∗Q (∆).
Restricting this to solutions of the difference equations gives a necessary condition for Q to be
a characteristic:
0 = D∗∆ (Q)|[∆]=0 .
(4.8)
In other words, the characteristics are members of the kernel of the adjoint of the linearized
symmetry condition (ALSC), restricted to solutions. Arriola [10] showed that (4.8) must be
satisfied for first integrals of autonomous ordinary difference equations.
We illustrate the above with an O∆E of the form
∆ ≡ u2 − ω(n, u0 , u1 ) = 0,
1 In
this paper it is called a Fréchet derivative.
58
Chapter 4: A New Approach to Finding CLaws
where ui = un+i . The linearized symmetry condition for the equation is
pr X(∆) ≡ Sn2 Q − (Sn1 Q)ω,2 (n, u0 , u1 ) − Qω,1 (n, u0 , u1 ) = 0,
when [∆] = 0.
Thus
D∆ =Sn2 − ω,2 (n, u0 , u1 )Sn1 − ω,1 (n, u0 , u1 )I,
D∗∆ =Sn−2 − Sn−1 (ω,2 (n, u0 , u1 ))Sn−1 − ω,1 (n, u0 , u1 )I.
Given a CLaw in characteristic form, C = Q(n, u0 , u1 , u2 − ω)(u2 − ω), applying the Euler
operator will give
∂
∂
∂
−1
−2
C + Sn
C + Sn
C
0 =E(C) =
∂u0
∂u1
∂u2
=(Q,1 − ω,1 Q,3 )(u2 − ω) + Sn−1 ((Q,2 − ω,2 Q,3 )(u2 − ω)) + Sn−2 (Q,3 (u2 − ω))
+ Sn−2 Q − Sn−1 (ω,2 Q) − ω,1 Q.
Restricting this to solutions of the equation then gives the desired result,
0 = Sn−2 Q − Sn−1 (ω,2 Q) − ω,1 Q |[∆]=0 = D∗∆ (Q)|[∆]=0 .
This expression now contains backwards shifts of both Q and ω, therefore u−2 and u−1 are
considered as the initial conditions, so when evaluated on solutions u0 = ω(n − 2, u−2 , u−1 ) etc.
The linearized symmetry condition for quad-graph equations that have been transformed into
Kovalevskaya form, u21 = ω(u00 , u10 , u11 ) (where the dependence on m and n is not shown), is
2 1
Sm
Sn Q−Qω,1 (u00 , u10 , u11 )−(Sm Q)ω,2 (u00 , u10 , u11 )−(Sm Sn Q)ω,3 (u00 , u10 , u11 )∆=0 = 0.
It is easiest to demonstrate (4.8) by changing co-ordinates:
zij =uij ,
i
Sm
Snj ∆
j ∈ Z, i ∈ {0, 1},
=u(2+i)(1+j) − ω(uij , u(1+i)j , u(1+i)(1+j) ) j ∈ Z, i ∈ N0 ,
The partial derivatives needed for the Euler operator transform to
∂
∂
∂
=
− ω,1 (u0j , u1j , u1(1+j) ) j ,
∂u0j ∂z0j
∂Sn ∆
∂
∂
∂
∂
=
−ω,2 (u0j , u1j , u1(1+j) ) j −ω,3 (u0(j−1) , u1(j−1) , u1j ) j−1 +
∂u1j ∂z1j
∂Sn ∆
∂Sn ∆
∂
−ω,1 (u1j , u2j , u2(1+j) )
,
∂Sm S2n ∆
∂
∂
∂
∂
−ω,2 (u1j , u2j , u2(1+j) )
−ω,3 (u1(j−1) , u2(j−1) , u2j )
+
=
∂u2j ∂Snj−1 ∆
∂Sm Snj ∆
∂Sm Snj−1 ∆
∂
− ω,1 (u2j , u3j , u3(1+j) )
.
2
∂Sm Snj ∆
Therefore given a CLaw in characteristic form (where w.l.o.g. the characteristic only depends on
shifts of the equation in the n direction) applying the Euler operator gives
0=
2
X
−i −j
Sm
Sn
i=0,j
=(−ω00,1 Q −
+
X
j
Sn−j
∂
(Q(n, z, [∆])∆)
∂uij
−1
Sm
(ω00,2 Q)
−
−1 −1
Sm
Sn (ω00,3 Q)
+
−2 −1
Sm
Sn Q)
+
X
j
−2
Sn−j Sm
∆
∂Q
∂Snj ∆
+
∂Q
∂Q
∂Q
∂Q
∂Q
−1
∆
− ω0j,1 j
+S1
∆
−ω0j,2 j − ω0(j−1),3 j−1
.
∂z0j
∂z1j
∂Sn ∆
∂Sn ∆
∂Sn ∆
59
Chapter 4: A New Approach to Finding CLaws
Therefore, restricting this to solutions of the difference equation gives
−1
−1 −1
−2 −1
0 = (−ω00,1 Q − Sm
(ω00,2 Q) − Sm
Sn (ω00,3 Q) + Sm
Sn Q)|[∆]=0 ,
which is the desired result.
This result suggests a new method for directly constructing CLaws of difference equations following that of [8, 9] for differential equations. Find functions Q that satisfy (4.8), using methods
similar to those used to find symmetries of difference equations. Then find additional constraints
on Q by applying the difference Euler operator to Q∆. Then reconstruct the densities using
homotopy operators or inspection.
4.3
Finding CLaws for Quad-Graph Equations
We now use the above theory to find CLaws for quad-graph equations. By working with the
equations on the quad-graph,
∆ ≡ u11 − ω(m, n, u00 , u10 , u01 ),
rather than in Kovalevskaya form, Rasin and Hydon’s method for finding symmetries of quadgraph equations [50] can be adapted to find characteristics on solutions. Shifting the ALSC (4.8)
gives
0 = Q̄ − Sn ω,2 Q̄ − Sm ω,3 Q̄ − Sm Sn ω,1 Q̄ [∆]=0 ,
(4.9)
and we search for solutions of (4.9) which are pulled back onto the initial conditions z =
{m, n, ui0 , u0j }, which we term the cross. To illustrate some of the assumptions we can make
about the form of the solutions, suppose
Q̄ = Q̄(m, n, u−20 , u0−2 , u−10 , u0−1 , u00 , u10 , u01 , u20 , u02 );
the ALSC for this choice of Q̄ is depicted in Figure 4.1 which helps us see which terms depend
on which initial conditions. To solve (4.9) on the cross, the quad-graph equation needs to be
solved for its different terms and shifted appropriately; we denote the solutions in the different
directions by
u11 =ω(m, n, u00 , u10 , u01 ),
u−11 =ΩL (m, n, u−10 , u00 , u01 ),
u−1−1 = Ω(m, n, u0−1 , u−10 , u00 ),
u1−1 = ΩR (m, n, u0−1 , u00 , u10 ).
Differentiating (4.9) w.r.t. u−20 and u0−2 gives the necessary condition
0=
∂2
Q̄(m, n, u−20 , u0−2 , u0−1 , u00 , u10 , u01 , u20 , u02 );
∂u−20 ∂u0−2
hence
Q̄ =A(m, n, u0−2 , u−10 , u0−1 , u00 , u10 , u01 , u20 , u02 )+
+ B(m, n, u−20 , u0,−1 , u−10 , u00 , u10 , u01 , u20 , u02 ).
Differentiating (4.9) w.r.t. u0−2 and u−10 now gives
0=
∂2
A(m, n, u0−2 , u−10 , u0−1 , u00 , u10 , u01 , u20 , u02 ),
∂u−10 ∂u0−2
60
(4.10)
Chapter 4: A New Approach to Finding CLaws
n
Sn Sm (ω,1 Q)
Sn (ω,2 Q)
ω
ΩL
00
−20
Q
m
Sm (ω,3 Q)
ΩR
Ω
0−2
Figure 4.1: Pictorial representation of the ALSC (4.9)
and differentiating (4.9) w.r.t. u0−1 and u−20 gives
0=
∂2
B(m, n, u−20 , u0,−1 , u−10 , u00 , u10 , u01 , u20 , u02 ).
∂u−20 ∂u0−1
Therefore (after relabeling)
Q̄ =A(m, n, u0−2 , u0−1 , u00 , u10 , u01 , u20 , u02 )+B(m, n, u−20 , u−10 , u00 , u10 , u01 , u20 , u02 )+
+ C(m, n, u−10 , u0−1 , u00 , u10 , u01 , u20 , u02 ).
Differentiating (4.9) w.r.t. u0−2 and u30 now gives
0=
∂2
(Sm ω,3 )A(m+1, n, Sn−1 ΩR , u1−1 , u10 , u20 , u11 , u30 , u12 ) ,
∂u30 ∂u0−2
so applying the chain rule (because Sn−1 ΩR depends on u0−2 ) and then shifting appropriately,
we obtain
0=
∂2
A(m, n, u0−2 , u0−1 , u00 , u10 , u01 , u20 , u02 ).
∂u20 ∂u0−2
Therefore
A = A1(m, n, u0−2 , u0−1 , u00 , u10 , u01 , u02 )+A2(m, n, u0−1 , u00 , u10 , u01 , u20 , u02 ).
The function A2 can be absorbed into C, so it can be set to be zero w.l.o.g. Now differentiating
(4.9) w.r.t. u0−2 and u20 gives
0=
∂2
ω,3 (m+1, n, u10 , u20 , u11 )A1(m+1, n, Sn−1 ΩR , u1−1 , u10 , u20 , u11 , u12 ) ,
∂u20 ∂u0−2
61
Chapter 4: A New Approach to Finding CLaws
which, after using the chain rule and shifting, becomes
0=
∂2
(ω,3 (m, n, u00 , u10 , u01 )A1(m, n, u0−2 , u0−1 , u00 , u10 , u01 , u02 )) .
∂u10 ∂u0−2
Hence
A1 =
A11(m, n, u0−2 , u0−1 , u00 , u01 , u02 )
+ A12(m, n, u0−1 , u00 , u10 , u01 , u02 ).
ω,3 (m, n, u00 , u10 , u01 )
As before, A12 can be absorbed into C and we note that A11 contains only terms on the vertical
axis (m = 0). By performing the same analysis for the B term (i.e. differentiating w.r.t. u−20
and the corresponding terms on the vertical axis) and relabeling we find that
Q̄ =
A(m, n, u0−2 , u0−1 , u00 , u01 , u02 ) B(m, n, u−20 , u−10 , u00 , u10 , u20 )
+
+
ω,3 (m, n, u00 , u10 , u01 )
ω,2 (m, n, u00 , u10 , u01 )
+ C(m, n, u−10 , u0−1 , u00 , u10 , u01 , u20 , u02 ).
Now C is the only term to depend on u−10 and u0−1 so it can be isolated by differentiating w.r.t.
these points. Thus, similar analysis to the above can be performed to show that
C=
C1(m, n, u0−1 , u00 , u01 , u02 ) C2(m, n, u−10 , u00 , u10 , u20 )
+
+
ω,3 (m, n, u00 , u10 , u01 )
ω,2 (m, n, u00 , u10 , u01 )
+ C3(m, n, u00 , u10 , u01 , u20 , u02 ).
Now C1 and C2 can be absorbed into A and B respectively. And so, after relabeling,
Q̄ =
A(m, n, u0−2 , u0−1 , u00 , u01 , u02 ) B(m, n, u−20 , u−10 , u00 , u10 , u20 )
+
+
ω,3 (m, n, u00 , u10 , u01 )
ω,2 (m, n, u00 , u10 , u01 )
+ C(m, n, u00 , u10 , u01 , u20 , u02 ).
Thus Q̄ has been split into three parts: a term just depending on vertical points divided by ω,3 ,
a term depending just on horizontal points divided by ω,2 , and a term that depends on points
in the upper right quadrant only. In fact Q̄ can always be split like this if it depends on points
on the cross. Therefore this is different to the search for symmetry generators, which in order
to be solutions of the linearized symmetry condition, splits into a horizontal and a vertical part.
Further simplifications can be made, without reference to any particular quad-graph equation,
by choosing the initial conditions to be on a cross centered at (1, 1) rather than at (0, 0). Differentiating w.r.t. u31 yields
∂
ω,1
0=
Sm Sn ω,1 C(m, n, u00 , u10 , u01 , u20 , u02 ) +
B(m, n, u−10 , u00 , u10 , u20 )
.
∂u31
ω,2
(4.11)
Differentiating (4.11) w.r.t. u13 isolates the C term and hence
C = C1(m, n, u00 , u10 , u01 , u20 ) + C2(m, n, u00 , u10 , u01 , u02 ).
Returning to (4.11) and then differentiating w.r.t. u12 now yields (after shifting back)
0=
∂
(ω,1 C1(m, n, u00 , u10 , u01 , u20 )) +
∂u01 ∂u20
∂
ω,1
∂
+
B(m, n, u−20 , u−10 , u00 , u10 , u20 ).
∂u01 ω,2 ∂u20
62
(4.12)
Chapter 4: A New Approach to Finding CLaws
Differentiating (4.12) w.r.t. u−20 or u−10 isolates the B term and so
B = B1(m, n, u−20 , u−10 , u00 , u10 ) + b2(m, n, u00 , u10 , u20 ).
Therefore (4.12) can now be solved for C1,
C1 =
B2(m, n, u00 , u10 , u20 )
b2(m, n, u00 , u10 , u20 )
+C12(m, n, u00 , u10 , u01 )−
.
ω,1
ω,2
By symmetry we can repeat the same analysis for the A and C2 terms. And so, after relabeling,
Q̄ =
A1(m, n, u0−2 , u0−1 , u00 , u01 ) B1(m, n, u−20 , u−10 , u00 , u10 )
+
+
ω,3 (m, n, u00 , u10 , u01 )
ω,2 (m, n, u00 , u10 , u01 )
A2(m, n, u00 , u01 , u02 ) + B2(m, n, u00 , u10 , u20 )
+
+ C(m, n, u00 , u10 , u01 ).
ω,1 (m, n, u00 , u10 , u01 )
(4.13)
Once again, it can be shown (using similar reasoning), that any Q̄ on the cross must have the
same form: a term with vertical points up to u01 divided by ω,3 , a term with horizontal points
up to u10 divided by ω,2 , a horizontal and vertical term both with only positive points divided
by ω,1 and finally a function containing just the points u00 , u10 and u01 .
n
Sn B1
ω,2
B1
−20
−1
Sm
ΩL
ω
ΩL
00
−10
10
m
Figure 4.2: Terms that depend on u−10 in the ALSC
From this point it is no longer possible to get generic information about the solution. The specific
quad-graph equation will affect the solution of the following PDEs which we are able to find by
differentiating the ALSC. Because the seed does not split into a horizontal and vertical part,
solving the ALSC is more complicated than solving the linearized symmetry condition. Therefore
we outline below some of the methods used for obtaining systems of differential equations for
the terms in the ALSC. For clarity, we will suppress the arguments of the functions.
Only B1 and Sn B1 depend on the point u−20 (see Figure 4.2). Hence we can differentiate w.r.t.
this point to obtain
∂
∂
B1
0=
(ALSC) =
+ Sn B1 ,
(4.14)
∂u−20
∂u−20 ω,2
∂B1
∂Sn B1
+ ω,2
.
0=
∂u−20
∂u−20
Now B1 is independent of u01 , thus
∂
∂
−1
0=
ω,2
B1(m, n + 1, Sm
ΩL , ΩL , u01 , ω) ,
∂u01
∂u−20
(4.15)
where all the arguments of B1 depend on u01 . Expanding out (4.15) leads to a PDE in B1
with functions as arguments. However, using (4.10), we can transform this to a PDE with
63
Chapter 4: A New Approach to Finding CLaws
u−21 , u−11 , u01 , u11 and u00 as the independent variables. Shifting this expression by Sn−1 gives
a PDE with arguments back on the original cross. The function B1 will not depend on u0−1 but
in general the expression will, so we can split the expression into a system of PDEs according to
the coefficients of u0−1 .
By applying a suitable differential operator to (4.14), a further system of PDEs is obtained for
B1. In order to construct quad-graph equation CLaws directly, differential operators have been
constructed to annihilate individual terms (see §1.3.2.1). We now extend this idea to construct
operators that annihilate multiple terms. The operator
∂
∂
∂
∂
LΩL :=
ΩL (m, n, u−10 , u00 , u01 )
−
ΩL (m, n, u−10 , u00 , u01 )
∂u−10
∂u00
∂u00
∂u−10
annihilates u−11 . We now use this operator to define
∂
∂
−
ω(m, n, u00 , u10 , u01 ) LΩL ,
LΩL ,ω := LΩL (ω(m, n, u00 , u10 , u01 ))
∂u10
∂u10
which by construction annihilates u11 , and because u−11 is independent of u10 , u−11 is also
annihilated. Continuing this process enables a differential operator to be defined recursively
that annihilates a whole row of dependent points. The operator we require is
∂
−1
LSm
+
ΩL ,ΩL ,ω,Sm ω :=LΩL ,ω (ΩL (m − 1, n, u−20 , u−10 , u−11 ))
∂u−20
∂
−
ΩL (m − 1, n, u−20 , u−10 , u−11 ) LΩL ,ω ,
∂u−20
which, because u11 and u−11 are independent of u−20 , annihilates u−21 , u−11 and u11 .
Having defined our differential operators we can return to (4.14),
−1
ΩL
∂
B1
∂Sm
−1
0=
+
B1,1 (m, n + 1, Sm
ΩL , ΩL , u01 , ω).
∂u−20 ω,2
∂u−20
In this equation Sn B does not explicitly depend on u−20 , u−10 , u00 , or u10 (see Figure 4.2),
∂S −1 Ω
−1
so dividing through by ∂um−20L , and then applying LSm
ΩL ,ΩL ,ω,Sm ω will annihilate this term
yielding
1
∂
−1
0 = LSm
B1(m, n, u−20 , u−10 , u00 , u10 .
(4.16)
ΩL ,ΩL ,ω,Sm ω
−1
ω,2 (Sm
ΩL ),1 ∂u−10
This expression contains u01 , but B1 does not depend on this term, so we can split (4.16) into a
system of PDEs for B1 according to the coefficients of u01 . Combining this system of equations
with the system of equations obtained from (4.15) gives a system of linear PDEs to solve for B1.
We now consider the initial conditions to be the cross centered at u10 . The only terms that
depend on u30 are Sm B2 and Sm Sn B2 (see Figure 4.3). Therefore, differentiating the ALSC
w.r.t. u30 yields
ω,3
∂
Sm
B2 + Sm Sn B2 .
(4.17)
0=
∂u30
ω,1
Shifting this back and rearranging gives
0=
∂B2
ω,1 ∂Sn B2
+
.
∂u20
ω,3 ∂u20
64
Chapter 4: A New Approach to Finding CLaws
n
11 Sm
ω,3
ω,1
Sm Sn B2
2
Sm
ω
Sm ω
10
20
Sm B2
30
m
Figure 4.3: Terms that depend on u30 in the ALSC
So differentiating w.r.t. u01 yields
0=
∂
∂u01
ω,1 ∂
B2(m, n + 1, u01 , ω, Sm ω)
ω,3 ∂u20
(4.18)
Expanding out (4.15) leads to a PDE for B2 with functions as arguments. Using (4.10), we can
transform this to a PDE with u01 , u11 , u21 and u00 as the independent variables. Shifting this
expression by Sn−1 gives a PDE with arguments back on the original cross. The function B2
will not depend on u0−1 , so we can split the expression into a system of PDEs according to the
coefficients of u0−1 .
Another system of equations is obtained by applying the differential operator
∂
∂
Lω,Sm ω :=Lω (ω(m+1, n, u10 , u20 , u11 ))
−
ω(m+1, n, u10 , u20 , u11 ) Lω ,
∂u20
∂u10
where
∂
∂
∂
∂
ω(m, n, u00 , u10 , u01 )
−
ω(m, n, u00 , u10 , u01 )
.
Lω :=
∂u00
∂u10
∂u10
∂u00
As Sn B2 doesn’t explicitly depend on u00 , u10 and u20 ,
ω,3
∂
0 = Lω,Sm ω
B2(m, n, u00 , u10 , u20 ) .
ω,1 Sm (ω,2 ) ∂u20
(4.19)
In (4.19) B2 does not depend on u01 so we can split this expression into a system of PDEs
according to the coefficients of u01 .
By symmetry we can carry out the same process as above, with corresponding differential operators, to obtain systems of differential equations that A1 and A2 must satisfy. We have used
this method on seeds that have extreme points u−10 , u0−1 , u20 , u02 (which are needed to find
five-point CLaws). Solving the resulting system of differential equations satisfies the ALSC’s
dependence on the most extreme points. It may, however, be necessary to solve a linear difference equation when going back up the hierarchy of equations, e.g. from (4.17), to achieve this.
This enables us to isolate terms that depend on the next most extreme points in the ALSC
and generate more differential equations for the remaining unknown functions. This suggests
that for more complicated seeds, such as the one used in the discussion above with extreme
points u−20 , u0−2 , u20 , u02 , the same methods can be applied. Solving the system of differential
equations generated should satisfy the ALSC’s dependence on the most extreme points. We can
then use the same techniques to generate more systems of differential equations to remove the
ALSC’s dependence on its new extreme points and so on. This process is continued until we are
left with only linear difference equations to solve. A maple worksheet for an example of this
process can be found in Appendix B.
65
Chapter 4: A New Approach to Finding CLaws
We now apply the above theory to the Hydon-Viallet equation (HV)
u11 = −
(u10 + 1)(u01 − 1)
,
u00 (u10 − 1)(u01 + 1)
(4.20)
the potential Lotka–Volterra equation (pLV)
u11
u01
−
= 1,
u00
u10
(4.21)
and the potential Hydon–Viallet (pHV) equation
u11 =
(−1)m (u01 − u10 )
.
u00 (u10 + u01 )
These three equations have known three-point CLaws and master symmetries [28]. Therefore
they should have five-point point CLaws which have not previously been found. A five-point
CLaw must have a seed of the form
Q̄ = Q̄(m, n, u−10 , u0−1 , u00 , u10 , u01 , u20 , u02 ),
(4.22)
so this is the ansatz (combined with (4.13)) that we use. The maple worksheet used for the
potential Lotka–Volterra equation is included in Appendix B. The solutions to the ALSC, which
Mikhailov et al. [41] term cosymmetries, are listed in Tables 4.1–4.3.
Having found solutions of the ALSC evaluated on solutions of the quad-graph equation we now
need to reconstruct the CLaws. The formula for reconstructing CLaws requires the difference
equation to be in Kovalevskaya form, so we transform the expressions for the seeds and the
quad-graph equation into this form using (3.15). For example, the pLV equation transformed to
Kovalevskaya form is
˜ ≡ ũ21 − ω00 = 0,
∆
ω00 = ũ00 +
ũ11 ũ00
,
ũ10
(4.23)
and its first cosymmetry in Table 4.2 is transformed to
−ũ22 ũ10 ũ20 ũ1−1 + ũ10 ũ221 ũ1−1 + ũ210 ũ221 − ũ22 ũ20 ũ1−1 ũ11 ñ
Q̄1 (m̃, ñ, ũ1 , ũ2 ) = −
+
ũ10 ũ1−1 ũ20 ũ221
2 ũ210 ũ21 + ũ10 ũ20 ũ1−1 + 2 ũ10 ũ21 ũ1−1 + ũ20 ũ1−1 ũ11
+
,
ũ10 ũ20 ũ1−1 ũ21
where we have used the difference equation to express Q̄1 with terms in ũ1 and ũ2 . Because the
difference equations are in Kovalevskaya form, we can follow the start of §4.1 and perform the
substitution ũ2j = λω0(j−1) + (1 − λ)g(m̃, ñ, ũ1 ). The density of the CLaw, which determines
the characteristic, is then obtained using (4.1). For the first cosymmetry of the pLV equation
we use g(m̃, ñ, ũ1 ) = 1, to yield
(ũ10 + ũ11 ) ñ(−ω00 + ω01 + ω00 2 − ω00 ω01 ) + ln (ω00 ) ω01 ω00
F̂ (m̃ + 1, ñ, ũ1 , [ω]) =
+ (4.24)
ω00 (ω00 − 1) ũ10
(ũ10 + ũ11 ) ln (ω00 ) ω00 2 − 2 ln (ω00 ) ω00
+
+ f (m̃ + 1, ñ, ũ1 ).
(4.25)
ω00 (ω00 − 1) ũ10
The other density Ĝ and the unknown function f can now be determined (if the cosymmetry
is the seed of a characteristic) by the direct construction method, provided we insist that they
depend on enough terms. Having reconstructed the densities we transform them back, using
66
Chapter 4: A New Approach to Finding CLaws
Table 4.1: Solutions of the ALSC for the Hydon–Viallet equation
Q̄1 = − 12
(−1)m+n u00 2 (u10 −1)2 (u01 +1)(u10 +u00 u10 −u10 u20 −u20 u00 u10 +u−10 u00 −u−10 −u−10 u20 +u−10 u00 u20 )
(u10 u20 −u10 +u20 +1)(u10 +1+u00 u10 −u00 )2 (u00 +1+u−10 u00 −u−10 )(u01 −1)
Q̄2 =
m+n
u00 2 (u01 +1)2 (u10 −1)(−u00 u01 +u01 +u01 u02 −u00 u01 u02 +u0−1 u00 u02 −u0−1 −u0−1 u00 +u0−1 u02 )
1 (−1)
2
(u01 u02 +u01 −u02 +1)(−u00 +1+u0−1 u00 +u0−1 )(−u01 +1+u00 u01 +u00 )2 (u10 +1)
Q̄3 =
1 (u00 u10 −u00 −u10 −1)u00 (u10 −1)(u01 +1)m
4
(u10 +1)(u01 −1)(u10 +1+u00 u10 −u00 )
Q̄4 =
1 u00 (u10 −1)(u01 +1)(u00 u10 −u10 −u00 −1)
4
(u01 −1)(u10 +1)(u10 +1+u00 u10 −u00 )
Q̄5 =
1 u00 (u10 −1)(u01 +1)(u00 u01 +u01 +u00 −1)
4 (−u01 +1+u00 u01 +u00 )(u01 −1)(u10 +1)
(m+n)π
u00 sin
(u10 −1)(u01 +1)
2
−
(u01 −1)(u10 +1)
(m+n)π
u00 cos
(u10 −1)(u01 +1)
2
−
(u01 −1)(u10 +1)
Q̄6 =
Q̄7 =
−
1 (u00 u01 +u00 +u01 −1)nu00 (u10 −1)(u01 +1)
4
(u10 +1)(u01 −1)(−u01 +1+u00 u01 +u00 )
Table 4.2: Solutions of the ALSC for the potential Lotka–Volterra equation
Q̄1 =
(−n+2)u10
u0−1 u00
Q̄2 = u00 −1 +
Q̄3 =
+
n−1
u00
+
nu02 u10
u00 2 (u10 +u01 )
u02 u10
u00 2 (u10 +u01 )
u10 (−m−1)u20
(u20 +u00 )u00 (u10 +u01 )
−
+
mu10
u00 u01
10 u20
+
Q̄4 = − (u20 +u00u)u
00 (u10 +u01 )
Q̄5 =
u10
u0−1 u00
mu10 2
(u10 +u−10 )u01 u00
−
u10
u00 u01
−
(−1)m+n (u00 2 u10 +u00 2 u01 +u10 2 u01 )
u00 2 (u10 +u01 )u01
Q̄6 = − u00
2
u10 +u00 2 u01 −u10 2 u01
u00 2 (u10 +u01 )u01
67
u10 2
(u10 +u−10 )u01 u00
Chapter 4: A New Approach to Finding CLaws
−1
−
m
−1
(−1)m
u00
− m (u10 + u01 ) u20 −
− n (u10 + u01 ) u02 +
(−1)m (u01 −u10 )
u00 (u10 +u01 )2
(−1)m (u01 −u10 )
u00 (u10 +u01 )2
(−1)
− u00 (u
−
10 +u01 )
(−1)m
u00 (u10 +u01 )
−1
−1
Table 4.3: Solutions of the ALSC for the potential Hydon–Viallet equation
(−1)m
u01
(−1)m
u10
(−1)−m u00 (u10 2 n−u10 2 −u01 2 m+u01 2 −u10 u01 m+u10 u01 n)(u10 +u01 )
(−u01 +u10 )u01 u10
2
m
2
1 u00 (u10 +u01 )(−u0−1 u01 u10 +u02 u00 u01 +u0−1 u10 +u02 u00 u10 +(u01 u10 u02 u00 u0−1 +u10 u02 u00 u0−1 +u01 −u10 )(−1) )
2
(u01 −u10 )u10 (u02 u00 +(−1)m )(u0−1 u01 +(−1)m )
− 21
m
− (m − 1) (−1) u10 −2 u−10 +
m
Q̄1 = (n − 1) (−1) u01 −2 u0−1 +
Q̄2 =
Q̄3 =
Q̄4 =
(
u01
2
2
u01 −u01 u10
2
u01 u10 (u01 −u10 )
u10 +u00
u01 u10 (u01 −u10 )
+u00
2
u10
) cos
+
(m−n)π
2
(u00 2 u10 2 +u00 2 u01 2 +u10 3 u01 −u01 3 u10 +2 u00 2 u01 u10 )(−1)n (−1)m
(u10 +u01 )(−1)m
(
(u10 +u01 )
u01
2
u00
2
2
u01 u10 (u01 −u10 )
u10 +u01 u00
u01 u10 (u01 −u10 )
−1
−1
−1
+
(u01 − u10 )
(u01 − u10 )
(−1)m
u00
(−u10 2 +2 u00 2 u01 2 u10 2 +u00 2 u10 3 u01 +u00 2 u01 3 u10 +u01 2 )(−1)n
−
2
m
2
1 u00 (u10 +u01 )(−u01 u−10 +u20 u00 u01 +u01 u10 u−10 +u20 u00 u10 +(u01 u00 u20 u−10 +u01 +u01 u10 u00 u20 u−10 −u10 )(−) )
2
(u01 −u10 )u01 (u−10 u10 +(−1)m )(−u00 u20 +(−1)m )
(m−n)π
(m−n)π
(u10 +u01 )(−1)m (u01 2 u10 +u00 2 u01 −u01 u10 2 +u00 2 u10 ) sin
(u10 +u01 )(u01 2 u00 2 u10 +u01 u00 2 u10 2 −u01 +u10 ) cos
2
2
−
u01 u10 (u01 −u10 )
u01 u10 (u01 −u10 )
(m−n)π
u10 2 −u01 +u10 ) sin
2
Q̄5 = −
Q̄6 = −
−1
+
68
Chapter 4: A New Approach to Finding CLaws
(3.20) and (3.21), to obtain densities for the original quad-graph equation. We can now simplify
these expression by adding trivial densities to obtain the densities in Tables 4.4–4.6.
For two of the cosymmetries, Q̄1 of the pLV and pHV equations, it was not possible to reconstruct
the densities using the direct method. Whatever choices we made for the unknown functions
f and Ĝ, we found a system of PDEs that could not be solved. Therefore, we calculated the
˜ is zero
‘characteristic’ for these solutions using (4.2) and sought to determine whether E(Q∆)
or not. For the pHV equations the expressions were to large to determine this condition, but for
˜ 6= 0. Using (4.25) and (4.2) we obtain the
the pLV equation we were able to show that E(Q∆)
‘characteristic’ which is too large an expression to display. Applying the Euler operator to the
˜ leads to an even larger expression which maple struggles to simplify. To
product of Q and ∆
show that this expression does not vanish, we differentiate it w.r.t. different variables until an
expression is obtained which is clearly non-zero. We begin by differentiating w.r.t. ũ21 and then
simplifying the expression. We then divide the numerator by the coefficient of
ln
ũ00 (ũ10 + ũ11 )
ũ10
and differentiate the numerator w.r.t. ũ21 and then simplify. We then divide the numerator of
this new expression by the coefficient of
Li2
ũ00 (ũ10 + ũ11 )
ũ10
,
and differentiate w.r.t. ũ21 and then simplify. We then divide this expression by the coefficient
of ln (ũ21 ) and differentiate w.r.t. ũ01 and simplify. Finally differentiating the numerator of this
expression four times w.r.t. ũ21 yields
3 ũ11 (−24 ũ11 − 24 ũ12 ) 6= 0.
Therefore this cosymmetry is not the seed of a CLaw. Hence there are solutions of the ALSC that
do not correspond to CLaws. Therefore an interesting question is: can an additional constraint
be found that guarantees a solution of the ALSC is a seed of a CLaw without needing to work
through the reconstruction process? This is especially relevant as Mikhailov et al. [41] use a
co-recursion operator to generate an infinite hierarchy of cosymmetries for the Viallet equation,
so a simple test to show that this corresponds to an infinite hierarchy of CLaws is apposite.
There are now two methods for finding CLaws of quad-graph equations: the direct construction
method of Rasin and Hydon [47, 48, 49] and the ALSC method outlined above. The question
now arises: which is the best method for directly constructing CLaws of difference equations?
The major disadvantage of the ALSC method is the need to transform the seeds and the quadgraph equation into Kovalevskaya form in order to reconstruct the densities and then having to
transform back to the quad-graph. Hence it is desirable to form a method to reconstruct CLaws,
from their seeds, for equations on the quad-graph. However, the major advantage of the ALSC
method is that any CLaws found are guaranteed to be distinct from one another. Apart from
these points it is not clear which method is the most efficient. Also, by using operators that
can annihilate more than one term, it may be possible to improve Rasin and Hydon’s method
to enable it to find higher order CLaws. In addition, Rasin and Hydon’s method is specific
to quad-graph equations so more generally the ALSC method needs to be adapted and then
compared with Hydon’s method [26] for equations in Kovalevskaya form.
69
Chapter 4: A New Approach to Finding CLaws
Table 4.4: CLaws of the Hydon–Viallet equation
F1 =
m+1−n
1 (u−10 u00 +u00 +u−10 +1)(−1)
4
u00 +1+u−10 u00 −u−10
G1 =
(u−10 u00 +u00 +u−10 +1)(−1)m−n u00 u10
1
2 (u00 +1+u−10 u00 −u−10 )(u10 +1+u00 u10 −u00 )
(u00 2 u01 u0−1 −u0−1 u00 u01 +u00 u01 +u0−1 u00 +u00 +u0−1 +1−u00 2 u01 )(−1)m+n
F2 = − 14
G2 =
(−u00 +1+u0−1 u00 +u0−1 )(−u01 +1+u00 u01 +u00 )
m+n
1 (u0−1 u00 +1)(−1)
4 −u00 +1+u0−1 u00 +u0−1
(u01 +1)(u01 −1)u00 2 (−1)n
F3 = n ln − (−u
+
m
ln
− 41
2
+1+u u +u ) u
01
G3 = (m + n) ln
F4 = ln
G4 = ln
F5 = ln
G5 = ln
G6 = sin
00
01
(u10 +1+u00 u10 −u00 )2
u00 (u10 +1)(u10 −1)
(u10 +1+u00 u10 −u00 )2
u00 (u10 −1)(u10 +1)
+ n ln 4 (u
2
m+n
1 (−u01 +1+u00 u01 +u00 ) u01 (−1)
4
(u01 −1)(u01 +1)u00 2
F6 = cos
01
m+n
(u01 −1)(u01 +1)
1 (−1)
4
u01
00
(u10 −1)u00 (u10 +1)
u10 2
(m+n)π
2
(m+n)π
2
u10 2 (−1)m+n
2
10 +1+u00 u10 −u00 )
n
ln u01 −1 + sin (m+n)π
ln − (u01u−1)(−1)
2
+1
01
ln
(−1)−n (u10 +1)
u00 (u10 −1)
n
cos
F7 = ln − (u01u−1)(−1)
+1
01
G7 = − cos
(u01 +1)(u01 −1)(−1)n
u01
(m+n)π
2
ln
1
2
π (m + n) + sin (m+n)π
ln (u01 )
2
u00 (u10 −1)(−1)n
u10 +1
70
+ ln u01 −1
(u00 2 −1)(−1)m+n
(m+n)π
2
u01 2 +1
u01
(u01 2 u00 2 −1)(−1)m
(m−n)π
+
+
−2 u01 (−1) +
cos
−
cos
u00
u01 u00
2
2
2
(u00 2 −1)(−1)n
(u01 2 +1)(−1)m
m+n
01
+ −
− 2 (−1)
u01 sin (m+n)π
+
+ u00u01−u
sin (m−n)π
u00
2
u01
u00
2
n
(u−10 u10 +(−1)m )(−1)−n
u10
u01 (−1)n
u01 u−10 +(−1)m
71
G5 =
+
u10 u00
u10 u00
(u10 2 u00 2 +u00 2 u10 +u00 +u10 +u10 2 u00 −1)(−1)m+n
u01 u00
cos
1
2
π (m + n) +
u10 u00
(1−u10 +u10 2 u00 2 +u10 u00 2 )(−1)m+n
u10 u00
u01 u00
(−u01 2 +u01 u00 2 +u01 +u00 u01 2 +u00 −u00 2 )(−1)n
(−u10 2 +u00 2 u10 +u10 +u00 2 −u00 −u10 2 u00 )(−1)n
−
+
(−u01 +u00 2 u01 2 −u00 +1−u00 u01 2 −u01 u00 2 )(−1)m+n
G6 = −
F6 = −
u10
m+n
(−u10 +u00 2 +u10 u00 2 +u10 2 )(−1)n
− 2 (−1)
(−1+u10 2 )(−1)m+n
(−1+u00 2 )(−1)n
2
(1+u01 2 )(−1)m
(m+n)π
u00 −u01 2
u
cos
+
+
cos (m−n)π
+
01
u00
2
u01 u00
u01
2
u 2 u 2 −1 (−1)m
(−1+u00 2 )(−1)m+n
( 00 01 )
n
(m+n)π
(m−n)π
1+u01 2
+ 2 (−1) u01 −
sin
+
−
sin
u00
2
u01 u00
u01
2
−
F5 =
+
u10
(−1+u10 2 )(−1)n
sin
(m+n)π
2
u 2 −1 (−1)m+n
(u10 u00 2 +u00 2 u10 2 −u10 +1)(−1)m+n
(u10 2 −1)(−1)n
( 10 )
(u10 u00 2 +u00 2 +u10 2 −u10 )(−1)n
(m+n)π
(m+n)π
−
+
+
G4 = −
cos
sin
u10 u00
u10
2
u10
u10 u00
2
F4 =
G3 = ln
F3 = ln
It is not known if Q̄1 is the seed of a CLaw or not.
u00 (−1)m
F2 = ln 2 u0−1
m
u01 +(−1)
m
)(−1)−m
G2 = ln 12 (u10 u0−1 +(−1)
u00
Table 4.5: CLaws of potential Hydon–Viallet equation
Chapter 4: A New Approach to Finding CLaws
Chapter 4: A New Approach to Finding CLaws
Table 4.6: CLaws of the potential Lotka–Volterra equation
The solution Q̄1 of the ALSC is not the seed of a characteristic.
)(u0−1 +u−10 )
F2 = − (u01 −u−10
u0−1 u−10
G2 = − uu−10
0−1
01
F3 = m ln uu−10
+ ln u00 −1
−1
−10
(m
+
1)
+
ln
(u
+
u
)
G3 = ln u10u+u
10
−10
10
G4 =
F5 =
G5 =
u01
u−10
(u10 +u−10 )
ln u0−1 uu00
10 −u0−1
F4 = ln
− ln (u0−1 ) − ln
u10 u00
u10 −u0−1
(−1)m+n u00
u01
(−1)m+n (−u10 2 +u00 2 )
u00 u10
u01
F6 = − u00u−2
01
u00 2 +u10 2
G6 = − u00 u10
4.4
Summary
We have demonstrated that the existence of a characteristic allows the direct construction of
CLaws for difference equations using the ALSC. This is because we have shown that characteristics of CLaw are members of the kernel of the ALSC, just as for continuous equations. Using
this method we have constructed new five-point CLaws, in addition to the known three-point
CLaws, for the HV, pHV and pLV equations, though their existence was predicted by the existence of a master symmetry. This raises the question: what is the most efficient method for
directly constructing CLaws of difference equations? We have also demonstrated that there are
objects in the kernel of the ALSC that are not seeds of CLaws, so Mikhailov et al.’s parenthetical
comment that cosymmetries are characteristics of CLaws (see [41]) is not always true, and care
should be taken to distinguish solutions of the ALSC and characteristics.
72
Chapter 5
Numerical Schemes
In this chapter we return to the problem of finding finite difference schemes of the KdV equation
∆ ≡ ut + uux + uxxx = 0 that preserve as many of its CLaws as possible. The brute-force
method of Chapter 2 is refined slightly by searching for CLaws in characteristic form. Instead
of discretizing the densities and fluxes, the characteristics and the PDE are discretized. The
discrete Euler operator is then applied to the product of the discrete characteristics and the
difference scheme, to find conditions that need to be satisfied so that the difference scheme has
CLaws with the desired characteristics. This eliminates the need to be able to solve the scheme
uniquely for a given point and the need to pull back the CLaw to some set of initial conditions.
We first show how the discrete gradient method for preserving first integrals exploits the characteristic form. We then outline the method for PDEs. The method is still very crude so by
making some observations we refine the method slightly. Finally we apply the method to KdV
to find some new and known finite difference schemes.
5.1
Discrete Gradients
In §1.2.2.1 the method of discrete gradients was outlined for preserving first integrals of ODEs
[38, 40]. In order to preserve the first integral of an ODE, the ODE is rewritten as a skew
gradient system
dx
= f (x) = S∇I,
dt
where S T = −S. We now note that ∇I is the characteristic of the first integral,
d
dx
dx
I=
· ∇I = ∇I ·
− S∇I + ∇I · (S∇I) = 0,
dt
dt
dt
as the last term vanishes by the skew-symmetry of S. The system is then discretized using a
¯ which is defined by the axioms
skew symmetric matrix S̃, and a discrete gradient, ∇,
¯ · (xn+1 − xn ),
I(xn+1 ) − I(xn ) =(∇I)
¯
∇I(x
n , xn+1 ) =∇I(xn ) + O(xn+1 − xn ).
This yields the numerical scheme
xn+1 − xn
¯
− S̃ ∇I(x
n , xn+1 ) = 0,
ν
73
Chapter 5: Numerical Schemes
where ν is a fixed time step. As the discrete gradient has been constructed to be a total
divergence, the method satisfies
¯ · (xn+1 − xn )) = E((Sn − I)I(xn )) = 0,
E(∇I
¯ T · S ∇I)
¯ = E(0) = 0,
E((∇I)
(5.1)
(5.2)
where E is the discrete Euler operator. Therefore the discrete gradient method gives a discretization for the characteristic and f that satisfies
xn+1 − xn
− f˜
,
(5.3)
0 = E Q̃
ν
˜
¯
where Q̃ is a discretization of the characteristic given by ∇I(x
n , xn+1 ) and f is a discretization
¯
for f given by S̃ ∇I(xn , xn+1 ). Hence if we were to search for discretizations of Q and f that
satisfy (5.3) then discrete gradient methods would be found. In fact more general methods may
be found because (5.2) is satisfied trivially by the skew-symmetric construction. Also, instead of
a one-step discretization of the derivative used in (5.3), higher order multi-step methods could
be found, though this would depart from the geometric integration philosophy (see §1.2.2.1) that
one-step methods should be used because these are maps on the phase space. Using discrete
gradients enables multiple first integrals to be preserved by writing the ODE as a skew gradient
system [40]. The main difficulty is to construct discrete gradients.
5.2
Method for PDEs
We now outline our method for finding discretizations of scalar PDEs with independent variables
x and t that preserve as many CLaws as possible. The basic method is similar to that used in
Chapter 2; however we now aim to preserve the CLaws in characteristic form. As in Chapter 2
we use tildes to denote discretizations of continuous terms.
1. Choose points for each term in the PDE to depend on. Then discretize each term in the
PDE, using Taylor series approximations centred at (xm , tn ) as in §2.2.1. For example, if
we desire that all the terms in KdV depend on the same points, then the terms in KdV
are discretized using (2.15),
ug
xxx =
with (2.16)
0=
B X
D
X
B
D
1 XX
αij uij ,
µ3
i=A j=C
αij , 0 =
i=A j=C
B X
D
X
iαij , 0 =
i=A j=C
0=
B X
D
X
uet =
B
D
1XX
βij uij .
ν
i=A j=C
B X
D
X
i2 αij , 3! =
i=A j=C
βij ,
1=
i=A j=C
B X
D
X
B X
D
X
i=A j=C
jβij .
i=A j=C
and (2.28)


D
B B
B X
D
B
X
X
1 X X X
uu
gx =
γijkj uij ukj +
γijkl uij ukl  ,
µ
j=C
i=A k=i
i=A l=j+1 k=A
74
i3 αij ,
Chapter 5: Numerical Schemes
with (2.29)
0=
X
γijkl ,
1=
X
(i + k)γijkl ,
P
where
is shorthand for the summation used for uu
gx . At this stage the discretization
may not be consistent. As there is not a unique way of imposing consistency, it seems best
to search for methods that preserve CLaws and then, if any methods are found, impose
consistency.
2. Choose the points on which each term of the characteristic of the desired CLaw will depend. Theorem 5.2.1 (see below) states that the discretization of the linear terms in the
characteristic and the PDE should be centred in the same place.
3. Form the most general discretization of the characteristic with the chosen points. For
example, KdV’s third CLaw’s characteristic is Q3 = u2 + 2uxx . This could be discretized
using


D
B X
B
B
D
B X
D
B
X
X
X
X
1
1 XX
f2 =

u
ijkj uij ukj +
ζij uij ,
ijkl uij ukl  , u
g
=
xx
µ
µ3
j=C
i=A k=i
i=A j=C
i=A l=j+1 k=A
(5.4)
with
1=
X
ijkl ,
0=
B X
E
X
ζij , 0 =
i=A j=C
B X
E
X
iζij ,
2! =
i=A j=C
B X
E
X
i2 ζij .
(5.5)
i=A j=C
In practice, one may not wish to seek the most general discretization of a term in the
characteristic or the PDE. The complexity of the problem can be reduced with an assumption about the discretization e.g. that the discretization is symmetric in space or time (see
Theorem 5.2.2).
4. Apply the discrete Euler operator, which for a scalar PDE with two independent variables
such as KdV is
X
−i −j ∂
,
E :=
Sm
Sn
∂uij
i,j
fi , and the discretized PDE, ∆.
e The
to the product of the discretized characteristic, Q
difference scheme has a conservation law with the given characteristic if
fi ∆).
e
0 = E(Q
(5.6)
Therefore split (5.6) according to the coefficients of the uij , ν and µ terms. This process
will result in an overdetermined system of quadratic equations in the coefficients of the
discretizations. For example, (5.6) may contain the term
(α−A−C βAC + αAC β−A−C )
u(2A)(2B)
,
µ3 ν
which would yield the condition
α−A−C βAC + αAC β−A−C = 0.
75
Chapter 5: Numerical Schemes
5. Repeat steps 3 to 4 for any additional CLaws that one wishes to preserve, to create a
system of constraints on the coefficients.
6. Calculate the Groebner basis of the large system to see if there are any solutions. See
§2.2.2 for information on Groebner bases and the Buchberger algorithm.
7. Solve the system. This may lead to several disjoint families of discretizations.
8. Use the direct construction method to construct the densities (this is effectively by inspection as the characteristic is known). Note that these densities may not be direct analogues
of the continuous densities as there may be terms that vanish in the limit. Also, it may be
necessary to add trivial densities to ensure that the discrete flux and densities tend to the
continuous flux and density as the step sizes tend to zero.
9. Check that the discretization of the PDE is consistent. If it is not consistent then impose
additional constraints from §2.2.1.
In principle this method should, for a given set of points, find any finite difference schemes
that have analogues of the desired CLaws. However, in practice the method is limited by the
amount of memory the computer has, as calculating the Groebner basis can require a large
amount of memory. Thus, rather than searching for the most general discretizations, we impose
assumptions on them to try and reduce the memory required to calculate the Groebner basis.
Doing this we may miss possible solutions; however, by choosing sensible ansätze the problem
can be considerably simplified.
Theorem 5.2.1. The discretization of the linear terms in the characteristic and the PDE must
share the same central point.
Proof. We first prove the theorem for O∆Es. Let
P =
A
X
j=−A
αj uj such that α−A , αA 6= 0, and Q =
C
X
βi ui .
i=B
Thus the expression P has an odd number of terms and is centred at the point j = 0. We prove
the case B < −A < 0 < C < A. The other cases follow by symmetry or similar reasoning.
Q
P
−A
B
0
m
A
C
Figure 5.1: The linear terms in the characteristic and O∆E
We require that
A
X
∗
0 =E(P Q) = DP∗ (Q) + DQ
(P ) =
=
A
X
k=−A
=
A
X
k=−A
αk

C
X
i=B
−C−1
X

j=B
βi ui−k
!
αk S −k Q +
k=−A
+
αk βj uj−k +
C
X
k=B

 βk
C
X
j=−C
C
X
βk S −k P
(5.7)
k=B
A
X
j=−A

αj uj−m  =
A
C
X
X
k=−A i=B
(αk βj + α−k β−j )uj−k +
76
−B
X
j=C+1
αk βj (uj−k + uk−j )

α−k β−j uj−k  .
(5.8)
Chapter 5: Numerical Schemes
The extreme shifts of P and Q occurring in (5.7) are depicted graphically in Figure 5.2.
B−A
A
α−A Sm
Q
−C
βC Sm
P
−A
αA Sm
Q
−A
B
0
C
−B
βB Sm
P
A−B
A
m
Figure 5.2: The Euler operator acting on a product of linear terms. The points in the extreme
shifts of P and Q in (5.7) are enclosed.
From this it clear that uA−B is the most extreme point in the positive direction in (5.8) (rightmost
−B
point in Figure 5.2). This point only occurs in the βB Sm
P term; hence its coefficient is βB αA .
Similarly uB−A is the most extreme point occurring in the negative direction and it only occurs in
−A
the αA Sm
Q term; thus its coefficient is also βB αA . Both of these terms must vanish, therefore
βB = 0. Provided that the most extreme point occurring in both directions occurs only in one
term, the above reasoning can now be repeated to show that βB+1 = 0. This process is continued
until the most extreme point occurring in both directions no longer occurs in just one term. This
A
will occur when the most extreme point in α−A Sm
Q coincides with the most extreme point in
−B
βB Sm P , i.e. when uC+A = uA−B . Thus C = −B, so Q must have the same centre point as P .
In addition, only the middle term from (5.8) remains. This equation is satisfied if αk = α−k and
β−j = −βj . Therefore if one of the linear terms is symmetric about the centre point and the
other is antisymmetric then their product is in the kernel of the Euler operator.
If P has an even number of terms then the above reasoning will still apply by considering
i + 21 , j + 12 , A + 12 , B + 21 , C + 12 ∈ Z (see Figure 5.3).
−A −
1
2
− 12 0
1
2
A+
1
2
Figure 5.3: A term with an even number of points
Having proved the theorem for P and Q containing terms on a line, we now do the case for P
and Q being a sum of terms on the plane. The proof will work in the same manner; the fact
that P is not centred at the same point as Q means that when applying the Euler operator the
extreme shifts of the extreme points will be isolated, so in order for E(P Q) to vanish Q cannot
depend on the extreme points. As in the O∆E case we will only show the proof for schemes
centred on (0, 0) and claim that the other cases (where in at least one direction there is an even
number of points) are similar.
PM
PN
Let P = i=−M j=−N αij uij so that it is a sum of points centred at (0, 0) (see Figure 5.4). In
addition let α−M N̂ and α−M Ñ be the extreme points on the leftmost edge with Ñ ≤ N̂ , let αM N̄
and αM Ṅ be the extreme points on the rightmost edge with Ṅ ≤ N̄ , let αM̂ N and αM̃ N be the
extreme points on the top edge with M̃ ≤ M̂ , and let αM̄ −N and αṀ −N be the extreme points
on the bottom edge with Ṁ ≤ M̄ . The extreme points mark the edge of the P term and so are
PB PD
non-zero. Similarly, let Q = k=A C βkl ukl where A < 0 < B < |A| and C < 0 < D < |C| so
that it is a sum of variables that does not have a centre on (0, 0). Other cases, where Q does not
77
Chapter 5: Numerical Schemes
P
Q
M̂ N
M̃ N
D
−M N̂
M Ṅ
00
00
−D
M N̄
−M Ñ
−B
A
Ṁ − N
B
C
M̄ − N
Figure 5.4: Linear terms with two independent variables. The blue dashed line marks the
boundary of the points included in each linear term.
have a centre at (0, 0) will follow by symmetry and similar reasoning. Now applying the Euler
operator to the product of P and Q gives
∗
0 =E(P Q) = DP∗ (Q) + DQ
(P ) =
M
X
N
X
−i −j
αij Sm
Sn Q +
i=−M j=−N
=
M
X
N
X
B
X
D
X
i=−M j=−N k=−B l=−D
+
M
X
N
X
B
X
B X
D
X
−k −l
βkl Sm
Sn P
k=A l=C
(αij βkl + α−i−j β−k−l )u(k−i)(l−j) +
−D−1
X
αij βkl (u(k−i)(l−j) + u(i−k)(j−l) )+
i=−M j=−N k=−B l=C
+
M
X
N
X
−B−1
D
X X
αij βkl (u(k−i)(l−j) + u(i−k)(j−l) )
(5.9)
i=−M j=−N k=A l=C
=
Ñ X
D
X
αM j βAl u(A−M )(l−j) + σ1 (uij ),
(5.10)
j=Ṅ l=C
where σ1 (uij ) denotes the sum of all the other uij terms with i > A − M . Equation (5.9) can
always be put in the form (5.10) provided A < −B. From this, we can see that the coefficient
of u(A−M )(C−N̄ ) is βAC αM N̄ and the coefficient of u(A−M )(D−Ṅ ) is βAD αM Ṅ . By assumption
αM Ṅ , αM N̄ 6= 0, therefore βAC = βAD = 0. Now (5.10) still holds with D replaced with D − 1
and C replaced with C + 1 thus βA(C+1) = βA(D−1) = 0. Continuing this process shows that
βAj = 0 for j ∈ Z. If A + 1 < −B then (5.9) can still be put in the form (5.10) only with
A replaced by A + 1, and from this it follows that β(A+1)j = 0 for j ∈ Z. Thus βij = 0 for
A ≤ i ≤ B − 1 and j = C ≤ j ≤ D. Therefore (5.9) simplifies to
0=
M
X
N
X
B
D
X
X
i=−M j=−N k=−B l=−D
+
M
X
N
X
B
X
(αij βkl + α−i−j β−k−l )u(k−i)(l−j) +
−D−1
X
αij βkl (u(k−i)(l−j) + u(i−k)(j−l) ),
(5.11)
i=−M j=−N k=−B l=C
=
M̂
B
X
X
αiN βkC u(k−i)(C−N ) + σ2 (uij ),
(5.12)
i=M̃ k=−B
where σ2 (uij ) denotes the sum of all the other uij terms with j > C − N . Equation (5.11)
can be put into the form (5.12) provided C < −D. From this, the coefficient of u(−B−M̂ )C is
β−BC αM̂ N and the coefficient of u(B−M̃ )(C−N ) is βBC αM̃ N . By assumption αM̂ N , αM̃ N 6= 0, so
78
Chapter 5: Numerical Schemes
βBC = β−BC = 0. The summation in (5.12) is now over k = −B + 1, . . . B − 1, so in the same
way β(B−1)C = β(−B+1)C = 0. Continuing this process shows that βij = 0 for −B ≤ i ≤ B and
PB
PD
C ≤ j ≤ −D − 1. Thus Q = k=−B −D βkl ukl and so it is centred on (0, 0).
We now observe that (5.9) simplifies to
0=
M
X
N
X
B
D
X
X
i=−M j=−N k=−B l=−D
(αij βkl + α−i−j β−k−l )u(k−i)(l−j) .
Thus if αij = α−i−j (P has 180 degree rotational symmetry about the centre) and β−k−l = −βk,l
(Q has 180 degree rotational antisymmetry about the centre) then E = (P Q) = 0 is satisfied.
This observation leads to the following theorem.
Theorem 5.2.2. The discrete Euler operator applied to the product of a linear sum of terms
with 180◦ rotational symmetry and a linear sum of terms with 180◦ antisymmetry is zero.
It must be noted that some products of linear terms that are not in this form are also in the
kernel of the Euler operator. For example, when there is only one independent variable, if
P = β−1 u−1 + (β−1 + β1 )u0 + β1 u1 ,
Q=
α(β−1 − β1 )
−αβ−1
u−1 +
u0 + αu1 ,
β1
β1
then E(P Q) = 0. Theorem 5.2.2 is analogous to the fact, from the continuous theory, that
E(u,m1 x,n1 t u,m2 x,n2 t ) =(−1)m1 +n1 Dxm1 Dtn1 u,m2 x,n2 t + (−1)m2 +n2 Dxm2 Dtn2 u,m1 x,n1 t
=0,
for (m1 + n1 ) odd and (m2 + n2 ) even, where u,mi x,ni t ≡ Dxmi Dtni u.
For KdV the first three CLaws in characteristic form satisfy the following conditions. From the
first CLaw:
(a) E(ut ) = 0,
(b) E(uux ),
(c) E(uxxx ) = 0.
(5.13)
From the second CLaw:
(a) E(u(ut )) = 0,
(b) E(u(uux )) = 0,
(c) E(u(uxxx )) = 0.
(5.14)
From the third CLaw:
(a) E(u2 ut ) = 0,
(b) E(2uxx ut ) = 0,
(d) E(u2 (uux )) = 0,
(c) E(2uxx uxxx ) = 0,
(e) E(u2 uxxx + 2uxx (uux )) = 0.
(5.15)
Thus if the discretizations for the linear terms are antisymmetric for odd derivatives and symmetric for even derivatives, equations (5.13a, 5.13c, 5.14a, 5.14c, 5.15b and 5.15c) are immediately
satisfied by Theorem 5.2.2.
In addition, we might wish to make assumptions about the nonlinear terms in the discretization. By assuming rotational symmetry and antisymmetry about the centre of the discretization some simplification is obtained but no equations from (5.14) and (5.15) are automatically satisfied. A k th order polynomial term is discretized by a sum of terms of the form
79
Chapter 5: Numerical Schemes
αi1 j1 i2 j2 ...ik jk ui1 j1 ui2 j2 . . . uik jk . The term u−i1 −j1 u−i2 −j2 . . . u−ik −jk is 180◦ opposite the previously mentioned term and we label its coefficient as α−ik −jk ...−i1 −j1 . The discretization is
symmetric if
αi1 j1 i2 j2 ...ik jk = α−ik −jk ...−i1 −j1
and antisymmetric if
αi1 j1 i2 j2 ...ik jk = −α−ik −jk ...−i1 −j1 .
This is illustrated in Figure 5.5 for a quadratic term discretized on a line. (Note that, just as
for linear terms, suitable adjustments must be made for schemes not centred at (0, 0) i.e. those
schemes with an even number of points in one direction).
α−22
α−12
α−21
−2
α−2−2
α−2−1
−1
α−10
0
1
α01
α−1−1
α00
α11
α−20
α−11
α02
α12
2
α22
Figure 5.5: A graphical representation of the discretization of a quadratic term of an ODE. Each
line represents the coefficient of the product of the two points the line is connecting. Coefficients
that are symmetric to each other are depicted by matching lines. Coefficients that are in red
will be zero in the antisymmetric case.
Before proceeding to study a general polynomial term let us consider a simple example. To avoid
subscripts, we use a hat to denote the coefficient of a term that is 180◦ symmetric to a term
already used in an expression, so for our example
P = αu−1 u0 + α̂u1 u0 + γu1 u3 + γ̂u−1 u−3 , and Q = βu1 + β̂u−1 ,
(5.16)
and
P · Q =(αβu−1 u0 u1 + α̂β̂u1 u0 u−1 ) + (αβ̂u2−1 u0 + α̂βu21 u0 ))+
+ (γβu1 u3 u1 + γ̂ β̂u−1 u−3 u−1 ) + (γ̂βu−1 u−3 u1 + γ β̂u1 u3 u−1 ).
(5.17)
The product of the two terms can be split into pairs of terms which are symmetric to one
another. We require that E(P · Q) = 0; expanding this out yields the following conditions on
the coefficients:
u−1 u1 :
(αβ + α̂β̂) = 0,
u−2 u2 :
(γ̂β + γ β̂) = 0,
u1 u2 :
(αβ + α̂β̂) = 0,
u2−1
u−1 u−2 :
(αβ + α̂β̂) = 0,
:
α̂β = 0,
u0 u−1 :
2α̂β = 0,
u21
:
αβ̂ = 0,
u0 u1 :
2αβ̂ = 0,
u2−2 :
γβ = 0,
u22 :
γ̂ β̂ = 0,
u0 u2 :
2γβ = 0,
u−2 u0 :
2γ̂ β̂ = 0,
u2 u4 :
γ̂β = 0,
u−2 u−4 :
γ β̂ = 0.
80
(5.18)
Chapter 5: Numerical Schemes
Adding and subtracting the coefficient of symmetric terms gives an equivalent system of equations:
(αβ + α̂β̂) = 0,
(γ̂β + γ β̂) = 0,
(αβ + α̂β̂) = 0,
αβ̂ + α̂β = 0,
αβ̂ − α̂β = 0,
γβ + γ̂ β̂ = 0,
γβ − γ̂ β̂ = 0,
γ̂β + γ β̂ = 0,
γ̂β − γ β̂ = 0.
(5.19)
Now if we assume that P is symmetric i.e. α̂ = α and γ̂ = γ and Q is antisymmetric i.e. β̂ = −β
or vice-versa (P is antisymmetric and Q is symmetric) then the left hand column of (5.19) is
satisfied so the number of equations that need to be solved has been more than halved. The
reason more than half of the equations are satisfied is because the coefficient of self-symmetric
terms e.g. u−1 u0 u1 in P · Q must vanish and the coefficient of self-symmetric terms, such as
u2 u−2 , in E(P · Q) must also vanish. If instead we assume that both P and Q are symmetric or
both are antisymmetric then the right hand column of (5.19) is satisfied. This is still a major
simplification but more than half the equations remain because the self-symmetric terms remain.
For the general case we suppose that P has coefficients denoted by α and Q has coefficients
denoted by β and, for convenience,
α = αi1 j1 i2 j2 ...ik jk ,
α̂ = α−ik −jk ...−i1 ij1 ,
β = βik+1 jk+1 ...iK jK ,
β̂ = β−iK −jK ...−ik+1 −jk+1 .
As seen in the example (5.17) the product of P and Q can be considered as consisting of pairs
of terms that are symmetric to one another. Applying the Euler operator to each term in one
of these pairs gives


!!
!
K
k Y
K
k
X
Y
Y


u(il −iq )(jl −jq )  ,
uil jl
=αβ 
uil jl
β
E
α
E
α̂
k
Y
l=1
u−il −jl
Thus the coefficient of
QK
l=1
l6=q
q=1
l=k+1
l=1
!
β̂
K
Y
l=k+1
u−il −jl
!!
l=1
l6=q


k Y
K
X


=α̂β̂ 
u(−il +iq )(−jl +jq )  .
q=1
l=1
l6=q
u(il −iq )(jl −jq ) must be a sum of multiples of the coefficients of the
form αβ which must vanish if the product is in the kernel of the Euler operator. Similarly the
QK
coefficient of its symmetric term, l=1 u(−il +iq )(−jl +jq ) , must be a sum of multiples of coefficients
l6=q
of the form α̂β̂ which must also vanish. An equivalent system of equations (which must also
vanish) is obtained by adding and subtracting these two coefficients together. This gives two sets
of equations one set is formed by a sum of terms of the form (αβ + α̂β̂) and the other set by sums
of terms of the form (αβ − α̂β̂). Thus if α = α̂ and β = −β̂, i.e. one of the polynomial terms
is a symmetric discretization and the other is an antisymmetric discretization, the first set of
equations is satisfied. Alternatively if α = α̂ and β = β̂, or α = −α̂ and β = −β̂, the second set
of equations is satisfied, i.e. both discretizations are symmetric or antisymmetric respectively.
Thus by assuming antisymmetry or symmetry on the appropriate polynomial terms the number
of equations that need to be satisfied is approximately halved. As in the example, terms that
81
Chapter 5: Numerical Schemes
are self-symmetric in P · Q and E(P · Q) will vanish for the symmetric antisymmetric case but
not the other cases. Therefore making symmetry assumptions on the polynomial terms in the
characteristic and PDE seems a sensible ansatz to make.
For discretizing KdV, this ansatz is imposed by insisting that uu
gx has 180◦ antisymmetry and
f2 has 180◦ symmetry. This ansatz combined with the previous ansatz on the linear terms
that u
is equivalent to imposing that the discretization preserves the discrete symmetry of KdV
t 7→ −t̂,
x 7→ −x̂,
KdV → −ut̂ − uux̂ − ux̂x̂x̂ = 0.
By adding additional restrictions (more severe ansätze) the number of variables is reduced, so the
computer may be able to find a solution to the Groebner basis. Some possible extra restrictions
are:
• If a term in (5.13, 5.14, 5.15) is in the kernel of the Euler operator because it a total
difference in either x or t then we can use the Euler operator just in that direction. Instead
of the usual discrete Euler operator we can use discrete Euler operators that treats the us at
different time or space levels as independent variables (see (1.18)), e.g. for two independent
variables we have
Em+i :=
X
j
Sn−j
∂
,
∂uij
and En+j :=
X
i
−i
Sm
∂
.
∂uij
For example, the condition (5.15a) is a consequence of the fact that u2 ut = Dt ( 13 u3 ) so
f2 uet ) = 0 we can require that Em+i (u
f2 u˜t ) = 0 for i ∈ Z. This
rather than insisting that E(u
f2 u˜t = (Sn − I)g for some function g([u]), rather than insisting
amounts to insisting that u
f2 u˜t = (Sn − I)g + (Sm − I)f for some functions
on the less restrictive requirement that u
f ([u]) and g([u]).
• A more restrictive assumption than 180◦ symmetry or antisymmetry is imposing symmetry in one direction (e.g. symmetry in the x direction if discretizing a t derivative) and
antisymmetry or symmetry in the other direction depending on whether an odd or even
derivative is being discretized.
• We could require that a given term is the result of discretizing along a line and then
imposing that the actual discretization is a weighted average of this discretization along a
line at the different space or time levels. This will lead to cubic equations in the coefficients
unless specific weights are imposed beforehand.
f2 factorize. This will again
• We could impose that the nonlinear terms such as uu
gx and u
lead to cubic equations in the coefficients rather than quadratic equations.
5.3
Numerical Schemes for KdV
Having outlined the method for finding finite difference schemes that preserve CLaws, we now
apply the method to find schemes that preserve some of the first three CLaws of KdV. We recall,
from §1.1, that the first three CLaws of KdV, ∆ ≡ ut + uux + uxxx = 0, in characteristic form,
82
Chapter 5: Numerical Schemes
are
0 =Dt (u) + Dx 12 u2 + uxx = ∆,
0 =Dt 21 u2 + Dx 13 u3 + uuxx − 12 u2x = u∆,
0 =Dt 31 u3 − u2x + Dx 14 u4 + u2 uxx + 2ux ut + u2xx = u2 + 2uxx ∆.
We want to look for schemes that are as compact as possible to eliminate the occurrence of
spurious waves [11] so we will focus on schemes with four and five spatial points and one step
in time; however, when searching for some schemes it will be necessary to have more than one
step in time. Below we present the schemes we have found, and the assumptions that we made
in order to find them.
5.3.1
Schemes that preserve the first and second CLaws
5.3.1.1
Eight-point one-step schemes
We begin by looking for schemes with only four spatial points, which should give the most
compact discretization. Three new eight-point schemes were found by discretizing the densities
of the first CLaw and then applying difference operators to obtain a discretization for KdV. The
discretization of uxx in the first CLaw’s flux was assumed to be a weighted average over two rows
f2 ,
of the discretization on just a single row. The discretization for the second characteristic, Q
was symmetric in time. In what follows we have included the local truncation error (LTE) about
the point (xm , tn ). However for the eight-point schemes the resulting discretization of KdV and
the characteristic should be considered as an approximation to u(xm − 1/2µ, tn + 1/2ν), the
densities to u(xm − 1/2µ, tn ) and the fluxes to u(xm − µ, tn + 1/2ν).
First scheme
f1 + 1 (Sm − I)F
f1 ,
e = 1 (Sn − I)G
∆
ν
µ
f2 = 1 (Sn + I) 1 u−10 + 1 u00 = u + O(ν) + O(µ),
Q
2
2
2
f1 = (u−20 − u−10 − u00 + u10 ) ψ + 1 u−10 + 1 u00 = u + O(µ),
G
2
2
2
1
1
1
1
1
f
F1 = 2 u00 + u01 u00 + 2 u−20 u00 + 2 u−21 u00 + 24
− 12 u−10 u00 + 24
− 21 u−11 u00 +
1
1
+ 12 u01 2 + 12 u−20 u01 + 12 u−21 u01 + 24
− 12 u−10 u01 + 24
− 12 u−11 u01 + 12 u−20 2 +
1
1
1
+ u−20 u−21 + 24
− 12 u−20 u−10 + 24
− 21 u−11 u−20 + 12 u−21 2 + 24
− 21 u−21 u−10 +
1
1
1
1
+ 24
− 12 u−11 u−21 + 24
− 12 u−10 2 + 12
− u−11 u−10 + 24
− 12 u−11 2 +
+
1
2µ2 (Sn
+ I) (u−20 − 2u−10 + u00 ) = 12 u2 + uxx + O(ν) + O(µ).
(5.20)
This discretization contains two parameters: ψ in the uet term and in the uu
gx term. From
Figure 5.6 it is clear that setting = 0 (removing the turquoise and green lines in Figure 5.6)
1
will give the most compact discretization and setting = 12
(removing the blue and navy lines in
1 2
f1 . Thus = 0 will give the most
Figure 5.6) will give the least compact discretization for 2 u in F
compact discretizations of uux in KdV. This suggests that = 0 may be the best discretization
and we will investigate this further in Chapter 6.
To find the densities of the second CLaw, given the characteristic, the direct construction method
f2 ∆
e can be split according to
is used. To simplify the construction, note that the expression Q
the coefficients µ and ν. As neither the density or the flux of the second CLaw contain any t
83
Chapter 5: Numerical Schemes
n
1
1
2( 24
− 12 )
m
1
( 24
− 12 )
0
−2
−1
0
1
2
1
f1 for the first scheme. A line joining two points
Figure 5.6: The discretization of 12 u2 in F
indicates that the product of the variable at the two endpoints is included in the discretization.
The colour of the line designates the coefficient of the product in the discretization.
derivatives, we desire that any ν terms must be the result of applying the difference operator in
f2 (u−20 , u−10 , u00 , u10 ) that satisfies
the n direction. Thus we search for a function G
f2 = coeff(∆,
e 1 ),
(Sn − I)G
ν
e 1 ) denotes the coefficient of 1 in the expression ∆.
e In order to satisfy this equation
where coeff(∆,
ν
ν
it is necessary to set ψ = 0. If ψ is a different value then ν terms will be introduced into the
f2 , and µ terms into the discrete density G
f2 , which will give terms that vanish
discrete flux, F
in the limit. It seems reasonable to want to avoid this situation; however, we will numerically
examine some schemes with ψ 6= 0 in Chapter 6. To find the discrete flux we search for a
f2 (u−20 , u−10 , u00 , u−21 , u−11 , u01 ), that satisfies
function, F
(Sm − I) f
e − (Sn − I) G
f2 .
F2 = ∆
µ
ν
The densities found by this process are
f2 = 1 (u00 + u−10 )2 = 1 u2 + O(µ),
G
8
2
1
f
F2 = 96 (u−11 + u−20 + u−21 + u−10 ) (u−10 + u00 + u−11 + u01 )
(12u00 + 12u−20 + u−10 − 24u−11 + 12u01 + 12u−21 − 24u−10 + u−11 ) +
+
1
8µ2
((u−10 + u00 + u−11 + u01 ) u−20 + (u−10 + u00 + u−11 + u01 ) u−21 +
−3u−10 2 − u−10 u00 + 6u−10 u−11 − u−10 u01 + u−11 u00 − 3u−11 2 + u−11 u01
= 13 u3 + uuxx − 12 u2x + O(ν) + O(µ).
The discretization for KdV is now a one-parameter family given by
e = 1 (Sn − I)
∆
ν
1
2 u−10
+ µ1 (Sm − I)
2
+ 21 u00 + µ1 (Sm − I) 2µ1 2 (Sn + I) (u−20 − 2u−10 + u00 ) +
2u01 u00 − u−11 u01 − u−11 u00 + u01 2 − 2u−10 u−11 + 2u−20 u−21 +
− u−10 2 − u−11 2 − u−10 u01 + u−21 u00 − u−10 u00 − u−21 u−11 + u−21 u01 − u−21 u−10 +
+ u−20 u01 + u−20 u00 + u00 2 − u−20 u−11 + u−21 2 − u−20 u−10 + u2−20 − u−20 u−11 +
1
(u−10 + u−11 ) (u−10 + u01 + u−11 + u00 + u−20 + u−21 )
(5.21)
+ 24
=ut + uux + uxxx + O(ν) + O(µ).
84
Chapter 5: Numerical Schemes
Subsequent analysis may help determine the value of that should be chosen.
Second scheme
e = 1 (Sn − I)G
f1 + 1 (Sm − I)F
f1 ,
∆
ν
µ
(5.22)
f = 1 (Sn + I)(u−20 + u−10 + u00 + u10 ) = u + O(ν) + O(µ),
Q2
8
f1 = (u−20 − u−10 − u00 + u10 ) ψ + 1 u−10 + 1 u00 = u + O(µ),
G
2
2
1
1
1
1
1
u
u
+
u
u
+
24 −11 00
24 −10 01
24 u−10 u00 + 24 u−21 u01 + 24 u−21 u00 +
1
1
1
1
1
1
24 u−21 u−11 + 24 u−21 u−10 + 24 u−20 u01 + 24 u−20 u00 + 24 u−20 u−11 + 24 u−20 u−10 +
1
1 2
2µ2 (Sn + I) (u−20 − 2u−10 + u00 ) = 2 u + uxx + O(µ) + O(ν).
f1 = 1 u−11 u01 +
F
24
+
+
f2 , it is
To construct the density and flux of the second CLaw so that no ν terms occur in F
1
necessary to set ψ = 4 . This gives
f2 = 1 (u−20 + u−10 + u00 + u10 )2 = 1 u2 + O(µ),
G
32
2
f2 = 1 (u−20 +u−21 +u−10 +u−11 ) (u01 +u−11 +u00 +u−10 ) (u00 +u−20 +u−21 +u01 )+
F
192
+
−2u−20 u−10 + 4u−20 u00 − 2u−20 u−11 + 4u−20 u01 + u−20 2 + 2u00 u01 + u01 2 +
1
16µ2
+ 2u−21 u−20 − 2u−21 u−10 + 4u−21 u00 − 2u−21 u−11 + 4u−21 u01 − 2u−11 2 + u00 2 +
+ u−21 2 − 2u−10 u01 − 2u−10 u00 − 4u−10 u−11 − 2u−10 2 − 2u−11 u01 − 2u−11 u00
= 13 u3 + uuxx − 12 u2x + O(µ) + O(ν).
The discretization for KdV is thus
e = 1 (Sn −I) (u−20 + u−10 + u00 + u10)+ 1 (Sm −I)
∆
4ν
µ
−
1
24µ
1
2µ2 (Sn
+ I) (u−20 − 2u−10 + u00 ) +
(u−20 − u11 − u10 + u−21 ) (u01 + u−11 + u00 + u−10 )
(5.23)
=ut + uux + uxxx + O(µ) + O(ν).
Third scheme
e = 1 (Sn − I)G
f1 + 1 (Sm − I)F
f1 ,
∆
ν
µ
(5.24)
f = 1 (Sn + I)( 1 u−20 + u−10 + u00 + 1 u10 ) = u + O(ν) + O(µ),
Q2
6
2
2
f1 = (u−20 − u−10 − u00 + u10 ) ψ1 + 1 u−10 + 1 u00 = u + O(µ),
G
2
2
f1 = 1 (u−10 + u−21 + u−20 + u−11 ) (u00 + u−11 + u−10 + u01 )+
F
32
+
1
2µ2 (Sn +I) (u−20
− 2u−10 + u00 ) = 12 u2 + uxx + O(ν) + O(µ),
f2 does not depend on ν it is necessary to set ψ = 1 . This gives
To ensure that F
6
f2 = 1 (u−20 + 2u−10 + 2u00 + u10 )2 = 1 u2 + O(µ),
G
72
2
f2 = 1 (u−11 + u−10 + u−21 + u−20 ) (u00 + u01 + u−10 + u−11 )
F
384
(u00 + u−21 + 2u−10 + u−20 + 2u−11 + u01 ) +
+
1
µ2
2
2
2
5
1
1
5
1
1
5
24 u−20 u00 − 24 u−20 u−10 − 24 u−20 u−11 + 24 u−20 u01 + 24 u00 + 24 u01 − 24 u−11 +
2
1
1
1
5
1
1
1
24 u−20 + 12 u−21 u−20 − 24 u−21 u−10 + 24 u−21 u00 − 24 u−21 u−11 − 24 u−11 u01 + 12 u00 u01+
5
1
1
5
5
1
1
+ 24
u−21 u01 + 24
u−21 2 − 24
u−10 u00 − 24
u−10 2 − 12
u−10 u−11 − 24
u−10 u01 − 24
u−11 u00
= 13 u3 + uuxx − 21 u2x + O(ν) + O(µ),
+
85
Chapter 5: Numerical Schemes
The discretization for KdV is now
e = 1 (Sn −I) (u−20 +2u−10 +2u00 +u10)+ 1 (Sm −I)
∆
6ν
µ
+
1
32µ
1
2µ2 (Sn
+ I) (u−20 − 2u−10 + u00 ) +
(u00 +u−11 +u−10 +u01 )(u00 −u−20 −u−21 +u01 +u11 +u10 −u−11 −u−10 ) ,
=ut + uux + uxxx + O(µ) + O(ν).
(5.25)
(5.26)
It is noteworthy that all three schemes, with the chosen values of ψ, preserve the density of
2
f
f2 = 1 G
the second CLaw in the same manner: G
2 1 . Therefore, we can take the results of
f1 , which will be an
each of the discretizations above and then form the average, ūm−1/2,n = G
alternative discretization of KdV that preserves the first and second conserved quantities exactly.
It is tempting to think that this will give unconditional stability in the `2 -norm with periodic
boundary conditions. However, the original schemes, from which the average is calculated, may
not themselves be unconditionally stable.
Additional schemes
It seems that the assumptions to find the above three schemes are very restrictive. Why insist
that the CLaws do not have additional terms in the density and flux that vanish as the step sizes
tends to zero? A more sensible restriction is to search for methods that have 180◦ symmetry
assumptions. Hence (5.13a, 5.13c, 5.14a, 5.14c) are automatically satisfied. Therefore we are free
to choose any discretizations for ut and uxxx that have 180◦ antisymmetry. So for our numerical
investigations in Chapter 6 we will choose
1
uet = ν1 (Sn − I) 12 u−10 + 12 u00 , ug
xxx = 2µ3 (Sm − I) ((Sn + I) (u−20 − 2u−10 + u00 )) ,
to make the discretizations as compact and symmetric as possible. This will ensure that the
resulting schemes are all consistent. The only discretizations that need to be determined are
those for Q2 and uux ; the results are shown in Table 5.1. To help visualize the schemes, stencils
are shown in Figures 5.7 and 5.8, in which different colours are used to represent terms with
different coefficients. To demonstrate how to reconstruct the uu
gx term from the stencil, let us
study the first scheme in Figure 5.7. There is a solid red line joining the points (1, 0) and (0, 1)
1
and so ( 12
− 1 )u10 u01 is a term in the discretization; a red dashed line joins (−1, 0) and (−2, 1)
1
hence −( 12 − 1 )u−10 u−21 is another term. Similarly, there is a purple solid loop attached to
(1, 0) and a purple dashed loop attached to (−2, 1); therefore 0 (u210 − u2−21 ) forms part of the
discretization. Thus, from the stencil, we can ascertain that
1
µuu
gx = ( 12
− 1 )(u10 u01 − u−10 u−21 ) + 0 (u210 − u2−21 ) + . . . ,
which we can compare with the first scheme in Table 5.1.
Using our chosen assumptions, we have found four families of schemes that preserve the first and
second CLaws of KdV in characteristic form. There are ten schemes shown in the table; however,
six of the schemes are special cases of the seventh scheme, which has three parameters, (κ, 1 , 2 ).
The parameter choices that yield the other schemes are as follows: the second scheme is given
by (1/4, 1 , 2 ), the third scheme by (1/2, 1 , 2 ), the fourth scheme results from (1/6, 1 , 2 ), the
, 2 ), the sixth scheme is obtained with (κ, 1 , 0) and
fifth scheme is obtained by (κ, −2 (2κ−1)
κ
finally the eighth scheme is obtained by (1/2 − 63 , 0, 0). The reason for including these schemes
in the table is to demonstrate the effects of some different parameter choices, and it should also
be clearer which parameter values to pick for numerical investigation.
The most striking observation about the schemes is that they seem to be skewed in two different
directions, and the characteristic and the uu
gx term are skewed in the same direction. We say
86
Chapter 5: Numerical Schemes
that a discretization of a linear term is skewed/angled upwards (downwards) if pairs of points
lying on a diagonal with positive (negative) gradient have a greater weighting than the pair of
points lying on the opposite diagonal. So in Table 5.1 the discretizations for Q2 for the first and
fourth schemes are angled downwards, the third scheme is angled upwards, the second and tenth
schemes are symmetric (no skew). The other schemes all involve a choice of parameters which
will affect how they are skewed (see Figures 5.7 and 5.8). The nonlinear terms are now seen to
be lying roughly along a diagonal lying in the same direction as the characteristic is skewed. It
should be noted that for the second scheme, even though the characteristic is symmetric, the
nonlinear part requires 1 = 22 to be symmetric.
We now observe that the schemes occur in pairs which are skewed in opposite directions; this
is most obvious with the first and third schemes and the fifth and sixth schemes (see Figure
5.7). The partner of a scheme can be obtained by a reflection in the line n = 1/2 (n̂ = −n + 1,
m̂ = m and ν 7→ −ν). This transformation is in GL(2, Z) and so, by Lemma 3.2.1, a divergence
e and
expression is mapped to a divergence expression, hence the product of the transformed Q
e must be mapped to a divergence expression and is thus in the kernel of the Euler operator.
∆
Thus the only remaining question is whether the reflected numerical scheme and characteristic
are discretizations of uux and Q2 respectively. To be a discretization of Q2 , the coefficients of
the uij s must add up to one; this will be unaffected by the transformation, so the transformed
characteristic is still an appropriate discretization. Now, for a one-step scheme,


1 X
B X
B
B X
B
X
X
1
uu
gx = 
γijkj uij ukj +
γi0k1 ui0 uk1  ,
µ j=0
i=A k=i
i=A k=A
with the necessary conditions
0=
1 X
B X
B
X
γijkj +
j=0 i=A k=i
0=
B X
B
X
γi0k1 ,
(5.27)
i=A k=A
1 X
B X
B
B X
B
X
X
(i + k)γijkj +
(i + k)γi0k1 .ui0 uk1 ,
j=0 i=A k=i
(5.28)
i=A k=A
The transformed equation is


1 X
B X
B
B X
B
X
X
1
γijkj ûi(−j+1) ûk(−j+1) +
γi0k1 ûi1 ûk0  ,
uu
dx = 
µ j=0
i=A k=i
i=A k=A


1 X
B X
B
B X
B
X
X
1
= 
γi(1−ĵ)k(1−ĵ) ûiĵ ûkĵ +
γi0k1 ûi1 ûk0  ,
µ
ĵ=0 i=A k=i
i=A k=A
so the necessary conditions for uu
dx → uux as µ, ν → 0 are
0=
1 X
B X
B
X
ĵ=0 i=A k=i
γi(1−ĵ)k(1−ĵ) +
B X
B
X
γi0k1 , 0 =
i=A k=A
1 X
B X
B
X
ĵ=0 i=A k=i
(i + k)γi(1−ĵ)k(1−ĵ) +
B X
B
X
γi0k1 .
i=A k=A
By relabeling the summation we retrieve the original conditions (5.27) and (5.28). So we have
mapped the discretization to discretizations which also preserve the second CLaw. So if a onestep scheme that preserves the second CLaw of KdV in characteristic form is reflected in the
line n = 1/2, another scheme that preserves the second CLaw is obtained.
We are now faced with a problem: which parameter values are best for each family of schemes?
So what choices of parameters should be numerically investigated? One of the things we can
87
Chapter 5: Numerical Schemes
investigate is how the schemes are skewed. An interesting question is whether being skewed in
one direction, or being symmetric, is preferable. It is also interesting to make the schemes as
compact as possible, to see if this makes the schemes more stable. Finally, where possible, we
make the discretization for ux in the linearized scheme (see §6.1 and Appendix A) as compact
as possible and symmetric in time, so that
u
fx = 21 (u01 + u00 − u−10 − u−11 ).
(5.29)
and then compare the schemes with this linearization (see Table A.1 for linearizations of the
relevant schemes).
Below are some parameter values for the different schemes that are noteworthy:
1
• First Scheme: (0 , 1 ) = (− 12
, 0) causes the linearization to be (5.29), (0, 0) makes the
scheme as compact as possible (this kills the purple and orange terms in Figure 5.7),
1
) gives the least compact scheme (kills the purple, red and blue terms).
(0, 12
• Third Scheme: this is the first scheme’s ‘partner’, i.e. it is angled in the opposite direction
and can be obtained by reflecting the first scheme in the line n = 1/2. For the linearization
1
we need (1 , 2 ) = (0, − 12
); compactness requires (0, 0) (kills the purple and orange terms)
1
, 0), (kills the blue, red and purple terms).
and the least compact case is given by ( 12
• Second Scheme: this scheme’s characteristic is symmetric in time and space. To make the
uu
gx term symmetric, set 1 = 22 (this is the first scheme (5.20) in the previous section with
1
1
= 1 ). To make the linearization as compact as possible choose (1 , 2 ) = (− 24
, − 48
); for
the actual scheme to be as compact as possible choose (0, 0) (this is the symmetric case
1
1
, 24
); this
of the eighth scheme in the table). The uu
gx discretization is angled up with ( 24
kills orange, blue, red and dark red terms. Its partner, given by the reflection, is given by
1
( 24
, 0) which kills the turquoise, red, dark red and light green terms.
1
1
, − 108
);
• Fourth scheme: to make the linearization compact and symmetric set (1 , 2 ) = (− 27
1
1
for compactness choose (0, 0); to make the scheme as symmetric as possible set ( 108 , 108
).
1
• Ninth Scheme: for symmetry κ = 12
(this is the third scheme (5.24) in the previous
1
section); it is skewed up if κ = 6 , and skewed down if κ = 0.
• Tenth Scheme: there are no parameters to choose (this is the second scheme (5.22) previously found).
88
89
Q21 = 12 u00 + 12 u−11
1
1
µuu
gx = 0 − 12
+ 1 u−10 2 − u01 2 + u−10 u−11 − u00 u01 + u−11 u−21 − u10 u00 + 12
− 1 (u01 u10 − u−21 u−10 ) +
2
2
+ u10 u−11 − u−10 u00 + u−11 u01 − u−21 + u10 − u00 u−21 0 + (u11 u01 − u−20 u−21 − u−21 u01 + u−10 u10 + u10 u11 − u−10 u−20 ) 1
Q22 = 14 u−10 + 14 u00 + 14 u−11 + 14 u01
1
1
µuu
gx = 24
− 2 (u10 u00 − u−21 u−10 + u01 u10 − u−11 u−21 ) + 24
− 1 2u00 u01 − 2u−10 u−11 + u01 2 − u−11 2 + u00 2 − u−10 2 +
2
2
1
+ (2 − 1 ) u−20 u00 + u00 u−21 + u−21 − u10 u−11 − u−11 u11 − u10 + (22 − 1 ) (u−10 u00 − u−11 u01 ) + 24 + 2 − 1 (u11 u00 + u11 u01 − u−10 u−20 − u−11 u−20 ) +
+ (u10 u11 − u−20 u−21 ) 1 + u11 u−10 − u01 u−20 + u11 2 − u−21 u01 + u−10 u10 − u−20 2 2
1
1
Q23 = 2 u−10 + 2 u01
1
µuu
gx = 12
− 2 − 1 u00 u01 + u00 2 − u−11 2 − u−10 u−11 + u11 u01 − u−10 u−20 +
1
+ 12
− 1 (u11 u00 − u−11 u−20 ) + (u10 u00 − u−11 u−21 − u−20 u00 + u10 u11 − u−20 u−21 + u−11 u11 ) 1 + −u−11 u01 + u−10 u00 + u11 2 − u01 u−20 + u11 u−10 − u−20 2 2
1
1
1
1
Q24 = 6 u−10 + 3 u00 + 3 u−11 + 6 u01
3
2
2
2
2
3
1
1
1
1
− 1 (u10 u00 − u−11 u−21 ) + (52 − 21 ) (u−10 u00 − u−11 u01 ) +
µuu
gx = 22 − 21 + 18
+ 18
2 u00 u01 − 2 u−10 u−11 + 2 u00 − 2 u−11 + u01 − u−10
2
2
1
1
+ (22 − 1 ) (u−20 u00 − u−11 u11 ) + (42 − 21 ) u00 u−21 − u10 u−11 + u−21 − u10 + 22 − 1 + 36
(u11 u00 − u−11 u−20 ) + 32 − 1 + 36
(u11 u01 − u−10 u−20 ) +
2
2
1
+ 18 − 22 (u01 u10 − u−21 u−10 ) + (u10 u11 − u−20 u−21 ) 1 + −u01 u−20 − 2u−21 u01 − u−20 + 2u−10 u10 + u11 u−10 + u11 2
Q25 = u−10 κ + u00 12 − κ + u−11 12 − κ + u01 κ
κ−2κ2 −62 +12κ2 u01 2 +u01 u10 +u10 u00 −u−11 u−21 −u−21 u−10 −u−10 2
(2κ−1)(u−20 u−21 −u10 u11 )
1
µuu
gx = 21 2
+
+ 61 κ − 2 u00 2 − u−11 2 + 61 u11 u00 − 61 u−11 u−20 κ+
κ
12
κ
2 −3
12κ
−κ
u
u
−u
u
(
)
(6
−κ)
u
u
−u
u
(2κ−1)
u
u
−u
u
( −10 −11 00 01 )
( −21 01 −10 10 )
2
2
11 01
−10 −20
2
1
+ 12
+ 12 2
+ 61
+ −u−11 u01 + u−10 u00 + u11 2 − u01 u−20 + u11 u−10 − u−20 2 2
κ
κ
κ
1
1
Q26 = u−10κ + u00 2 − κ + u
u01 κ
−11 2 − κ +
−2κ2 +κ−61 +12κ1 u01 2 −u−10 2
1
+ 16 κ − 1 u00 2 − u
u + u11 u00 − u−11
u−20 − u−11 2 + u11 u01 +
µuu
gx = 12
κ
−10 −20
2
2
2
1 (2κ−1) u00 u−21 +u−10 u00 −u−11 u01 −u10 u−11 +u−21 −u10
κ−61 +24κ1 −2κ (u10 u00 −u−11 u−21 )
1
1 (−κ+61 )(u−10 u−11 −u00 u01 )
+ 21
+ 12
+ 12
+
κ
κ
κ
1
1
+ 12 − 6 κ (u01 u10 − u−21 u−10 ) + (−u−20 u00 − u−20 u−21 + u−11 u11 + u10 u11 ) 1
Q27 = u−10 κ + u00 (1/2 − κ) + u−11 (1/2 − κ) + u01 κ
−4κ2 +2 −2κ1 +42 κ2 +4κ2 1 u10 2 −u−21 2
−18κ2 +32 −6κ1 +242 κ2 +12κ2 1 +κ2 −2κ3 u01 2 −u−10 2
(2κ−1)(u01 u−21 −u10 u−10 )
+ 2 u11 2 − u−20 2 −
+
+
µuu
gx = 2
2
2
2κ
4κ
12κ
12κ2 −κ2 −32 +6κ1 u00 2 −u−11 2
κ−2κ2 −62 +12κ2 (u01 u10 −u−21 u−10 )
12κ2 −κ2 −32 +6κ1 (u−11 u−10 −u01 u00 )
(2κ2 −2 +2κ1 )(u11 u−11 −u−20 u00 )
−
+
+
+
+
6κ
2κ
12κ
12κ
2
−κ2 +18κ2 −62 +6κ1 (u−10 u−20 −u11 u01 )
6κ2 −κ2 −32 +6κ1 (u−11 u−20 −u11 u00 )
+2 (u11 u−10 − u01 u−20 ) + (u11 u10 − u−21 u−20 ) 1 +
+
+
6κ
6κ
82 κ2 −4κ2 +2 −2κ1 +4κ2 1 (u00 u−10 −u01 u−11 )
−2κ3 −24κ2 +32 −6κ1 +362 κ2 +24κ2 1 +κ2 (u10 u00 −u−11 u−21 )
−4κ2 +2 −2κ1 +42 κ2 +4κ2 1 (u−21 u00 −u−11 u10 )
+
+
+
2
2
2
4κ
12κ
4κ
Q28 = u−10 12 − 61 + 6u00 1 + 6u−11 1 + u01 21 − 61
1
1
1
µuu
gx = 12
− 1 u11 u01 + u11 u00 + u00 2 − u−11 u−20 − u−10 u−20 − u−11 2 + u01 2 + u01 u10 + u10 u00 − u−11 u−21 − u−21 u−10 − u−10 2 1 + 12
u00 u01 − 12
u−10 u−11
1
1
1
1
1
1
Q29 = u−20 κ + 6 u−10 + 6 u00 + u10 −κ + 6 + u−21 −κ + 6 + 6 u−11 + 6 u01 + u11 κ
µuu
gx = − 81 (−6κu−10 + 6κu−11 − u00 + 6κu00 − 6κu01 − u−11 )
(6κu00 − 6u−20 κ − 6κu10 + 6κu−21 + 6u11 κ − 6κu01 + 6κu−10 − 6κu−11 − u−10 + u10 − u−21 + u01 )
Q210 = 18 u−20 + 81 u−10 + 81 u00 + 18 u10 + 18 u−21 + 81 u−11 + 18 u01 + 81 u11
1
µuu
gx = − 24
(u−20 − u10 − u11 + u−21 ) (u−10 + u−11 + u01 + u00 )
Chapter 5: Numerical Schemes
Table 5.1: Eight-point schemes with 180◦ antisymmetry that preserve the first and second CLaws of KdV in characteristic form
Chapter 5: Numerical Schemes
First Scheme
uu
gx
u
e
1
2
1
12
0
1
12
− 1 − 0
1
− 1
Second Scheme
1
4
1
24
1
24
− 2
− 1
1 − 2
1
24
+ 2 − 1
22 − 1
1
12
1
12
− 2 − 1
1
1
2( 24
− 1 )
1 2
Third Scheme
1
2
− 1
2
Fourth Scheme
1
6
1
3
22 − 21 +
52 − 21
1
18
1
2 (22
− 21 +
22 − 1
22 − 1 +
1
36
−κ
2 (1−2κ)
2κ
1
6 κ − 2
κ−2κ2 −62 +12κ2
12κ
62 −κ
1
2
6 κ 12κ
κ2 +32 −12κ2
6κ
−κ
12κ1 −61 +κ−2κ2
12κ
1 (1−2κ)
2κ
61 −κ
1
12κ
1
18
− 22
1
32 − 1 +
1
3
18 ) 2 (22
2(22 − 1 )
1
36
1
18
2 22
− 21 +
1
18 )
− 1
Fifth Scheme
κ
1
2
Sixth Scheme
κ
1
2
1
6 κ − 1
κ−61 +24κ1 −2κ2
12κ
1
1
12 − 6 κ
Figure 5.7: Stencils for the eight-point discretizations of Q2 and uux . Lines indicate that the
multiple of the two end points is included in the scheme. Different colour lines represent different
coefficients and a dashed line represents the negative of its solid counterpart.
90
Chapter 5: Numerical Schemes
Seventh Scheme
1
2
κ
uu
gx
u
e
−κ
2 (2κ−1)
4κ2 −2 +2κ1 −42 κ2 −41 κ2
2κ
4κ2
18κ2 −32 +6κ1 −242 κ2 −12κ2 1 −κ2 +2κ3
2
12κ
4κ2 −82 κ2 −2 +2κ1 −4κ2 1
4κ2
3
2κ +24κ2 −32 +6κ1 −362 κ2 −24κ2 −κ2
12κ2
1 2
κ2 +32 −6κ1 −6κ2
2 −2κ1 −2κ2
6κ
2κ
κ2 +32 −6κ1 −12κ2
κ2 +32 −6κ1 −12κ2
12κ2
6κ
κ−2κ2 −62 +12κ2
κ2 −18κ2 +62 −6κ1
12κ
6κ
Eighth scheme
1
2
1
− 61
1
1
12
− 1
1
12
Factors of uu
gx
Ninth Scheme
κ
1
6
−κ
1
6
− 12 (6κ − 1)
− 12 κ
1
4 (1
3
2κ
− 6κ)
Tenth Scheme
1
8
1
6
1
4
Figure 5.8: More stencils for the eight-point discretizations of Q2 and uux . Lines indicate that
the multiple of the two end points is included in the scheme. Different colour lines represent
different coefficients and a dashed line represents the negative of its solid counterpart. A cross
represents the negative value of the coefficient (depicted by the colour) for a linear term.
91
Chapter 5: Numerical Schemes
5.3.1.2
Ten-point one-step schemes
We now search for schemes that have five points in space instead of four. The wider stencil may
cause problems in the numerics, but schemes with good preservation properties may be found by
using a wider stencil, such as the norm-preserving scheme in [11]. All the eight-point schemes we
have found used all the points to discretize uux , so by using an odd number of points, we may
be able to find schemes that have a more compact discretization for the uux term than the eight
point schemes. However, the ug
xxx term will have a wide stencil. The following two schemes were
found by making the following assumptions: uet must satisfy the consistency condition (2.24);
f
ug
gx depend only on the central
xxx must satisfy the consistency condition (2.18), and Q2 and uu
six points.
First scheme
e = 1 (u01 − u00 ) −
∆
ν
1
8µ
(−u11 − u10 + u−11 + u−10 ) (u01 + u00 ) +
+ µ13 −ηu−20 − u−10 + 12 + 2η + 2 u00 + (−1 − ) u10 + 21 − η u20 +
+ − 12 + η u−21 + ( + 1) u−11 + − 21 − 2 − 2η u01 + u11 + ηu21 ,
(5.30)
=ut + uux + uxxx + ( −
1
2
2
+ 4η)uxxt µν + O( νµ ) + O(ν) + O(µ),
f2 = 1 u−10 + 1 u00 + 1 u10 + 1 u−11 + 1 u01 + 1 u11 = u + O(ν) + O(µ2 ).
Q
6
6
6
6
6
6
Unlike the eight-point schemes, this discretization was not found by discretizing the density and
flux of the first CLaw. Therefore they need to be constructed using the direct construction
method. The result of this is
f1 =u00 ,
G
f1 = 1 (u−10 + u−11 ) (u01 + u00 ) +
F
8
− 12 − η − u00 + (η + ) u01 + ηu−20 +
+ (η + ) u−10 + − 12 − η − u−11 + 12 − η u10 + ηu11
= 12 u2 + uxx + ( −
1
2
1
µ2
1
2
− η u−21 +
2
+ 4η)uxt µν + O( νµ ) + O(ν) + O(µ),
and so the mass is conserved exactly. When constructing the densities of the second CLaw, it is
f2 does not depend on ν. The
not possible in this case to choose parameter values so that the F
resulting density and flux for the second CLaw are
f2 = 1 u10 u00 + 1 u00 2 + ν 3 2u00 2 + (−1 − 2η − 2) u10 u00 + (1 + 4η + 4) u20 u00 +
G
3
6
12µ
+ (−1 + 4η) u20 u−10 + (1 − 4η − 2) u20 2 + (−2η − 2) u20 u10
= 12 u2 + ( 13 − η − 23 )uux µν2 + O( µν ) + O(µ),
f2 = 1 (u−10 + u−11 ) (u01 + u00 ) (u00 + u01 + u−10 + u−11 ) +
F
48
+
1
12µ2
2ηu11 u−10 + (−2 − 2) u−11 2 + (−2η − 1 − 4) u11 u−11 + (1 − 4η − 2) u10 2 +
+ (−1 + 4η + 2) u11 2 − 2u−11 u−10 + (1 − 2η) u01 u−21 + (6η + 6 + 1) u−10 u01 +
+ (2η + 2) u01 u−11 + (1 − 2η) u01 u10 − 2u00 u01 + (−1 − 4η − 4) u00 2 +
+ 2ηu−20 u00 + 2ηu11 u00 + (1 − 2η) u11 u−21 + 2u−10 2 + (1 − 2η) u00 u−21 +
+ (−1 − 2η − 2) u−10 u00 + (−6η − 2 − 6) u00 u−11 + (1 − 4η − 2) u10 u00 +
+ (1 − 2η) u−21 u−10 + (1 − 2η) u−21 u−11 + (1 − 2η) u−21 u10 + (2 + 4 + 2η) u−10 u10 +
+ (1 − 2η) u10 u−11 + (4 − 1 + 4η) u01 2 + 2ηu−20 u01 + (4η + 2) u11 u01 + 2ηu−20 u−10 +
+ 2ηu−20 u−11 + 2ηu−20 u10 + 2ηu11 u−20 ) +
= 31 u3
+ uuxx −
1 2
2 ux
+ (η −
1
3
+
2
ν
3 )uut µ2
+
µ
6ν
(u00 − u01 ) (u−10 + u−11 )
2
O( µν 2 )
92
+ O( µν ) + O(µ).
Chapter 5: Numerical Schemes
In order to minimize the LTE of the discretization of KdV and F1 , set − 21 + 4η = 0. To
minimize the LTE of the density and flux of the second CLaw, set η − 13 + 23 = 0. Solving these
equations simultaneously yields η = 0 and = 12 and so the ug
xxx term becomes symmetric in
time. Therefore, this choice of parameters will be investigated in Chapter 6.
Second scheme
1
0 = ν1 (u01 −u00 )− 24µ
(−u11 −u10 +u−11 +u−10)(u−10 +u−11 +u01 +u00 +u11 +u10) +
1
+ µ2 −ηu−20 + (1 − ξ) u−10 + − 32 + 2η + 2ξ u00 − ξu10 + 12 − η u20 +
(5.31)
+ − 12 + η u−21 + ξu−11 + 32 − 2ξ − 2η u01 + (−1 + ξ) u11 + ηu21
=ut + uux + uxxx + ( −
1
2
2
+ 4η)uxxt µν + O( νµ ) + O(ν) + O(µ2 ),
f2 = 1 (u00 + u01 ) = u + O(ν).
Q
2
The density and flux of the first CLaw are
f1 =u00 ,
G
f1 = 1 u−10 2+2u−10 u−11 +u−10 u01 +u−10 u00 +u−11 2+u−11 u01 +u−11 u00 +u00 2+2u01 u00 +u01 2
F
24
+
1
2µ2
(2ηu−20 + (2η − 2 + 2ξ) u−10 + (1 − 2η − 2ξ) u00 + (1 − 2η) u10 ) +
+ (1 − 2η) u−21 + (1 − 2η − 2ξ) u−11 + (2η − 2 + 2ξ) u01 + 2ηu11 )
= 21 u2
+ uxx + (4η −
3
2
2
+ )uxt µν + O( νµ ) + O(ν) + O(µ).
The density and flux of the second CLaw are
f2 = 1 u00 2 + ν3 − 1 + η u20 u00 + ξ − 1 u20 u10 + −η − ξ + 3 u20 2 ,
G
2
µ
4
2
4
2
=u2 + (1 − 2η − )uux µν2 + O( µν 2 ) + O( µν ),
f2 = 1 (u−10 + u−11 ) (u00 + u01 ) (u00 + u01 + u−10 + u−11 ) +
F
48
+ µ12 − 12 + 12 ξ u−10 u01 + − 21 + 12 ξ u−10 u00 − 12 u−11 u01 ξ − 12 u−11 u00 ξ +
+ 12 ηu01 u−20 + −η−ξ + 34 u10 2 + − 43 +η+ξ u11 2 + − 34 +η+ξ u01 2 + 21 ηu00 u−20 +
+ 12 ηu10 u−10 + −η − ξ + 34 u00 2 + 12 ηu11 u−10 + − 21 η+ 14 u00 u−21 + − 12 η+ 14 u11 u−11 +
+ − 12 η+ 14 u01 u−21 + ξ − 12 u10 u00 + 12 −ξ u11 u01 + − 12 η+ 41 u10 u−11
2
= 13 u3 − 12 u2x + uuxx + (2η − 1 + )uut µν2 + O( µν 2 ) + O(µ).
If the parameters are chosen to be η = 41 and = 12 then the ug
xxx term becomes symmetric in
time and antisymmetric in space and the local truncation errors are all reduced. In addition,
the densities become
1
G1 = u00 , G2 = u200 ,
2
so that both the mass and the momentum are conserved exactly. The exact conservation of the
momentum implies non-linear stability in the `2 -norm. In fact this choice of parameters gives
the norm-preserving scheme in [11] and we can now see it preserves the second CLaw locally and
not just globally.
Additional schemes with rotational symmetry
As for the case with eight points, it seems that our assumptions to find the above two schemes
are very restrictive. So we relax them to only require that the schemes have 180◦ rotational
symmetry assumptions. With this we are free to choose
uet = ν1 (Sn − I)u00 ,
ug
xxx =
1
4µ3 (Sn
+ I)(Sm − I)(u−20 − u−10 − u00 + u10 ),
93
Chapter 5: Numerical Schemes
to make the discretizations as compact and symmetric as possible. Due to computational considerations, we remove the points u−21 and u20 from the discretization of uux so that it only
uses eight points which are sheared to the right1 . From the numerics (see Chapter 6), we observe
that, for the single soliton problem, the schemes which are skewed upwards appear to produce
solitons that travel faster than symmetric schemes (which generally travel slower than the actual
soliton), so it seems sensible to focus on schemes sheared to the right rather than to the left.
However, schemes that are sheared to the left can be obtained from the schemes presented here
by reflecting the characteristic and uu
gx terms in the line n = 1/2, as we discussed in §5.3.1.1.
The schemes found with these assumptions are shown in Table 5.2 and the stencils of the schemes
are shown in Figures 5.9 and 5.10. It appears that there are ten different schemes; however, six of
these schemes are consequences of specific parameter choices for the seventh scheme in Table 5.2,
which has three parameters (κ, 3 , 4 ). The second scheme in the table is given by (1/4, 3 , 4 );
the third scheme is the choice (1/2, 3 , 4 ); the fourth scheme is obtained by (1/6, 3 , 4 ); the
2
4 +κ
3
fifth scheme results from (κ, 34 −12κ
, 4 ); the sixth scheme is the choice ( 12 3+
, 3 , 0); and
6κ
2
finally the eighth scheme is obtained by (63 , 3 , 0). We have chosen to keep these schemes in
Table 5.2 so that the effects of the different parameter choices can be seen.
If one compares Figures 5.9 and 5.10 with Figures 5.7 and 5.8, it should be immediately apparent
that all of these schemes are generated by shearing the eight-point schemes and changing the
parameters. It is interesting that no schemes have been found that aren’t generated by shearing
an eight-point scheme. From Lemma 3.2.1, if the shear m̂ = m + n, n̂ = n is applied to the
product of the discrete Q2 and uu
gx , the result will still be in the kernel of the Euler Operator.
The transformed discretization of uux for a one-step scheme is


B X
B
1
B B
X
1 X X X
uu
dx =
γi0k1 ûi0 û(k+1)1  ,
γijkj û(i+j)j û(k+j)j +
µ j=0
i=A k=A
i=A k=i


B+j B+j
1
B
B+1
X
X
1 X X X
=
γ(î−j)j(k̂−j)j ûîj ûk̂j +
γi0(k̂−1)1 ûi0 ûk̂0  ,
µ j=0
i=A k̂=A+1
î=A+j k̂=î
so the first necessary condition for uu
dx → uux as µ, ν → 0 is
0=
B+j
1
X
X B+j
X
γ(î−j)j(k̂−j)j +
j=0 î=A+j k̂=î
B
B+1
X
X
γi0(k̂−1)1 ,
i=A k̂=A+1
which by relabeling the indicies yields (5.27) and so must be satisfied. The second necessary
condition is
0=
B+j
1
X
X B+j
X
(î + k̂)γ(î−j)j(k̂−j)j +
j=0 î=A+j k̂=î
=
1 X
B X
B
X
B X
B
X
(i + k + 2j)γijkj +
i=A k=i
2γi1k1 +
(i + k̂)γi0(k̂−1)1 ,
i=A k̂=A+1
j=0 i=A k=i
=
B
B+1
X
X
B X
B
X
(i + k + 1)γi0k1 ,
i=A k=A
B X
B
X
γi0k1 ,
(using (5.28)).
(5.32)
i=A k=A
1 If
these points are left in, calculating the Groebner basis with the total degree ordering was quick to accomplish. However, when transforming the resulting basis to one with lexicographic ordering, the computer was left
running for a week without completing the task.
94
Chapter 5: Numerical Schemes
Hence, in general, applying a shear to a one-step discretization will not yield an alternative
discretization for uux . However, if we assume that uu
gx has 180◦ antisymmetry then (5.32) will
be satisfied. In this case, the second term in (5.32) must vanish as it contains all the coefficients
of the products of terms of the form ui0 uk1 and so if it contains γi0k1 it must also contain
γ−i1−k0 . The remaining term is
A X
A
X
2γi1k1 =
i=−A k=i
A X
A
X
i=A k=i
γi1k1 −
A X
A
X
i=−A k=i
γ−i0−k0 =
1 X
A X
A
X
γijkj = 0.
j=0 i=−A k=i
Thus, because we have imposed that the uux terms have 180◦ antisymmetry, there is a oneto-one correspondence between the eight-point schemes we found in Table 5.1 and the schemes
found in Table 5.2.
As for the eight-point schemes, we will want to investigate parameter choices that make the
schemes as compact and symmetric as possible, and compare the cases where the linearization
yields a discretization for ux that is as compact and symmetric as possible (the linearizations
for the schemes are shown in Table A.2), so that
u
fx = 14 (−u−11 − u−10 + u10 + u11 ).
Therefore the following parameter choices will be of interest:
To make the first scheme symmetric in time and give the desired linearization, set (2 , 3 ) =
1
1
, 0) (this is the second scheme (5.31) in the previous section). The choice ( 12
, 0), for the first
( 24
scheme kills the the orange and purple terms (see Figure 5.9), so the scheme is skewed upwards;
the resulting linearization for this is
u
fx = − 16 u−11 − 13 u−10 − 16 u01 + 16 u00 + 13 u11 + 16 u10 .
To obtain the opposite scheme, which is skewed downwards, killing the blue and orange terms,
we need (0, 0) and its linearization is
u
fx = − 13 u−11 − 16 u−10 + 16 u01 − 16 u00 + 16 u11 + 13 u10 .
1
) to include the orange terms and also kill the O(µν) term and the
Finally, we choose ( 18 , − 24
2
O(µ ) term in the LTE. This gives a linearization of
u
fx =
1
12 u−20
− 16 u−11 − 21 u−10 − 14 u01 + 14 u00 + 12 u11 + 16 u10 −
1
12 u21 .
We wish to investigate the second, third and fourth schemes as well. As it is not clear which
parameter values to choose, we shall pick the parameter values that give the schemes the same
1
linearizations as we have found above. So for the second scheme we need (3 , 4 ) = (0, − 48
)
1
1
1
1
for the symmetric case, (− 48 , − 48 ) for the downwards case, ( 48 , − 48 ) for the upwards case and
1
1
1
1
, − 24
) for the elongated case. For the third scheme we require (3 , 4 ) = ( 24
, − 12
) for the
( 48
1
1
1
1
symmetric case, (0, − 12 ) for the downwards case, ( 12 , − 12 ) for the upwards case and ( 8 , − 18 ) for
1
1
the elongated case. And for the fourth scheme (3 , 4 ) = (− 216
, − 108
) gives the symmetric case,
1
1
1
1
1
5
, − 216
) the elongated
(− 54 , − 108 ) the downwards case, ( 108 , − 108 ) the upwards case and (− 216
case. The ninth and tenth schemes both have characteristics depending on more points, so they
too should be investigated. Fortunately, the tenth scheme doesn’t have any parameters that
need choosing. So finally, choosing κ = 0 causes the ninth scheme to become symmetric (this is
the first scheme (5.30) in the previous section); the choice κ = 16 kills the central terms.
95
96
Q21 = 12 u00 + 12 u01
1
uu
gx = − 12
+ 2 + 3 u−11 u00 − u01 u10 + u00 u−10 − u10 2 + u−11 2 − u01 u11 +
2
2
1
+ 12 − 3 (u11 u10 − u−11 u−10 ) + u00 u10 + u11 − u−11 u01 − u−10 + u00 u11 − u−10 u01 2 + (u11 u21 + u10 u−10 + u10 u21 − u−20 u−10 − u−20 u−11 − u11 u−11 ) 3
1
1
1
1
Q22 = 4 u−10 + 4 u00 + 4 u01 + 4 u11
1
1
uu
gx = 4 − 24
(u−11 u−10 − u00 u10 − u11 u10 + u−11 u01 ) + 24
− 3 u10 2 − u00 u−20 − u−11 u00 − u−11 2 + u01 u10 + u01 u21 +
2
2
2
2
1
+ 24 − 4 − 3 (u01 u11 − u00 u−10 ) + (4 − 3 ) u−10 + u01 − u00 − u11 − 2u00 u11 + 2u−10 u01 +
1
+ (u11 u21 + u21 u00 − u−20 u−10 − u01 u−20 ) 3 + −u−20 u11 + u−10 u21 + u21 2 + u10 u−10 − u−20 2 − u11 u−11 4 + 24
+ 4 − 3 (u21 u10 − u−11 u−20 )
1
1
Q23 = 2 u−10 + 2 u11
1
uu
gx = 12
− 4 − 3 (u10 u21 − u−11 u01 − u00 u−20 + u00 u10 − u−20 u−11 + u01 u21 ) + (4 + 3 ) (u21 u00 − u01 u−20 ) +
+ u00 u11 + u00 2 − u01 2 − u−10 u01 − u−20 u−10 + u11 u21 3 + −u−20 u11 + u−10 u21 − u−20 2 − u01 u11 + u21 2 + u00 u−10 4
1
1
1
1
Q24 = 6 u−10 + 3 u00 + 3 u01 + 6 u11
1
1
− 34 + 3 (u00 u10 − u01 u−11 ) + 36
+ 4 − 3 u01 u21 − u00 u−20 − 2u−11 u00 + 2u10 2 + 2u01 u10 − 2u−11 2 +
uu
gx = 36
1
1
1
+ 36 −3 +34 (u10 u21 − u−20 u−11 )+ 18 −23 +4 (u01 u11 − u00 u−10 )+ 18 −24 (u11 u10 − u−11 u−10 )+ −u−20 u11 + u−10 u21 − u−20 2 + u21 2 − 2u11 u−11 + 2u10 u−10 4+
2
2
2
2
+ (24 − 3 ) u01 + 2u−10 − 2u11 − u00 − 3u00 u11 + 3u−10 u01 + (4 − 3 ) (u01 u−20 − u21 u00 ) + (−u−20 u−10 + u11 u21 ) 3
1
Q25 = u
κ + u00 −κ +
+ u01 −κ + 21 + u11 κ
−10
2
2
κ+12κ4 −2κ −64
(−κ+64 )
u11 2 − u−10 2 + u00 u10 − u−11 u01 − u−11 u−10 + u
u
+
uu
gx =
(u−10 u01 − u00 u11 ) + −u−20 u11 + u−10 u21 − u−20 2 − u01 u11 + u21 2 + u00 u−10 4 +
12κ
12κ
11 10
2 +12κ −3
−κ
4
4
(2κ−1)
+ 4 2κ
(u−20 u−11 − u10 u−10 + u11 u−11 − u10 u21 ) +
(u−20 u−10 − u11 u21 ) + 61 κ − 4 u00 2 − u01 2 + − 61 u01 u−20 + 16 u21 u00 κ
6κ
2
3
2
3
Q26 = 12 +
u00 + 12 +
u−10 + 12 +
u01 + 12 +
u11
2
3
2
3
2
3
2
3
1 3 (−1+122 +123 )
1 2 (−1+122 +123 )
u−11 u00 − u01 u10 + u00 u−10 − u10 2 + u−11 2 − u01 u11 + 12
(u00 u−20 − u01 u21 + u−20 u−11 − u10 u21 ) + u11 2 − u−10 2 2 +
uu
gx = 12
2 +3
2 +3
122 2 +3 −123 2
2
2
2
1
1
+ 12
(u00 u10 − u−11 u01 ) + 12
2 +3
2 +3 (u11 u10 − u−11 u−10 ) + (2 + 3 ) (−u−10 u01 + u00 u11 ) + −u01 u−20 + u11 u21 − u−20 u−10 − u01 + u00 + u21 u00 3
Q27 = u−10 κ + u00 −κ + 12 + u01 −κ + 21 + u11 κ
κ2 −2κ3 −18κ4 −6κ3 +34 +244 κ2 +12κ2 3
κ2 −2κ3 −18κ4 −6κ3 +34 +244 κ2 +12κ2 3
(2κ4 +2κ3 −4 )
2
2
u
+
uu
gx =
(−u
u
+
u
u
)
+
−
u
(u01 u10 − u−11 u00 ) +
−10
01
00
11
10
−11
4κ2
12κ2
12κ2
−κ2 −34 +12κ4 +6κ3
−2κ3 −18κ4 −6κ3 +34 +364 κ2 +24κ2 3
(−4 +4κ4 +2κ3 )
(2κ4 +2κ3 −4 )
2
2
(u−11 u01 − u00 u10 ) +
+
u00 − u01 +
(u00 u−20 − u01 u21 ) +
(u21 u00 − u01 u−20 ) +
2
2κ
6κ
2κ
12κ
κ+12κ4 −2κ2 −64
−κ2 −64 +18κ4 +6κ3
(2κ−1)
(u11 u10 − u−11 u−10 ) +
(u
u
− u10 u21 ) + 4 2κ
(u11 u−11 − u10 u−10 ) +
12κ
6κ
−20 −11
2
3
2
2
2
2
κ −2κ −18κ4 −6κ3 +34 +124 κ +12κ 3
44 κ +4κ 3 +4 −4κ4 −2κ3
+
(u01 u11 − u00 u−10 ) +
u−10 2 − u11 2 + u21 2 − u−20 u11 + u−10 u21 − u−20 2 4 + (u11 u21
12κ2 4κ2
Q28 = 6u−10 3 + u00 12 − 63 + u01 12 − 63 + 63 u11
1
1
1
uu
gx = 12
− 3 u11 2 − u−10 2 + u00 u10 − u−11 u01 − u−11 u−10 + u11 u10 + −u01 u−20 + u11 u21 − u−20 u−10 − u01 2 + u00 2 + u21 u00 3 + 12
u00 u11 − 12
u−10 u01
Q29 = u−20 κ + 16 u−10 + 16 u00 + u10 61 − κ + u−11 16 − κ + 16 u01 + 16 u11 + u21 κ
uu
gx = − 81 (−u00 + 6u01 κ − 6u11 κ − 6u−10 κ − u01 + 6u00 κ) (6u00 κ − 6u−20 κ + 6u21 κ + 6κu−11 − 6u11 κ − 6κu10 − 6u01 κ + 6u−10 κ − u−11 − u−10 + u10 + u11 )
1
Q210 = 18 u−20 + 81 u−10 + 18 u00 + 18 u10 + 18 u−11 + 81 u01 + 18 u11 + 18 u21
uu
gx = − 24
(u−20 − u10 + u−11 − u21 ) (u11 + u01 + u00 + u−10 )
− u−20 u−10 ) 3
Chapter 5: Numerical Schemes
Table 5.2: Ten-Point schemes with 180◦ antisymmetry that preserve the first and second CLaws in characteristic form
Chapter 5: Numerical Schemes
uu
gx
u
e
First Scheme
1
2
1
12
2
− 2 − 3
1
12
3
− 3
Second
Scheme
1
4
1
24
− 4
4 + 3 −
1
24
1
24
3
− 3
3 − 4
4
1
24
+ 4 − 3
1
12
− 4 − 3
4 + 3
− 34 − 3
3 − 4 −
2(3 − 4 )
Third Scheme
1
2
3
4
Fourth Scheme
1
3
1
6
1
36
1
36
1
18
− 3 + 34
− 24
3 − 24
Fifth Scheme
1
2
1
18
1
36
− 23 + 4
2(3 − 4 −
1
36 )
4 24 3 − 4
2(3 − 24 ) 3(3 − 24 )
3
−κ
κ
κ+12κ4 −2κ2 −64
12κ
64 −κ
1
6κ
12κ
2
κ +34 −12κ4
6κ
4
4 (1−2κ)
2κ
1
6 κ − 4
Figure 5.9: Stencils for the ten-point discretizations of Q2 and uux . Lines indicate that the
multiple of the two end points is included in the scheme. Different colour lines represent different
coefficients and a dashed line represents the negative of its solid counterpart.
97
Chapter 5: Numerical Schemes
uu
dx
u
b
Sixth Scheme
2
2(2 +3 )
3
2(2 +3 )
2 (1−122 −123 ) 3 (1−122 −123 )
12(2 +3 )
12(2 +3 )
122 2 +3 −123 2
2
12(2 +3 )
12(2 +3 )
−(2 + 3 ) 2
3
Seventh Scheme
κ
1
2
−κ
2κ4 +2κ3 −4
κ2 −2κ3 −18κ4 −6κ3 +34 +244 κ2 +12κ2 3
4κ2
12κ2
12κ4 +6κ3 −κ2 −34
2κ4 +2κ3 −4
6κ
2κ
4κ4 +2κ3 −4
24κ2 3 −2κ3 −18κ4 −6κ3 +34 +364 κ2
2κ
12κ2
κ+12κ4 −2κ2 −64
18κ4 +6κ3 −64 −κ2
4
12κ
6κ
4 (2κ−1)
44 κ2 +4κ2 3 +4 −4κ4 4 −2κ3
3
2
2κ
4κ
κ2 −2κ3 −18κ4 −6κ3 +34 +124 κ2 +12κ2 3
12κ2
− 63
1
12
Eighth Scheme
3
1
2
− 3
3
1
12
Ninth Scheme
κ
1
6
−κ
1
6
1
2 (1
− 6κ)
3κ
1
4 (1
− 6κ)
3
2κ
Tenth Scheme
1
8
1
6
1
4
Figure 5.10: More stencils for the ten-point discretizations of Q2 and uux . Lines indicate that
the multiple of the two end points is included in the scheme. Different colour lines represent
different coefficients and a dashed line represents the negative of its solid counterpart. A cross
represents the negative value of the coefficient (depicted by the colour) for a linear term.
98
Chapter 5: Numerical Schemes
5.3.1.3
Two-step schemes
With appropriate assumptions the three-parameter explicit family of schemes from §2.3, which
includes the Zabusky-Kruskal scheme, can be found. Note that the uu
gx term in the threeparameter family is not antisymmetric in space; this is also true for the (0, 1/6, 0) scheme which
was found to perform very well.
5.3.2
Schemes that preserve the first and third Claws
5.3.2.1
Eight-point one-step schemes
As we desire our discretizations to be compact, we searched for discretizations that preserve the
first and third CLaws using a stencil that was only four points wide. In order to find a scheme
with eight points we assumed that all the linear terms had 180◦ symmetry if they were even
f2 ), where
derivatives or antisymmetry if they were odd derivatives, that uu
gx = µ1 (Sm − I)( 12 u
f2 had no additional assumptions, and finally, the discretization for u2 in the
the discretization u
characteristic was assumed to be symmetric is space and time. With these assumptions the
following novel scheme was found:
1
0 = 2ν
(Sn − I) (u−10 +u00 )+ µ1 (Sm − I)
+
+
2
1
96 u−20
+
1
24 u−20 u−10
+
1
48 u−20 u00
+
2
1
24 u−10 +
2
1
1
1
1
1
1
24 u−10 u00 + 96 u00 + 96 u−20 u−21 + 48 u−20 u−11 + 96 u−20 u01 + 48 u−10 u−21 +
2
1
1
1
1
1
1
24 u−10 u−11 + 48 u−10 u01 + 96 u00 u−21 + 48 u00 u−11 + 96 u00 u01 + 96 u−21 +
2
2
1
1
1
1
1
+
24 u−21 u−11 + 48 u−21 u01 + 24 u−11 + 24 u−11 u01 + 96 u01
1
+µ3 (−2αu−20 −ηu−10 −(η+3) u00 −(2α−1) u10 +(2α−1) u−21 +(η+3) u−11 +ηu01 +2αu11 )
+
=ut + uux + uxxx + (4α + 2η + 2) µν3 + O
2
ν
µ3
+O
ν
µ2
(5.33)
+ O(µ) + O(ν),
with characteristic
f3 = 5 u00 2 +
Q
96
+
+
+
2
2
2
2
2
2
2
1
1
1
5
5
5
1
96 u−21 + 96 u10 + 96 u−20 + 96 u−10 + 96 u−11 + 96 u01 + 96 u11 +
1
1
1
1
1
1
12 u−10 u00 + 48 u−10 u10 + 24 u00 u10 + 48 u−20 u−11 + 96 u−20 u01 + 48 u−10 u−21 +
1
1
1
1
5
5
96 u−10 u−11 + 24 u−10 u01 + 96 u−10 u11 + 96 u00 u−21 + 24 u00 u−11 + 96 u00 u01 +
1
1
1
1
1
1
48 u00 u11 + 96 u10 u−11 + 48 u10 u01 + 24 u−21 u−11 + 48 u−21 u01 + 12 u−11 u01 +
1
1
1
1
1
1
48 u−11 u11 + 24 u01 u11 + 48 u−20 u00 + 96 u−20 u−21 + 96 u10 u11 + 24 u−20 u−10 +
+ µ22 αu−20 + α− 12 u−10 −αu00 + 12 −α u10 + 21 −α u−21 −αu−11 + α− 21 u01 +αu11
+
=u2 + 2uxx + (8α − 2)uxt
ν
µ
+O
ν2
µ
+ O(ν) + O(µ).
In order to reconstruct the density and flux for the first CLaw, we set η = −1 − 2α; this prevents
99
Chapter 5: Numerical Schemes
f1 . The reconstruction yielded
any ν terms occurring in F
f1 = 1 u−10 + 1 u00 = u + O(µ) + O(ν),
G
2
2
1
f
F1 = µ2 (2αu−20 + u−21 − 2αu−21 − u−10 − u−11 + (1 − 2α) u00 + 2u01 α) +
2
2
2
1
1
1
1
1
1
96 u−20 + 24 u−20 u−10 + 48 u−20 u00 + 24 u−10 + 24 u−10 u00 + 96 u00 +
1
1
1
1
1
u−20 u−21 + 48
u−20 u−11 + 96
u−20 u01 + 48
u−10 u−21 + 24
u−10 u−11 +
+ 96
2
1
1
1
1
1
1
u−21 u01 +
+ 96 u00 u−21 + 48 u00 u−11 + 96 u00 um,n + 96 u−21 + 24 u−21 u−11 + 48
1
1
1
1
+ 24
u−11 2 + 24
u−11 u01 + 96
u01 2 + 48
u−10 u01
ν
1 2
= 2 u + uxx + (4α − 1)uxt µ + O(ν) + O(µ).
+
The density and flux for the third CLaw are
f3 = 1 u10 18u00 u10 + 18u00 2 + 10u10 2 + 3u−10 u10 + 3u−10 2 + 12u−10 u00 +
G
192
(u00 − u−10 ) (u00 − u−20 ) + µ2ν5 α2 (u00 − u−10 ) (u00 + u−10 )
2
= 13 u3 − u2x + 4α2 uux µν4 + O µν 4 + O µν3 ,
−
1
2µ2
f3 =− µ (2u−20 u−10 u−11 −6u−20 u−10 u00 +3u−20 u−10 u01 +u−20 u00 u01 +3u−20 u−11 u01 +
F
192ν
−3u−21 u−10 u00 −3u−21 u00 u−11 −u−21 u00 u01 −2u−21 u−10 u−11 +6u−21 u−11 u01 +
+u−10 u00 u01 +2u−10 u−11 u01 −u−11 u00 u01 +u−11 u−20 2−u00 u−20 2−2u−10 u00 u−11+
+ u−20 2 u01 − 14u−10 2 u00 − u−21 2 u−10 + 2u−10 2 u01 + 14u−11 2 u01 + u−21 2 u−11 +
− u−21 2 u00 − 2u−11 2 u00 + u−21 2 u01 + 5u−11 3 − 10u00 3 − 5u−10 3 + 10u01 3 +
+u−20 u−11 u−21 −u−20 2 u−10 −u−20 u00 u−21 −4u−20 u−10 2−2u−20 u00 2 +
+ 2u−20 u−11 2 −u−20 u−10 u−21 +u−20 u01 2−u−21 u00 2−2u−21 u−10 2+4u−21 u−11 2 +
+ 2u−21 u01 2 +u−10 u01 2−17u−10 u00 2+17u−11 u01 2 −u−11 u00 2+u−20 u01 u−21 +
1
u01 αu−21 −u01 αu−20 +u−11 αu01 −u−10 αu01 + 21 −α u00 u−21 + α− 12 u−20 u00 +
+ µν
+ 12 −α u00 u−11 + α− 12 u−10 u00 +(α−1) u−21 u−11 + 12 −α u−10 u−21+
+ αu−11 2 + 12 −2α u−10 u−11 + −12 +α u−10 2 + α+ 12 u−10 u−20 −αu−20 u−11 +
+
1
48
u00 2 +u00 u01 +u00 u−21 +2u−20 u00 +2u00 u−11 +4u−10 u00 +u01 2 +4u−10 2 +
+4u−11 2 +2u−21 u01 +u−20 u01 +4u−11 u01 +2u−10 u01 +u−20 2 +u−21 2 +2u−20 u−11 +
+ 2u−10 u−21 + 4u−21 u−11 + 4u−10 u−11 + 4u−20 u−10 + u−20 u−21 )
1
µ2
(−u−10 − u−11 + u−21 + u00 + 2αu−20 − 2αu−21 − 2αu00 + 2u01 α) +
+
1
µ4
4α2 u01 2 + 8u−20 u01 α2 − 4α (−1 + 2α) u−21 u01 − 4u−10 αu01 − 4u−11 αu01 +
2
2
−4α (2α− 1) u00 u01 +(2α− 1) u00 2 −4α (2α−1) u−20 u00 +2 (2α−1) u−21 u00 +
2
+ (−2+4α) u−10 u00 +(−2+4α) u−11 u00 +(−1+2α) u−21 2 − 4α (−1+2α) u−20 u−21 +
+ (−2 + 4α) u−10 u−21 + (−2 + 4α) u−11 u−21 + 4α2 u−20 2 − 4u−10 αu−20 +
−4αu−20 u−11 + 2u−10 u−11 + 2α2 + 1 u−10 2 + 1 − 2α2 u−11 2 +
+
1
9216
u00 2 + u00 u01 + u00 u−21 + 2u−20 u00 + 2u00 u−11 + 4u−10 u00 + u01 2 +
+ 4u−10 2 +4u−11 2 +2u−21 u01 +u−20 u01 +4u−11 u01 +2u−10 u01 +u−20 2 +u−21 2+
2
+ 2u−20 u−11 + 2u−10 u−21 + 4u−21 u−11 + 4u−10 u−11 + 4u−20 u−10 + u−20 u−21 )
2
=uxx u2 + 2ut ux + 14 u4 + u2xx − 4α2 ut u µν4 + O( µν 4 ) + O( µν3 ) + O(µ).
Two interesting choices are α =
1
4,
in which case the ug
xxx term becomes symmetric in time
100
Chapter 5: Numerical Schemes
and antisymmetric in space (which minimizes the local truncation error), and α = 0, which
minimizes the truncation error of the third CLaw’s density and flux.
Additional schemes with rotational symmetry
Just as we did for the schemes that preserve the first and second CLaws, we relax the assumptions
to just 180◦ rotational symmetry assumptions. Because the uet and ug
xxx expressions occur in
(5.15a) and (5.15e), we are no longer free to choose them as we desire. Therefore, we need to
take into account consistency when choosing any parameters in the linear terms.
Five families of schemes result from the above considerations. The first scheme is the scheme
we explored above (5.33), which is as symmetric as possible. The other schemes are all skewed
and occur in pairs: one skewed up and the other skewed down. This is a consequence of the
fact that, for one-step schemes, the transformation n̂ = −n + 1, m̂ = m and ν 7→ −ν is linear
and maps each discretization of KdV and Q3 to another discretization of KdV and Q3 . For a
f2 terms are all skewed in the same direction (see Figure 5.11).
given scheme, the uet , uu
gx and u
Because the uet term is skewed, for the scheme to be consistent if ν = λµr , we require the ug
xxx
term to be antisymmetric in space and symmetric in time (which we can choose the parameters
in this term to accomplish) and r < 1. Thus, the time steps will need to be of a similar size or
larger than the space steps.
Only the second and third schemes have a parameter in the nonlinear term that we are free to
choose. Here are some suggested choices of the parameters for numerical investigation:
Second scheme:
β = 1/72 - symmetric in time except for the orange diagonals which have no opposites. The
linearization of its uu
gx term is
u
fx =
1
36 (−5u−21
− 4u−20 − 4u−11 − 5u−10 + 5u01 + 4u00 + 4u11 + 5u10 ).
β = 1/96 - gives the same linearization for uu
gx as the first scheme, which is
u
fx = 18 (−u−21 − u−20 − u−11 − u−10 + u01 + u00 + u11 + u10 ).
β = 1/48 - gives the same linearization for uu
gx as the fourth scheme,
u
fx =
1
12 (−2u−21
u
fx =
1
12 (−u−21
− u−20 − u−11 − 2u−10 + 2u01 + u00 + u11 + 2u10 ).
β = 0 - (kills orange terms) gives the same linearization for uu
gx as the fifth scheme,
− 2u−20 − 2u−11 − u−10 + u01 + 2u00 + 2u11 + u10 ).
f2 or uu
β = 1/24 - kills the red, dark red and pink terms so that neither u
gx depend on the points
u−20 and u11 .
β = −1/24 - kills the central blue and dark terms.
Third scheme:
β = 1/72 - symmetric in time except for the orange diagonals which have no opposites. The
linearization for this is
u
fx =
1
36 (−4u−21
− 5u−20 − 5u−11 − 4u−10 + 4u01 + 5u00 + 5u11 + 4u10 ).
β = 1/64 - gives the same linearization for uu
gx as the first scheme.
β = 1/48 - (kills the orange terms) gives the same linearization for uu
gx as the fourth scheme.
β = 1/96 - gives the same linearization for uu
gx as the fifth scheme.
101
Chapter 5: Numerical Schemes
f2 or uu
β = 0 - kills red, dark red and pink terms so that neither u
gx depend on the points u−21
and u10 .
β = 1/24 - kills the central blue and dark blue terms.
102
1
1
1
1
1
1
1
5
5
1
1
1
1
1
Q31 = 96
u01 u−20 + 96
u11 u−10 + 24
u−11 u00 + 24
u01 u−10 + 48
u−20 u00 + 96
u10 u11 + 96
u−20 u−21 + 96
u01 2 + 96
u−10 2 + 48
u01 u10 + 24
u11 u01 + 24
u10 u00 + 48
u11 u00 + 48
u−21 u−10+
1
1
1
1
1
1
1
1
1
1
1
1
5
+ 48
u−11 u−20 + 24
u−11 u−21 + 24
u−10 u−20 + 96
u10 u−11 + 96
u00 u−21 + 96
u11 2 + 48
u−11 u11 + 48
u−21 u01 + 48
u−10 u10 + 96
u−20 2 + 12
u−11 u01 + 12
u−10 u00 + 96
u00 u01 +
1
1
1
1
η
u
+u
−
η
u
+u
+
(η−1)
u
+u
−
(η−1)
u
+u
(
)
(
)
(
)
(
)
11
00
01
10
−20
−11
−10
−21
5
5
1
1
5
2
2
2
u−10 u−11 + 96
u−11 2 + 96
u00 2 + 96
u−21 2 + 96
u10 2 + 2 2
+ 96
µ2
−
1
u
−
1
u
+
1
u
+
1
u
103
1
1
1
1
1
1
1
1
1
1
1
1
0 = 2 −10 2 00ν 2 −11 2 01 + µ
− 96
u01 u−20 + 96
u11 u−10 − 48
u−20 u00 + 96
u10 u11 − 96
u−20 u−21 + 32
u01 2 − 32
u−10 2 + 48
u01 u10 + 24
u11 u01 + 24
u10 u00 + 48
u11 u00 +
2
1
1
1
1
1
1
1
1
1
1
1
1
1
− 48 u−21 u−10 − 48 u−11 u−20 + − 24 u−11 u−21 − 24 u−10 u−20 + 96 u10 u−11 − 96 u00 u−21 + 96 u11 + 48 u−11 u11 − 48 u−21 u01 + 48
u−10 u10 − 96
u−20 2 + 32
u00 u01 − 32
u−10 u−11 +
η (u11 −u−20 )+(η−1)(u−21 −u10 )+(α+3)(−u00 +u−11 )+α(−u−10 +u01 )
1
1
1
1
− 32
u−11 2 + 32
u00 2 − 96
u−21 2 + 96
u10 2 +
µ3 2
2
1
1
1
Q32 = 24 + β u−11 u00 + u−11 + u00 + 2 u01 u10 + 21 u−21 u−10 + 48
− 12 β u11 2 + u−20 2 + u−21 u01 + u−20 u00 + u−20 u−21 + u−11 u11 + u−11 u−20 + u−10 u10 + u10 u11 +
1
1
1
+u11 u00 + 2u01 u−10 ) + 16
− 12 β u00 u01 + u−10 2 + u−10 u−11 + u01 2 + 16
− 3/2β (u11 u01 + u−10 u−20 ) + 48
+ 3/2β (u−11 u−21 + u10 u00 ) +
1
1
1
1
η
u
+u
−
η
u
+u
+
(η−1)
u
+u
−
(
)
(
)
(
)
11
00
01
−20
−11
−10
1
1
2
2
2 (η−1)(u−21 +u10 )
u−11 u01 + 12
u−10 u00 + 2 2
+ u−21 2 + u10 2 + u00 u−21 + u10 u−11 β + 12
µ2
−u00 +u−11
1
1
1
1
1
1
1
1
1
1
(24β + 1) 48
0=
+ µ
u01 u10 − 48
u−21 u−10 − 48
u−10 u−11 + 48
u00 u01 + 48
u01 2 − 48
u−10 2 + (−1 + 24β) 48
u−21 u01 − 48
u11 2 + 48
u−11 u−20 +
ν
1
1
1
1
1
1
1
1
1
1
1
1
1
− 48
u−11 u11 − 48
u11 u00 − 48
u−10 u10 + 48
u−20 2 + 48
u−20 u00 + 48
u−20 u−21 − 48
u10 u11 − 24
u00 2 + 24
u−11 2 + 16
u−10 u−20 − 16
u11 u01 + (72β − 1) 24
u−11 u01 − 24
u−10 u00 +
(η−1)(u−21 −u10 )+(η+1)(u−10 −u01 )+η (u11 −u−20 )+(−2+η)(u00 −u−11 )
2
2
1
+ 48 (72β + 1) (u10 u00 − u−11 u−21 ) − β u00 u−21 − u10 u−11 + u−21 − u10
+
µ3
1
1
Q33 = 24
− 2β u11 2 + u−20 2 + u11 u−10 + u01 u−20 + 24
+ β u−11 2 + u00 2 + u−10 u−11 + u00 u01 +
1
1
− β u11 u00 + u−11 u−20 + 2u01 2 + 2u−10 2 + 2u01 u−10 + 12
− 3β (u11 u01 + u−10 u−20 ) +
+ 24
1
1
+ 2u−11 u00 + u01 u10 + u−21 2 + u−21 u−10 + u−20 u00 + u−10 u10 + 3u10 u00 + 3u−11 u−21 + u−21 u01 + u10 2 + u10 u11 + u−11 u11 + u−20 u−21 β + 12
u−10 u00 + 12
u−11 u01 +
1
1
(η−1)
u
+u
−u
−u
+
η
u
+u
−u
−u
(
)
(
)
01
10
11
00
−10
−21
−20
−11
2
+2 2
µ2
−u−10 +u01
1
1
1
1
1
1
0=
+ µ
(−1 + 48β) 24
u−20 2 − 24
u11 u−10 − 24
u11 2 + 24
u01 u−20 + 24
(−1 + 24β) u−11 2 − u00 u01 − u11 u00 + u−11 u−20 − u00 2 + u−10 u−11 +
ν
2
2
1
1
+ 12 (36β − 1) (u−10 u−20 − u11 u01 ) + 12 (72β − 1) (u−11 u01 − u−10 u00 ) + 3u10 u00 + u01 u10 − u−21 u−10 + 2u01 − u−20 u00 − u−21 + u−10 u10 +
(η−1)(u−21 −u10 )+(η+1)(u−10 −u01 )+η (u11 −u−20 )+(−2+η)(u00 −u−11 )
−3u−11 u−21 + u−11 u11 − 2u−10 2 + u10 2 + u10 u11 − u−20 u−21 − u−21 u01 β +
µ3
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
+ 24
u10 u−11 + 24
u00 u
+ 1 u
u + 24
u−21 u01+ 24
u−10
u10+
−10 u−20
24 u−20 u00 + 24 u01 + 24 u−10 + 24 u01 u10 + 24 u11 u01 + 24 u10 u00 + 24 u−21 u−10 + 24 u−11 u−21 + 24 u
−21 24 −11 11
1
1
1
1
1
1
1
1
1
1
1
1
ηu
+
η−
u
−
ηu
+
−
η+
u
+
−
η+
u
−
ηu
+
η−
00
10
−20
−10
−21
−11
2
2
2
2
2
2
2
2
2
2 u01 + 2 ηu11
1
1
1
1
1
1
1
u−11 u01 + 12
u00 u−11 + 12
u−10 u00 + 24
u00 u01 + 24
u−10 u−11 + 24
u−11 2 + 24
u00 2 + 2 2
+ 12
µ2
−u10 +u−21
1
1
1
1
1
1
1
1
1
1
1
1
1
0=
+ µ
− 24
u−20 u00 + 24
u01 2 − 24
u−10 2 + 24
u01 u10 + 24
u11 u01 + 24
u10 u00 − 24
u−21 u−10 − 24
u−11 u−21 − 24
u−10 u−20 + 24
u10 u−11 − 24
u00 u−21 + 24
u−11 u11 +
ν
−ηu−20 −(−η−1)u−10 −(−η+2)u00 −(η−1)u10 +(η−1)u−21 +(−η+2)u−11 +(−η−1)u01 +ηu11
1
1
1
1
1
1
− 24
u−21 u01 + 24
u−10 u10 + 24
u00 u01 − 24
u−10 u−11 − 24
u−11 2 + 24
u00 2 +
µ3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
u−10 u11 + 24
u−20 u01 + 24
u−20 u00 + 24
u01 2 + 24
u−10 2 + 24
u11 u01 + 24
u10 u00 + 24
u11 u00 + 24
u
u + 24
u−11 u−21
+ 24
u−10 u−20
+ 24
u
u + 24
u−21
u + 24
u−10 u10+
Q35 = 24
−11 −20
−11 11
01
1
1
1
1
1
1
1
1
1
1
1
1
2 ηu−20 + 2 η− 2 u−10 − 2 ηu00 + − 2 η+ 2 u10 + − 2 η+ 2 u−21 − 2 ηu−11 + 2 η− 2 u01 + 2 ηu11
2
2
1
1
1
1
1
1
1
+ 12 u−11 u01 + 12 u−10 u01 + 12 u−10 u00 + 24 u00 u01 + 24 u−10 u−11 + 24 u−11 + 24 u00 + 2
µ2
−u−20 +u11
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
0=
+ µ
ν
24 u−10 u11 − 24 u−20 u01 − 24 u−20 u00 + 24 u01 − 24 u−10 + 24 u11 u01 + 24 u10 u00 + 24 u11 u00 − 24 u−11 u−20 − 24 u−11 u−21 − 24 u−10 u−20 + 24 u−11 u11 +
−ηu−20 −(−η−1)u−10 −(−η+2)u00 −(η−1)u10 +(η−1)u−21 +(−η+2)u−11 +(−η−1)u01 +ηu11
2
2
1
1
1
1
1
1
− 24 u−21 u01 + 24 u−10 u10 + 24 u00 u01 − 24 u−10 u−11 − 24 u−11 + 24 u00 +
µ3
Q34 =
Chapter 5: Numerical Schemes
Table 5.3: Eight-Point schemes with 180◦ antisymmetry that preserve the first and third CLaws in characteristic form
Chapter 5: Numerical Schemes
f2
u
First Scheme
uu
gx
uet
1
2
1
96
1
24
1
48
5
96
1
12
1
96
1
24
1
48
1
32
Second
Scheme
1
+ β)
+ β 12 ( 24
1
1
1
− 12 β)
− 12 β) 3( 48
− 2 β 2( 48
1
3
1
1
1
β 12 16 − 2 β 48 + 2 β
1
24
1
48
1
48 (24β + 1)
1
24 (1 − 24β)
1
24 (72β − 1)
1
48 (1 − 24β)
1
β
16 (1 − 24β)
1
48 (72β
1
+ 1)
Third
Scheme
1
24
1
24
+β
− 2β
β 2β
3β
1
24
1
12
1
12
1
− β)
− β 2( 24
− 3β
1
24 (1
1
24 (1
− 24β)
− 48β)
3β
β 2β
1
12 (1
1
12 (1
− 72β)
1
− 36β)
Fourth
Scheme
1
24
1
12
1
24
1
12
1
24
1
Fifth
Scheme
1
24
1
Figure 5.11: Stencils of the eight-point discretizations of u2 , uux and ut . Lines indicate that
the multiple of the two end points is included in the scheme. Different colour lines represent
different coefficients and a dashed line represents the negative of its solid counterpart. A cross
represents the negative value of the coefficient (depicted by the colour) for a linear term.
104
Chapter 5: Numerical Schemes
5.3.2.2
Ten-point one-step schemes
We now extend our search to discretizations with a wider stencil, but with a more compact
discretization for the uux term. The following scheme was found by making the following asf and uu
sumptions: uet has one consistency condition (2.24), Q2
gx depend only on the central six
points and ug
xxx has one consistency condition (2.18).
1
u11 2 + u10 u11 + u10 2 − u−11 2 − u−10 u−11 − u−10 2 +
0 = ν1 (u01 − u00 ) + 12µ
+ µ13 − 14 + 12 η u−20 + 12 u−10 − ηu00 − 21 u10 + 14 + 21 η u20 +
+ − 14 − 12 η u−21 + 12 u−11 + ηu01 − 12 u11 + 14 − 12 η u21
(5.34)
2
=ut + uux + uxxx − 2ηuxxt µν + O( νµ ) + O(ν) + O(µ2 ),
f3 = 1 u01 2 + 1 u01 u00 + 1 u00 2 +
Q
3
3
3
+ µ22 12 − η u−10 − u00 +
1
2
2
+ η u10 +
1
2
+ η u−11 − u01 +
=u2 + 2uxx − 4uxt η µν + O( νµ ) + O(ν) + O(µ2 ).
1
2
− η u11
The density and flux for the first CLaw are
f1 =u00 ,
G
f1 = 1 u−10 2 + u−10 u−11 + u−11 2 + u00 2 + u01 u00 + u01 2 +
F
12
+
1
4µ2
((−1 + 2η) u00 + (−1 − 2η) u01 + (1 − 2η) u−20 + (1 + 2η) u−21 +
+ (−1 − 2η) u−10 + (−1 + 2η) u−11 + (1 + 2η) u10 + (1 − 2η) u11 ) ,
2
= 12 u2 + uxx − 2uxt η µν + O( νµ ) + O(ν) + O(µ).
The density and flux for the third CLaw are
−2u00 2 + u00 u−10 + u00 u10 = 13 u3 + uuxx + O(µ2 ),
u−10 2 + u−10 u−11 + u−11 2 u01 2 + u01 u00 + u00 2 +
f3 = 1 u00 3 +
G
3
f3 = 1
F
36
+
1
µ2
− 16 u01 u00 u−11 − 16 u01 u00 u−10 + 16 u10 ηu−11 2 − 16 u11 ηu−11 2 +
1
µ2
− 16 u01 2 u−11 − 16 u01 2 u−10 − 61 u00 2 u−11 − 16 u00 2 u−10 +
1
+ 12
u01 2 + u01 u00 + u00 2 (1 + 2η) u−21 + 16 u01 2 u00 + 16 u00 2 u01 +
+ 16 ηu00 3 + 16 u−11 2 u−10 + 61 u−10 2 u−11 − 16 ηu−10 3 +
2
1
12 u10 u−10 +
1
1
6 u10 ηu−11 u−10 − 6 u11 ηu−11 u−10 +
− 16 u01 u−10 2 +
+ 16 ηu−11 3 −
+
+
−
+
−
2
2
1
1
12 u11 u−10 − 6 u00 u−10 +
2
2
2
1
1
1
6 u−11 u00 + 12 u10 u−11 − 6 u01 u−11 +
2
3
2
2
1
1
1
(−1 + 2η) u−20 +
12 u11 u−11 + 12 u−11 − 12 u01 + u01 u00 + u00
3
3
2
1
1
1
1
1
12 u00 + 12 u−10 + 12 u10 u−11 u−10 − 6 u01 u−11 u−10 − 6 u11 ηu−10 +
2
3
1
1
1
6 u00 u−11 u−10 + 1/12u11 u−11 u−10 + 6 u10 ηu−10 − 12 u01 (−1 + 2η) +
1
νµ
(−2ηu−11 u01 − 2ηu00 u−10 − (−1 − 2η) u−11 u00 − (1 − 2η) u01 u−10 ) +
1
4µ4
(−u−10 + 2u−10 η + 2u00 − u10 − 2u10 η − u−11 − 2u−11 η + 2u01 − u11 + 2u11 η)
(−2u−20 η + 2ηu00 − 2ηu01 + 2u−21 η − 2u−10 + u−20 + u−21 + u00 + u01 − 2u−11 )
2
= 14 u4 −uuxt +u2xx +u2 uxx + ux ut −η(2u2t +2u2 uxt −4uxt uxx ) µν +O( νµ )+O(ν)+O(µ).
By setting η = 0, the local truncation errors are minimized and the ug
xxx term becomes symmetric in time and antisymmetric in space. Doing this results in the scheme that Furihata [23]
constructed.
105
Chapter 5: Numerical Schemes
Additional schemes with rotational antisymmetry
Once again, we relax our previous assumptions and search for schemes that have 180◦ rotational
antisymmetry. However, as for ten-point schemes that preserve the first and second CLaws,
due to computer limitations we have omitted the points u−21 and u20 from the uu
gx term. Two
one-parameter families of schemes were found which are shown in Table 5.4; their stencils are
shown in Figure 5.12. Unlike the eight-point schemes, the ut term is symmetric in space and so
consistency is not an issue. Also, in both families, we can choose ug
xxx to be symmetric in time.
For the one-step schemes that preserve the third CLaw, there is no correspondence between the
eight-point schemes and the ten-point schemes found here. This is because the shear transformation, though it preserves the divergence expressions, does not map a discretization of uxx in
the characteristic to another discretization of uxx . For
u
g
xx =
1 X
B
X
αij uij ,
j=0 i=A
to be a discretization of uxx , the necessary conditions that the coefficients must satisfy are
0=
1 X
B
X
αij ,
0=
j=0 i=A
1 X
B
X
iαij ,
2! =
j=0 i=A
1 X
B
X
i2 αij .
(5.35)
j=0 i=A
If in addition we wish that when the scheme is sheared, with m 7→ m + n and n 7→ n, that it too
is a discretization of uxx , then the coefficients must also satisfy
0=
B
1 X
X
αij ,
0=
1 X
B
X
(i + j)αij ,
2! =
j=0 i=A
j=0 i=A
1 X
B
X
(i + j)2 αij .
j=0 i=A
Using the original conditions, the additional conditions simplify to
0=
B
X
αi1 ,
1=
i=A
B
X
iαi1 .
(5.36)
i=A
Thus in general it is not possible to generate a one-step scheme that preserves the third CLaw
of KdV by shearing another scheme that already does. If one compares the first scheme in Table
5.4 with the second scheme in Table 5.4, even though the nonlinear terms are related by a shear
(β 7→ −1 ), the u
g
xx and ug
xxx terms are not.
The following are some suggested choices of parameters to investigate. Setting 1 = −1/24
1
in the second family (this is Furihata’s scheme, which we
in the first family and 2 = − 12
studied above) causes both schemes to be symmetric in time and as compact as possible. The
linearization of both schemes coincide as
u
fx = − 14 u−11 − 41 u−11 + 14 u11 + 14 u10 .
To remove the central points of the second family, choose 2 = 0, which gives a linearization of
u
fx = − 16 u−20 −
1
12 u−11
u
fx = − 19 u−20 −
5
36 u−11
−
1
12 u−11
− 61 u01 + 16 u00 +
1
12 u11
+
1
12 u10
+ 16 u21 .
The first scheme has the same linearization if 1 = 0, which removes the orange terms. To cause
the u2−10 to have the opposite sign to the u200 term in the second family, so that the interior
1
(red) and exterior terms terms (blue) have the same weighting, choose 2 = − 36
. The resulting
linearization is
−
5
36 u−11
− 91 u01 + 19 u00 +
5
36 u11
+
5
36 u10
+ 19 u21 .
1
The scheme from the first family with the same linearization is given by 1 = − 72
.
106
−u
+u
2
2
2
2
1
00
01 + 1
0=
− u−20 u−11 +
ν
µ
48 (241 + 1) u21 u00 − u00 u−20 − u01 u−20 − u11 u−11 − u−20 + u01 u21 + u10 u21 + u10 u−10 + u21 + 2u00 − 2u
01 1
1
1
(721 + 1) (u00 u−10 − u01 u11 ) + 48
(241 − 1) u−10 u01 − u00 u11 + u2−10 + u−10 u−11 − u11 u10 − u11 2 + 48
(721 − 1) (u−11 u01 − u00 u10 ) +
−3u−20 u−10 + 3u11 u21 ) + 24
+ −u10 2 + u−11 u00 + u−11 2 − u01 u10 1 +
1
1
2(α−η)(u11 −u−10 )+η (u21 −u−20 )+ 2 (1+8α−4η)(u00 −u01 )+ 2 (−1+2η)(u−21 −u20 )+(−2η+2α+1)(u−11 −u10 )
+
µ3
α(u−10 +u11 )−(−1+α)(u−11 +u10 )−u00 −u01
Q32 = 22 + 16 u00 u−10 + u00 u11 + u−10 2 + u−10 u01 + u01 u11 + u11 2 + −4u00 2 − 4u01 2 − 4u01 u00 2 + 2
µ2
1
2
2
2
2 + u
2
2
2
2 2
−11 u−10 −u11 u10 −u10 +u−11 −u11 +u−10
−u00 +u01
24 (122 +1) u00 u10 +u00 u11 −u−20 +u10 u21 +u00 −u−10 u01 −u01 −u−20 u−10 −u−11 u01 +u11 u21 −u−20 u−11 +u21
+
ν
µ
1
1
1
1
1
1
1
(−1+2α)(u
−u
)+
(−1+α)
u
−u
α
u
−u
+
u
+
u
−
u
−
u
+
( −21 20 ) 2 ( 21 −20 ) 2 −10 2 −11 2 10 2 11
00
01
2
2
µ3
0=
+
107
First
Scheme
c2
u
1
1
2( 48
+ 12 1 ) 3( 48
+ 12 1 )
1
1
48 + 2 1
1
1
48 − 2 1
1
24 − 1
1
48 −
1
1
12
16 +
3
2 1
1
2 1
−1
uu
dx
1
24 (241
+ 1)
1
16 (241
+
+ 1)
1
1
(1 − 721 )
48 (241 + 1) 48
1
1
− 24
(721 + 1)
(1
−
24
)
1
48
−1
Second
Scheme
22 +
42
1
6
1
24 (122
+ 1)
2
Figure 5.12: Discretizations of u2 and uux using ten points. Lines indicate that the multiple of the two end points is included in the scheme. Different
colour lines represent different coefficients and a dashed line represents the negative of its solid counterpart.
Chapter 5: Numerical Schemes
Table 5.4: Ten-Point schemes with 180◦ antisymmetry that preserve the First and Third CLaws in characteristic form
1
Q31 = 48
+ 12 1 u21 u00 + u10 u21 + u01 u21 + u00 u−20 + u−20 u−11 + u01 u−20 + u11 u−11 + u21 2 + u−20 2 + u10 u−10 + 2u11 u−10 + 3u−20 u−10 + 3u11 u21 +
2
2
2
2
1
1
1
1
1
+ 48 − 2 1 u11 u10 + u−11 u−10 + 2u01 u00 + 2u00 + 2u01 + 48 − 3/21 (u00 u10 + u−11 u01 ) + 16 + 2 1 u00 u11 + u−10 + u−10 u01 + u11 +
1
1
1
1
η (u−20 +u21 )+α(u11 +u−10 −u−11 −u10 )− 4 (−1+2η)(u−21 +u20 )− 4 u00 − 4 u01
1
1
u00 u−10 + 12
u01 u11 + 2 2
+ −u01 u10 − u−11 2 − u10 2 − u−11 u00 1 + 12
µ2
Chapter 5: Numerical Schemes
5.3.2.3
The 3-step explicit scheme
Since it is possible to preserve the first and second CLaws with an explicit scheme, it was an
interesting question to see whether an explicit scheme could be found to preserve the first and
third CLaws. We found that instead of two time steps it was necessary to have three time
steps to find such a scheme. To find this scheme, the density and flux for the first CLaw were
discretized and then the appropriate difference operators were applied. The only restrictions on
the discretizations were the number of points each term could depend on:
uet = ν1 (Sn − I)e
u where u
e used the points u0j with j = −2, −1, 0,
uu
gx =
ug
xxx =
1
µ (Sm
1
µ (Sm
f2 where u
f2 depended on points with i = −1, 0 and j = −1, 0
− I)u
− I)g
uxx where u
g
xx depended on i = −2, . . . , 1 and j = −1, 0.
Both terms in the characteristic for the third CLaw used the central six points with i = −1, 0, 1,
and j = −1, 0, but u
g
xx was assumed to be a total divergence in the m direction. The resulting
difference scheme is
1
1
(u01 − u0−2 )+ 4µ
(u1−1 u10 − u−1−1 u−10 )+ µ13 21 u−10 − 12 u1−1 − 12 u10 + ηu−2−1 − u20 η +
0 = 3ν
+ − 12 − η u−20 + 12 + η u2−1 + 12 + 2η u00 + −2η − 21 u0−1 + 12 u−1−1
(5.37)
2
=ut + uux + uxxx − (1 + 4η)uxxt µν + O( νµ ) + O(ν) + O(µ).
The characteristic is
f3 =u0−1 u00 +
Q
1
µ2
(−2u00 − 2u0−1 − 4u−1−1 η + (2 + 4η) u−10 + (2 + 4η) u1−1 − 4ηu10 )
2
=u2 + 2uxx − 2(1 + 4η)uxt µν + O( νµ ) + O(ν) + O(µ2 ).
The density and flux for the first CLaw are
f1 = 1 u0−2 + 1 u0−1 + 1 u00 = u + O(ν),
G
3
3
3
1
1
f
F1 = 4 u0−1 u00 + 4 u−1−1 u−10 + µ12 u0−1 η −ηu−2−1 − 12 +η u−1−1 + 12 +η u1−1 + 12 +η u−20+
2
+ ηu−10 − 12 +η u00 −ηu10 = 12 u2 + ux x − (1 + 4η)uxt µν + O( νµ ) + O(ν) + O(µ).
108
Chapter 5: Numerical Schemes
The density and flux for the third CLaw are
f3 = 1 u0−1 u00 u0−2 +
G
3
1
µ2
− 23 u00 u0−2 − 23 u0−1 u00 − 43 u−1−2 ηu00 −
2
3
1
2
3
(−1 − 2η) u00 u−1−1 +
(−1 − 2η) u1−2 u00 − 43 ηu00 u1−1 − 13 u0−1 u0−2 + − 43 η − 3 u−1−1 u0−2 +
+ 13 + 43 η u−1−2 u0−1 + 31 u1−1 u0−2 + 13 u0−1 u1−2 − 13 u1−1 u1−2
−
2
= 13 u3 + 23 uuxx − 13 u2x + (1 + 4η)(−uxt u − 13 ux ut ) µν + O( νµ ) + O(ν),
+ 43 u−1−2 ηu00 − 31 u00 u−1−1 − 13 u0−1 u0−2 + 23 + 43 η u−10 u0−2 +
+ 13 u−1−1 u0−2 + − 43 η − 13 u0−1 u−10 + − 43 η − 13 u−1−2 u0−1 +
f3 = 1
F
µν
1
3 u0−1 u00
+ µ12 −ηu0−1 u00 2 +
1
2
(1 + 2η) u00 u0−1 2 +
1
2
(1 + 2η) u−20 u00 u0−1 − 12 u−1−1 u−10 u0−1 +
− ηu−2−1 u0−1 u00 − 12 u0−1 u00 u−10 − 12 u−1−1 u0−1 u00 − 12 u−1−1 u−10 u00 +
+ 12 (1+2η)u−1−1 u−10 2 −ηu−1−1 2 u−10 −ηu−1−1 u10 u−10 + 12 (1+2η)u1−1 u−1−1 u−10 +
+ 14 u0−1 u00 u−1−1 u−10 +
−
1
µ4
(u1−1 + 2u1−1 η − u00 − 2ηu10 − 2u−1−1 η − u0−1 + u−10 + 2ηu−10 )
(2ηu00 + 2ηu−2−1 − 2u0−1 η − 2u−20 η + u−10 − u0−1 − u−20 + u−1−1 ) ,
= 14 u4
+ u2xx + u2 uxx + 43 ux ut − 23 uxt u+
2
+ (1 + 4η)(−u2 uxt + 13 utt u − 2uxt uxx − 23 u2t ) µν + O( νµ ) + O(ν) + O(µ).
f3 contains ν terms and to ensure these terms did not blow up in the limit, as the step
Note that F
sizes tend to zero, trivial CLaws were added during the reconstruction. Setting η = − 41 causes
the ug
xxx term to become symmetric in time and antisymmetric in space and this minimizes the
LTE of the difference scheme.
5.3.3
Preserving all three CLaws together?
Ultimately we would like to find a difference scheme that preserves the first three CLaws of
KdV together. However we have been unsuccessful. Such a method may exist but by using the
brute force approach outlined in this chapter we have been unable to find it due the amount
of time and memory that the Groebner basis calculations require (see §2.2.2). For example,
when looking for a one-step method with 22 points (i.e. m = −5, . . . , 5) with no assumptions
on the terms other than that they are discretizations, calculating2 the Groebner basis (using
the total degree ordering) required 8.1 GiB of RAM and 5796.5s (≈ 97mins). In contrast, when
searching for the additional schemes that preserve the first and third CLaws, using 5 × 2 points
(which resulted in 688 equations in 51 unknowns) calculating the Groebner basis required 8.6s
and 26.6MB. Because we were unable to find a one-step scheme with eleven spatial points, that
preserves all three CLaws it suggests that, if it is possible to find a method that preserves all
three CLaws, it will require at least two time steps. However, once we investigate two-step
schemes we start to run out of memory. We were unable to calculate the Groebner basis for a
scheme with 4 × 3 points, with 180◦ symmetry and antisymmetry assumptions on all the terms,
as we ran out of memory (63 GiB). However by assuming that the uet term was symmetric in
space we could quickly show there were no solutions. So if solutions do exist for this case they
will have a counter-intuitive discretization for ut . Similarly, when using 5 × 3 points, we run out
of memory even when we assume that the uet term is symmetric in space and the ug
xxx term is
2 These
calculations were conducted on the departmental server which has 63.0 GiB of RAM and 4 AMD
Opteron(tm) 6134 Processors (2.3 GHz) giving 32 cores.
109
Chapter 5: Numerical Schemes
symmetric in time. Thus it is desirable to develop a more efficient method of solving the system
of equations.
The system of equations that need to be solved to find schemes that preserve CLaws has a
special structure. To demonstrate this we return the simple example of the effects of the Euler
operator on nonlinear terms. If we order the coefficients of P and Q (from (5.16)) in vectors
p = (α, α̂, γ, γ̂)T and q = (β, β̂)T respectively and let Aij be the four by two matrix [Aij ]kl =
δik δjl then we can rewrite the system of equations (5.18), that results from the condition that
E(P · Q) = 0, as
0 = αβ̂ = pT A12 q,
0 = γβ = pT A31 q,
0 = (αβ + α̂β̂) = pT (A11 + A22 )q,
0 = γ̂β = pT A41 q,
0 = α̂β = pT A21 q,
0 = (γ̂β + γ β̂) = pT (A41 + A32 )q,
0 = γ̂ β̂ = pT A42 q,
0 = γ β̂ = pT A32 q.
Clearly, this system has the trivial solutions p = 0 and q = 0; however, for our problems this is
not possible as P and Q are discretizations for some terms. So not all of their coefficients can
be zero, i.e. p and q must also satisfy some conditions of the form


 
0
0
 . 
 . 
 .. 
 .. 
,

Bp = 
Cq = 


 ,
 0 
 0 
1
1
for some matrices B and C. So if we seek to preserve a given CLaw of an equation with
polynomial nonlinearities, we will obtain a system of equations with a similar structure. To
f2 and u
illustrate this, the the coefficients of u
g
xx for the discretized characteristic Q3 can be
ordered into vectors β and ζ respectively, and the parameters in the discretizations of ut , uux
and uxxx can be ordered into vectors γ, and α respectively. Then from (5.15), to find a
discretization of KdV that preserves the structure of the third CLaw, with 180◦ symmetry
assumptions, we need to find vectors α, β, γ, , ζ that solve the system of equations of the form
0 = βAi γ,
i = 1, . . . , na ,
(5.38)
0 = βBi ,
i = 1, . . . , nb ,
(5.39)
0 = βCi α + ζDi ,
i = 1, . . . , nc ,
(5.40)
along with the conditions that come from discretizations




0
!
!
0


0
0
 0 


Dut γ =
, Duux =
, Duxxx α = 
 , Duxx ζ =  0  , Du2 β = (1) ,


1
1
0
2!
3!
where Dut , Duux , Duxxx , Duxx , Du2 are matrices containing the information in (2.16, 2.29, 5.5).
Clearly the above system can have solutions, as we showed in the previous sections by finding
the Groebner basis for the resulting polynomial equations. However finding the Groebner basis
will work for any system of polynomial equations and is very expensive, as we have found, to
use. Therefore can we exploit the above structure to find a more efficient method of finding
solutions?
A splitting strategy does speed the process up. We can assume that a given variable is either
nonzero or zero and thus gain two systems of equations with fewer variables. The aim is to chose
110
Chapter 5: Numerical Schemes
a variable that by being nonzero or zero immediately makes several other equations linear. Then
we can solve them and either find that there are no possible solutions or else we have further
reduced the number of variables. This in turn may cause other equations to become linear, etc.
Once all the consequences of our assumption have been worked through, we can either choose
another variable to make assumptions on, or else calculate the Groebner basis for the two new
systems we have found. To illustrate this we show that our example system of equations, (5.16),
only has the trivial solutions p = 0 or q = 0. To begin, assume α 6= 0; this implies that β = 0
and so β̂ = 0 as well. Therefore, for a nontrivial solution α = 0. If we now, in addition, assume
γ 6= 0 then this too implies β = 0 which in turn implies β̂ = 0. Therefore, for nontrivial solutions,
α = 0 and γ = 0. Continuing this process by making assumptions on α̂ and γ̂, we can show that
this system of equations only has trivial solutions.
5.4
Direct construction of explicit schemes to preserve one
CLaw
In this section we use the theory from Chapter 3 to illustrate problems that can arise when we
construct explicit discretizations that preserve a single chosen CLaw. Given an explicit P∆E,
∆ ≡ u01 − ω(z), where z denotes all the terms of the form uij with j ≤ 0, the characteristic for
a CLaw
(Sn − I)G(z) + (Sm − I)F (z) = 0,
is given by
Q=
Z
1
λ=0
X
−i
Sm
i
∂Sn G i ∆ ∂Sm
dλ.
∆7→λ∆
So the characteristic typically depends on the difference equation. However, if G is linear in the
P
variables ui0 , i.e. G = h + i ui0 gi , where the functions gi and h depend on uij with j ≤ −1,
then the characteristic no longer depends on the equation:
X
−i
Q=
Sm
Sn gi ,
i
so the characteristic equals the seed. Using this observation, we can now construct schemes to
preserve a given CLaw.
Given a CLaw for an PDE in characteristic form,
Dt (G) + Dx (F ) = Q∆,
if the LHS is discretized as
(Sm − I)
(Sn − I)
g
G̃ +
F̃ = Q∆,
ν
µ
g factors to give Q̃∆
˜ where Q̃ is a discretization
then a discretization for Q∆ is obtained. If Q∆
˜
of the characteristic and ∆ is a discretization of the PDE then a discretization of the PDE has
been found that preserves the CLaw. This is the form of discretization that is sought by the
method of §5.2. However, by imposing that
G̃ =
u00 g
+ h,
λ
111
F̃ = F̃ (z),
Chapter 5: Numerical Schemes
where g and h are functions depending on variables of the form uij with j ≤ −1 and λ is a
non-zero constant, we obtain
Sn g
Sn g
λνω(z)
u01
ω(z)
g
Q∆ =
u01 − ω(z) =
u01 −
= Sn g
−
.
λν
λν
Sn g
λν
Sn g
Thus the discretization of the CLaw is a difference CLaw for the explicit difference equation
ng
u01 − Sνω
= 0. Therefore Sλν
is its characteristic which cannot be the zero function on solutions
ng
of the difference equation (see Chapter 3). If Sn g is a discretization for the PDE’s characteristic
01
then uλν
− Sωn g must be an explicit discretization of the PDE. Therefore to construct an explicit
finite difference scheme that preserves a given CLaw, we use the following approach.
1. Write the CLaw in characteristic form.
2. Add trivial CLaws of the second kind so that the density, G, is written as uQ and some
additional terms. For example, the density of the third CLaw of KdV can be written as
G3 = u(u2 + 2uxx ) − 32 u3 − uuxx .
3. Replace the u that multiplies the characteristic by u00 and discretize all the other terms in
G with points of the form uij (with j ≤ −1) to construct G̃. Thus, for the above example
we might choose
u0−1 (u−1−1 − 2u0−1 + u1−1 )
u−1−1 − 2u0−1 + u1−1
2
f
.
G3 =u00 u0−1 + 2
− 32 u0−1 3 −
2
µ
µ2
4. The discretization of the characteristic Q̃ is given by Sn applied to the coefficient of u00 in
G̃.
5. Discretize the flux using points of the form uij with j ≤ 0 to get F̃ .
6. The discretization for the PDE is given by
1 (Sn − I)
(Sm − I)
e
∆=
G̃ +
F̃ .
ν
µ
Q̃
(5.41)
This method inevitably leads to schemes which involve dividing by the discrete characteristic.
This may cause difficulties because, even though the characteristic cannot be the zero function,
the characteristic could be zero or close to zero at a specific grid-point. The Zabusky-Kruskal
(2.3) and the three-step explicit scheme (5.37) both could be constructed in this way; however
in both these cases the discretization was found by searching for discretizations of the CLaw of
˜ and hence no division is necessary.
the form Q̃∆
We used the above method to discretize KdV so as to preserve given CLaws. The resulting
schemes are given below.
5.4.1
Preserving the second CLaw of KdV
A discretization for KdV that preserves the second CLaw is given by (5.41) with
f2 = 1 u00 u0−1 = 1 u2 + O(ν),
G
2
2
f2 = 1 u00 3 +
F
3
1
µ2
(u00 (u10 − 2u00 + u−10 )) −
= 13 u3 + uuxx − 12 u2x + O(µ2 ),
f =u00 .
Q2
112
1
((u00 − u−10 ) (u10 − u00 ))
2µ2
Chapter 5: Numerical Schemes
As the discretization for the flux is not determined by the discretization for the density, an
alternative discretization is to use
−1 f
1 f
1 3
1 2
g
F
2a = 2 F2 + Sm F2 = 3 u + uuxx − 2 ux + O(µ),
f2 . The discretization of KdV using F
f2 has a local truncation error of O(ν) + O(µ),
instead of F
2
g
whereas the discretization using F
2a has a local truncation error of O(ν) + O(µ ).
In order to apply the method, all that is necessary is that the resulting discretization can be put
in Kovaleveskya form. Thus we could choose to make the flux linear in its highest shifted term
and create a method that is not explicit. For example, insisting that Fe2 is linear in u10 and all
other terms have i ≤ 0 will give a scheme that can be solved uniquely for u20 ,
f2 = 1 u200 ,
G
2
f
F2 = 16 u3−10 + u300 + µ12 (u10 − 2u00 + u−10 ) u−10 + 2µ1 2 (u00 − u−10 )2 = 13 +uuxx − 12 u2x +O(µ),
f =u00 .
Q2
The resulting discretization of KdV preserves the second CLaw exactly. However it can only
be solved uniquely for u201 , not u01 , so the solutions are not unique; however, when modeling a
problem where u(x, t) > 0 (such as soliton solutions), the positive square root could be chosen to
give an explicit scheme. However, in §2.5 the numerical soliton solutions developed wave trains
that followed the soliton which at times are negative.
5.4.2
Preserving the third CLaw of KdV
In order to construct a scheme that preserves the third CLaw, the CLaw needs to be written as
G3 = 31 u3 + uuxx ,
F3 = 14 u4 + u2 uxx + ux ut − uuxt + u2xx ,
Q3 = u2 + 2uxx .
A scheme that preserves the third CLaw is then given by (5.41) with
f3 =u00 u0−1 2 + 22 (u−1−1 − 2u0−1 + u1−1 ) − 2 u0−1 3 − 12 u0−1 (u−1−1 − 2u0−1 + u1−1 )
G
µ
3
µ
= 13 u3 + uuxx + O(ν) + O(µ2 ),
f3 = 1 u00 4 +
F
4
−
1
νµ u00
2
1
µ2 u00
(u10 − 2u00 + u−10 ) +
1
νµ
(u00 − u−10 − u0−1 + u−1−1 ) +
(u00 − u−10 ) (u00 − u0−1 ) +
1
µ4
2
(u10 − 2u00 + u−10 )
= 14 u4 + u2 uxx + ux ut − uuxt + u2xx , +O(ν) + O(µ)
f3 =u00 2 + 2 u10 − 2u00 + u−10 = u2 + 2uxx + O(µ2 ).
Q
µ2
5.4.3
Preserving the fourth CLaw of KdV
To demonstrate that higher order CLaws can also be constructed in this way, a scheme that
preserves the fourth CLaw of KdV is constructed. The fourth CLaw for KdV is
G4 =5u4 + 60uu2x + 60u2 uxx + 72uuxxxx − 36u2xx ,
F4 =4u5 + 20u3 uxx + 30u2 u2x + 72uux uxxx + 36u2xxx +
+ 24uu2xx − 12u2x uxx − 72uutxxx + 72ux utxx − 60u2 utx ,
Q4 =20u3 + 60u2x + 120uuxx + 72uxxxx ..
113
Chapter 5: Numerical Schemes
A finite difference scheme, of the form (5.41), that preserves it is constructed as follows
f4 =20u00 u0−1 3 − 15u0−1 4 +
G
+
120
µ2 u00 u0−1
+
72
µ4 u00
−
36
µ4
60
µ2 u00
(u1−1 − u0−1 ) (u0−1 − u−1−1 ) +
(u−1−1 − 2u0−1 + u1−1 ) −
2
60
µ2 u0−1
(u−1−1 − 2u0−1 + u1−1 ) +
(u2−1 − 4u1−1 + 6u0−1 − 4u−1−1 + u−2−1 ) +
2
(u−1−1 − 2u0−1 + u1−1 ) ,
f4 =4u00 5 +
F
3
20
µ2 u00
(u10 − 2u00 + u−10 ) +
2
30
µ2 u00
(u10 − u00 ) (u00 − u−10 ) +
2
+ µ724 u00 (u00 −u−10 ) (u10 −3u00 +3u−10 −u−20 )+ µ366 (u10 −3u00 +3u−10 −u−20 ) +
2
+
24
µ4 u00
−
72
νµ3 u00
+
72
νµ3
−
2
60
νµ u00
12
µ4
(u00 − u−10 ) (u10 − u00 ) (u10 − 2u00 + u−10 ) +
(u10 − 3u00 + 3u−10 − u−20 − u1−1 + 3u0−1 − 3u−1−1 + u−2−1 ) +
(u00 − u−10 ) (u10 − 2u00 + u−10 − u1−1 + 2u0−1 − u−1−1 ) +
f4 =20u00 3 +
Q
+
(u10 − 2u00 + u−10 ) −
72
µ4 (u20
(u00 − u−10 − u0−1 + u−1−1 ) ,
60
µ2
(u00 − u−10 ) (u10 − u00 ) +
120
µ2 u00
(u10 − 2u00 + u−10 ) +
− 4u10 + 6u00 − 4u−10 + u−20 ).
Unfortunately, all these schemes proved to be unstable when attempting to implement them.
Even though the discrete characteristic function cannot be the zero function, it may become zero
or be close to zero at a given point, causing the scheme to blow up. Thus, we have demonstrated
that having a numerical scheme that algebraically satisfies the CLaw may lead to a useless
numerical method.
5.4.4
Summary
In this chapter we have developed a method for searching for CLaws that preserve CLaws. This
method is limited by computing power and the amount of memory the resulting Groebner basis
requires to be calculated. So far, the method has only been tested on KdV; however, the method
should be capable of finding discretizations for other PDEs that have polynomial nonlinearities.
The method has found schemes that preserve the first and second CLaws and the first and third
CLaws simultaneously for KdV. Some of these schemes are novel and others are known. In
Chapter 6 we will study these schemes further.
In addition we have directly constructed explicit schemes that preserve a single CLaw of KdV
using the characteristic. This method appears limited to finding methods that preserve only a
single CLaw at a time and the discretization may be a rational function which will be problematic
to implement.
114
Chapter 6
Analysis and Numerical
Experimentation
In this chapter we analyze the numerical schemes for the KdV equation that we found in the
previous chapter. We begin by linearizing the schemes and studying their linear stability so as
to find any step size restrictions. We then use the exact conservation of the second conserved
quantity for the eight-point schemes (5.21, 5.23, 5.25) to give a nonlinear stability result. Having
studied the linearized schemes, we implement the schemes using matlab to model two different
problems to compare the behavior of the schemes. We then study the numerical dispersion
relations of the discretized schemes to seek to better understand the behaviour we have observed.
6.1
Linear Stability Analysis of Schemes
As in Chapter 2 we linearize the schemes about the constant solution umn = ρ. The linearized
numerical schemes are thus discretizations of
0 = ut + ρux + uxxx .
The stability analysis is performed by assuming vmn = Aξ n eiωm and so the scheme is stable if
|ξ| ≤ 1.
Lemma 6.1.1. If a one-step linear discretization only consists of terms with 180◦ symmetry or
terms with only 180◦ antisymmetry then the discretization is unconditionally stable.
Proof. We prove the lemma for when there are an odd number of points in each row, so that
each row is centred at m = 0. If a linear one-step scheme has such a symmetry then it can be
written as
0=
X
j
aj vj0 ± aj v−j1 ,
where aj ∈ R; the plus is used for a symmetric scheme and the minus for an antisymmetric
115
Chapter 6: Analysis and Numerical Experimentation
scheme. Therefore if vmn = Aξ n eiωm
X
X
0 =A
aj eiωj ± Aξ
aj e−iωj + Aa0 (1 ± ξ),
j
j6=0
(6.1)
j
j6=0
=z ± z̄ξ,
(6.2)
where z is a complex function of ω and z̄ is its conjugate. Hence
z̄ |ξ| = ∓ = 1,
z
so the scheme is unconditionally stable. For the case where there is an even number of points,
1
with each row centred at m = − 12 , multiplying through (6.1) by e 2 iω will reveal the conjugate
pairs. In addition, there will be no purely real terms, as there are no points at the centre of the
grid.
Lemma 6.1.2. If a polynomial term has been discretized so that the discretization is either 180◦
symmetric or 180◦ antisymmetric then the discretization’s linearization is also 180◦ symmetric
or 180◦ antisymmetric respectively.
Proof. If a k th order polynomial term has been discretized in a symmetric or antisymmetric
manner then the discretization is a sum of terms of the form
k
Y
l=1
uil jl ±
k
Y
l=1
u−il −jl .
The linearization of this expression about the solution umn = ρ is given by
! k
k
k
Y
Y
X
∂
(uil jl + vil jl ) ± (u−il −jl + v−il −jl ) = ρk−1
(vil jl ± v−il −jl ).
∂
l=1
l=1
=0 umn =ρ
l=1
Therefore the linearization inherits the symmetry of the original scheme.
When searching for numerical schemes that preserved the CLaws of KdV in §5.3, we required
that the discretizations for KdV had to have 180◦ antisymmetry. Therefore, by combining the
above two lemmas:
Corollary 6.1.3. The linearizations of all the one-step schemes found in §5.3 are unconditionally stable.
Hence the only remaining scheme, that we need to study the linear stability of, is the three-step
explicit scheme that preserves the first and third CLaws (5.37). The linearization of that scheme
is
1
1
1/3v01 − 1/3v0−2
2 + 2η v00 + −2η − 2 v0−1
1 ρ (v−1−1 + v−10 − v1−1 − v10 )
−4
+
+
0=
ν
µ
µ3
− 12 − η v−20 + ηv−2−1 + 12 v−10 + 21 v−1−1 − 12 v10 − 12 v1−1 − v20 η + 12 + η v2−1
+
.
µ3
(6.3)
Note that η = − 14 makes uxxx symmetric and minimizes the LTE. The trial solution vmn =
Aξ n eiωm results in the condition
0 = ξ 3 + ξ 2 (A + Bi) + ξ(−A + iB) − 1,
116
(6.4)
Chapter 6: Analysis and Numerical Experimentation
where
A = µ3ν3
1
2
+ 2η (1 − cos(2ω)) ,
− 2ρ
B = 3ν
µ
4 sin(ω) +
So
|A| ≤
3ν
µ3
|B| ≤
3ν
µ3
1
µ2
− sin(ω) +
1
+ 2η ,
2
2ρ µ2 +
4
1
2
sin(2ω)
.
√ 3 3
.
4
When η = − 14 , i.e. the case when the discretization for uxxx is symmetric in time and each row
is antisymmetric (and the LTE is minimized), A = 0. The roots of (6.4), ξi ∈ C, must satisfy
a) − ξ1 ξ2 ξ3 = 1,
b) ξ1 ξ2 + ξ1 ξ3 + ξ2 ξ3 = −A + iB,
c) − (ξ1 + ξ2 + ξ3 ) = A + Bi.
(6.5)
We want to know what conditions A and B need to satisfy to ensure that |ξ| ≤ 1. Clearly
A = 0 and B = 0 provides such a condition but this is not possible. To find a necessary but
not sufficient condition for stability, we assume that stability holds, so |ξi | ≤ 1, and find the
conditions A and B must, as a consequence, satisfy. Equation (6.5 a) implies that |ξi | = 1.
Hence (6.5 b) and (6.5 c) give the necessary conditions that |A + iB| ≤ 3 and |A − iB| ≤ 3. So
if stability holds then
A2 ≤ 9,
B 2 ≤ 9 − A2 .
Clearly when η = − 41 , the first of these conditions is satisfied and the condition on B is less
stringent, suggesting that this is the most stable parameter choice. So assuming this, a necessary
but not sufficient condition for stability is that
√ 3 3
3ν 2umax 2
≤ 3,
(6.6)
µ
+
3
µ
4
4
where we have assumed that |ρ| ≤ umax . This is very similar to the stability condition found
for the three-parameter family of explicit schemes (2.33), even though this is a three-step rather
than a two-step scheme. However, if a sufficient condition can be found, it may be far more
restrictive.
6.2
A Nonlinear Stability Consideration
We now consider the stability of the eight-point schemes (5.21), (5.23) and (5.25) on a periodic
domain so that u1j = uM j for all j. Each of these schemes exactly preserve the structure of the
second CLaw so that it can be written in the form

!2 
p
X
f2 = 0,
(Sn − I)  12
αi uij  + (Sm − I)F
(6.7)
i=1
where 0 < αi ∈ R and p < M and in these cases p is either 2 or 4. Summing over the domain
then yields the conserved quantity
!2
p
M
−1 X
X
αi um+i,n+j
= A,
(6.8)
m=0
i=1
117
Chapter 6: Analysis and Numerical Experimentation
where A is a constant. We now seek to show that as a consequence of (6.8) and the periodic
boundary conditions, if p divides M then |uij | must be bounded.
The case p = 1 is trivial and occurs for the ten-point norm-preserving scheme. So let us begin
with p = 2. We wish to show that if u1j ≥ B1 > 0 becomes large enough then there exist
numbers Bi > 0 such that uij ≤ −Bi for i even, and uij ≥ Bi for i odd. Therefore if M is
even, uM j < 0. But the periodic boundary conditions imply that uM j = u1j > 0; hence, the
assumption that u1j ≥ B1 must be false.
For p = 2 we can use α and β in preference to α1 and α2 and, as we are at a fixed time level,
the js have been dropped, so (6.8) implies that
√
√
− A − αum ≤ βum+1 ≤ A − αum , ∀m.
(6.9)
Let us assume that m is odd and um ≥ Bm > 0. Then (6.9) implies that
√
√
um+1 ≤ β1 ( A − αum ) ≤ β1 ( A − αBm ).
So, to ensure that um+1 ≤ −Bm+1 ≤ 0, Bm must satisfy
√
1
so Bm ≥
β ( A − αBm ) ≤ −Bm+1 ,
√
1
α(
A + βBm+1 ).
Similarly, suppose that m is even and that −um ≥ Bm > 0. Hence (6.9) yields
√
√
1
1
β − A + αBm ≤ β − A − αum ≤ um+1 ,
so for um+1 ≥ Bm+1 , Bm must satisfy
Bm ≥
√
1
α(
A + βBm+1 ).
As the condition is the same for both cases, we have a well defined sequence to satisfy to ensure
that uij oscillates. Given Bm > 0, we can then find a suitable Bm−1 > 0, etc. until I find B1 > 0.
This sequence yields
!
M
−2 i
√
M −1
√
√
X
√
β
β
β
β
1
A
A
A
B1 ≥ α ( A + βB2 ) ≥ α + α α + α B3 ≥ . . . ≥ α
BM .
+
α
α
i=0
Hence if B1 ≥
√
PM −2 β i
A
α
i=0
α
then BM ≥ 0, and so if M is even then uM j ≤ 0, which is a
contradiction. Thus we have shown the solution must be bounded by
M −1

β

√ 1−

α
 A
, α 6= β;
α
β
|uij | <
1− α


 √A
α = β.
α (M − 1)
For p > 2, we only suggest how the proof should work. Equation (6.8) implies that
!2
p
X
αi um+i,n+j
≤ A,
∀m.
(6.10)
i=1
If u1j → ∞ then
p
X
i=2
αi uij → −∞,
118
(6.11)
Chapter 6: Analysis and Numerical Experimentation
2
P
p+1
α
u
≤ A. Because
in order to satisfy (6.10). Now we also have the condition that
i−1
ij
i=2
Pp
all the αi are positive, (6.10) implies that i=2 αi−1 uij → −∞, and so u(p+1)j → ∞. In the
same manner, we can move along the domain and use (6.10) to show that u(p+i)j must have the
same asymptotic behaviour as uij , either tending to plus infinity, minus infinity, or remaining
bounded. We can now relabel our periodic domain so that u1j → ∞ and upj → −∞ or is
bounded. Thus if p divides M we have a contradiction, because uij = uM j . Therefore, the
solution must remain bounded. However this gives us no information on the size of this bound,
which could be very large.
Therefore, the presence of a CLaw in the form (6.7), combined with periodic boundary conditions
with the correct number of points, prevents the modes that cause the scheme to blow up from
growing.
6.3
Numerics
In this section, we compare some of the different numerical schemes we have found by using
them to solve the one-soliton initial value problem for KdV, 0 = ut + uux + uxxx . The exact
solution to this problem on an infinite domain is
√
u(x, t) = 3c sech2 2c (x − ct) .
For the numerics we use a periodic domain with x ∈ [−20, 20] and let t = [0, 2]; M denotes the
40
number of spatial steps and N denotes the number of time-steps, so µ = M
and ν = N2 . The
resulting solution profiles are compared with that of the exact, infinite domain, solution. This is
fine provided that ct is small enough that the soliton does not get close to the boundary. Since
we are interested in how the different schemes preserve the CLaws, we will use fairly coarse
discretizations to emphasize the qualitative differences between the different schemes. In order
to solve the implicit schemes we used fsolve in matlab, which was allowed to run until the
error reached the default tolerance, so that the difference between the schemes should be due to
their different discretizations rather than due to the nonlinear solver. We calculate the error in
preserving the different conservation laws by approximating the conserved quantities by
X
X
X
1 3
1
µ
um,n ,
µ
u2m,n ,
µ
3 um,n + µ2 um,n (um−1,n − 2um,n + um+1,n )
m
m
m
at each time step.
Not only do we compare the schemes with each other but also with the eight-point multisymplectic method of Ascher and McLachlan [12, 11],
1
0 = 8ν
(Sn − I)(u−20 + 3u−10 + 3u00 + u10 ) +
+
1
64µ
1
2µ3 (Sn
+ I)(−u−20 + 3u−10 − 3u00 + u10 )+
(u00 + u10 + u01 + u11 ) − (u−20 + u−10 + u−21 + u−11 )2 ,
2
and their narrow box scheme
1
1
0 = 2ν
(Sn −I)(u−10 +u00 )+ 8µ
(Sm −I)((u−11 +u−10 )2 )+ 2µ1 3 (Sn +I)(−u−20 +3u−10 −3u00 +u10 ),
which was found using a finite volume discretization. In particular they conducted numerics for
the alternative form of KdV,
0 = ut + uux + δuxxx ,
119
Chapter 6: Analysis and Numerical Experimentation
with δ = 0.0222 and initial condition cos(πx) on a periodic domain with x ∈ [0, 2]. They showed
that the multisymplectic scheme stayed remarkably smooth despite a very coarse discretization
whereas the other schemes couldn’t cope with the discretization and the norm-preserving scheme,
even though it didn’t blow up, resembled noise. They concluded that this was due to the
compactness of the discretization used for the multisymplectic scheme and that its linearization
is the most accurate unconditionally stable box semi-discretization at the steady state [12]. Thus
it will be interesting to replicate this experiment for the schemes that we have found. In addition
this problem is interesting because the nonlinear part of KdV will be more significant than in the
soliton problem (δ = 1), and the problem becomes close to Burger’s equation, which develops
shocks. This should be a good problem to see how the different schemes handle the nonlinear
part of KdV. (Note that for this form of KdV the density of the 3rd CLaw is 31 u3 + δuuxx ).
The first case to examine is the original eight-point schemes found in §5.3.1.1 (the first scheme
(5.20), second (5.22) and third (5.24)), with different values of ψ which affect how ut is discretized. For ψ = 0, the ut term uses only the four central points and so is the most compact
discretization possible. However, what is of interest is whether it is important to discretize the
structure of second CLaw exactly (i.e. so that there are no additional terms in the discrete density and flux that tend to zero as the step sizes tend to zero) or just in characteristic form. For
ψ = 0, ψ = 1/4, ψ = 1/6, the first, second and third schemes conserve the second CLaw with no
additional terms in the fluxes respectively, but for all other values additional ν terms will appear
in the flux and µ terms in the density which will need to vanish in the limit. Finally, ψ = 1/8
will cause the schemes to differ from the multisymplectic scheme only in how they discretize
the uux term. The value ψ = −1/24 was suggested by Ascher and McLachlan, to increase the
accuracy of the uet term. For the single soliton solution the most accurate of the choices of ψ
for each scheme is shown in Figure 6.1. The only case where preserving the structure of the
CLaw exactly yields the best discretization is for the first scheme. For the other two schemes
this is not the case, the most remarkable example being the second scheme with ψ = 1/6, which
appears to preserve the second and third CLaws better than all the other schemes, even though
the discretization only preserves the CLaw in characteristic form. It is not clear why this scheme
is performing so well. It is also of note that as ψ increases, the speed of the numerical soliton
increases; in §6.4 we seek to explain this using the dispersion relation of the linearized schemes.
Finally, for the δ = 0.0222 example with a very coarse grid (for which the narrow box scheme
blows up) all the schemes except the second and third schemes with ψ = 1/4 coped with the
discretization and produced relatively smooth solutions. The most accurate schemes in this case
all had ψ = −1/24 (shown in Figure 6.2), though the compact choice of ψ = 0 was not much
worse at preserving the conserved quantities. The first scheme out-performs the third which in
turn out-performs the second scheme. Having ψ = 1/6 in the second scheme no longer performs
as well. If we double the number of time and spatial steps then the narrow box scheme is able to
out-perform all the schemes at conserving the second and especially the third CLaw (see Figure
6.3).
It appears that preserving the structure of the second CLaw exactly does not makes a significant
difference compared to just preserving it in characteristic form. However, the presence of this
conservation law along with the compactness of the discretization seems to enable our finite
difference schemes, like the multisymplectic scheme, to cope with very coarse discretizations and
remain smooth. What is also clear is that the two different problems we have examined favoured
different schemes - the narrow box scheme was the most accurate for the δ = 0.0222 case but
the second scheme with ψ = 1/6 was the best at preserving the second CLaw for the soliton
120
Chapter 6: Analysis and Numerical Experimentation
problem.
Having said all this, in §5.3.1.1 we noted that when the structure of each CLaw was preserved
f2 =
exactly, the density of the second CLaw is half the density of the first CLaw squared, i.e. G
2
1f
1
f
2 G1 . So by taking the results of each scheme and forming the average, u(xm − 2 µ, tn ) ≈ G1 ,
we should have an alternative scheme for KdV that preserves the first and second conserved
quantities exactly. The results of doing this are shown in Figures 6.4 and 6.5. It is clear that the
second conserved quantity has been conserved exactly by all three averaged schemes. From the
image of the profiles, we can see that the solutions from the averaged schemes have a slightly lower
soliton that is traveling slightly slower than the actual solution, but this is the correct qualitative
behaviour. In addition all three averaged schemes preserve the third CLaw slightly better than
the non-averaged schemes. However the narrow box scheme still significantly outperforms them
at this. Of particular note is the second scheme (yellow) in Figure 6.5; the original scheme fails
to cope with the coarse discretization and has become a wave that resembles noise with a large
amplitude, but the averaged scheme is still exactly preserving the second conserved quantity
and its amplitude remains correct. However, it doesn’t look smooth. This illustrates that even
though the averaged scheme looks like it is unconditionally nonlinearly stable, the scheme it is
derived from may become unstable, in which case it will be impossible to calculate the averaged
scheme.
121
Chapter 6: Analysis and Numerical Experimentation
122
Figure 6.1: Single soliton solution of KdV (c = 8) with eight-point schemes, that preserve the 1st and 2nd CLaws in characteristic form. The schemes
displayed have the values of ψ (shown in brackets) that resulted in the most accurate preservation of the second CLaw. M = 400, N = 100, t = 2.
Chapter 6: Analysis and Numerical Experimentation
123
Figure 6.2: KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 2nd CLaws, with the values of ψ (shown in brackets) that resulted in
the most accurate preservation of the second CLaw; M = 60, N = 1500, t = 10
Chapter 6: Analysis and Numerical Experimentation
124
Figure 6.3: KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 2nd CLaws, with the values of ψ (shown in brackets) that resulted in
the most accurate preservation of the second CLaw; M = 120, N = 3000, t = 10
Chapter 6: Analysis and Numerical Experimentation
125
Figure 6.4: Single soliton solution of KdV (c = 8) with eight-point schemes that exactly preserve the structure of the 1st and 2nd CLaws. The dashed and
f1 . The initial condition should provide an estimate
dotted lines in the discretization are formed by taking the average of each scheme using um−1/2,n = G
of the position of the actual soliton. The values of ψ are shown in brackets. M = 800, N = 1000, t = 10.
Chapter 6: Analysis and Numerical Experimentation
126
Figure 6.5: KdV (δ = 0.0222 ) with eight-point schemes that exactly preserve the structure of the 1st and 2nd CLaws. The dashed lines in the discretization
f1 . The values of ψ are shown in brackets. M = 800, N = 2000, t = 10.
are formed by taking the average of each scheme using um−1/2,n = G
Chapter 6: Analysis and Numerical Experimentation
Having examined the original three schemes from §5.3.1.1, we now examine the output from
the additional schemes found in Table 5.1. We reiterate that the original three schemes (just
discussed) are special cases of these schemes: the first scheme (5.20) is the second scheme in the
table with = 1 = 22 , the second scheme (5.22) is the ninth scheme in the table with κ = 1/12
and the third scheme (5.24) is the tenth scheme in the table. For the numerics we use
1
uet = ν1 (Sn − I) 12 u−10 + 12 u00 , ug
xxx = 2µ3 (Sm − I) ((Sn + I) (u−20 − 2u−10 + u00 )) ,
so that both expressions are as compact and symmetric as possible. Hence any differences
between the schemes are due to how they handle the uux term. However, as we have seen, this
may not be the optimal discretization of ut for the given problem.
The most intriguing question about the schemes is: what is the effect of being skewed? We
examined the schemes with the parameter choices we suggested in §5.3.1.1. We found that
different parameter choices resulted in the numerical solitons traveling at different speeds and
the CLaws were conserved differently. From the numerical experiments we see that schemes that
are angled in the upwards direction travel faster than equivalent schemes angled downwards.
In Figure 6.6 we demonstrate the effect of the different skews on the schemes. The first four
schemes and the narrow box scheme, all share the same linearization; so the differences between
the different schemes are due to the nonlinearity. The first and third schemes are partners (the
nonlinear term in one scheme can be obtained from the other by a reflection in the line n = 1/2),
the first scheme is angled downwards and the third scheme is angled upwards. We can see that
the third scheme travels faster than the second and narrow box schemes, which are symmetric,
and these, in turn, travel faster than the first and fourth schemes which are angled downwards.
The multisymplectic, ninth and tenth schemes are all symmetric but have different linearizations;
they are traveling at different speeds but they are all slower than the actual solution. This picture
shows that angling a scheme upwards increases the speed of a soliton. It should also be noted
that, for this example, it is the fourth scheme that is preserving the second CLaw the best, even
though other solution profiles are closer to the correct position. The other noticeable property
of the skewed schemes is that the first conserved quantity is no longer exactly conserved. This
is because we have no longer constructed the schemes to exactly preserve the structure of the
first CLaw, but rather, have only insisted on the discretization for KdV being in the kernel of
the discrete Euler operator.
We also attempted to adjust the parameters to find schemes that had the soliton traveling close
to the exact solution, and parameter choices that minimized the error in the second CLaw for the
soliton solution. However, unlike for the explicit schemes we found in §2.5, matching the soliton
speed, did not result in the best conservation properties. In Figure 6.7 we show the schemes, with
the parameters we have found by experimentation, that preserve the second conserved quantity
the best for the soliton with c = 8. However if we change the speed of the soliton, or the ratio of
the step sizes, then these are no longer the best schemes. This can be seen by comparing Figure
6.6 to Figure 6.8. The relative performance of the different schemes has changed and the only
difference between the two cases is the speed of the soliton. Therefore it is far from clear how
the parameters should be chosen. It is also worth noting, that in Figure 6.8, schemes with the
same linearization are traveling at approximately the same speed.
When studying the case with δ = 0.0222 , the schemes that did badly, and failed to cope with
the very coarse discretization (M = 60 and N = 1500), were those schemes that had the same
linearization as the narrow box scheme. The rest produced relatively smooth solution profiles.
The schemes which now conserve the second CLaw the best are those that are as compact
127
Chapter 6: Analysis and Numerical Experimentation
as possible. (We only examined the parameter choices that were used for the investigation of
the soliton problem). This is shown in Figure 6.9 where we have refined the mesh to include
the narrow box scheme, which now preserves the CLaws better than any of the schemes we
have found. The schemes that have the same linearization as the narrow box scheme (Figure
6.10), conserve the second CLaw slightly worse than the compact schemes, but they they are
significantly better at preserving the third Claw in this situation, though again, not as well as
the narrow box scheme.
Overall, the best schemes seem to be better at preserving the CLaws than the multisymplectic
scheme for the soliton case and only slightly better for the δ = 0.0222 problem. For the soliton
solution, we can find parameters so that the schemes conserve the second CLaw more accurately
than the narrow box scheme, for given step sizes and speed. However, for the δ = 0.0222 case the
narrow box scheme is significantly better at preserving the second and especially the third CLaws,
but it is less able to cope with coarse discretizations than the conservative schemes. Hence being
compact appears to be an important property for dealing with very coarse discretizations or
when the nonlinear term is large.
128
Chapter 6: Analysis and Numerical Experimentation
129
Figure 6.6: Single soliton solution of KdV (c = 8) with eight-point schemes skewed in different directions that preserve the 1st and 2nd CLaws in
characteristic form; M = 400, N = 100, t = 2
Chapter 6: Analysis and Numerical Experimentation
130
Figure 6.7: Single soliton solution of KdV (c = 8) with eight-point schemes with the parameters found to best preserve the second CLaw; M = 400,
N = 100, t = 2
Chapter 6: Analysis and Numerical Experimentation
131
Figure 6.8: Single soliton solution of KdV (c = 1) with eight-point schemes; M = 400, N = 100, t = 2
Chapter 6: Analysis and Numerical Experimentation
132
Figure 6.9: KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 2nd CLaws in characteristic form; M = 60, N = 1500, t = 10
Chapter 6: Analysis and Numerical Experimentation
133
Figure 6.10: KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 2nd CLaws in characteristic form with the same linearization as the
narrow box scheme; M = 120, N = 3000, t = 10
Chapter 6: Analysis and Numerical Experimentation
We now investigate the ten-point schemes that preserve the first and second CLaws in characteristic form. We reiterate that for all these schemes we use
uet = ν1 (Sn − I)u00 ,
ug
xxx =
1
4µ3 (Sn
+ I)(Sm − I)(u−20 − u−10 − u00 + u10 ),
so that we can compare how they handle the nonlinear term. However, as we demonstrated
when examining the eight-point schemes, a different discretization for the ut term can make a
significant (potentially beneficial) difference for a specific problem.
The most striking result is that all the parameter choices for the first scheme preserve the
second conserved quantity exactly (see Figures 6.11 and 6.13), though they do display different
sensitivities to the coarseness of the mesh, for example the blue scheme (the elongated scheme)
in Figures 6.11 has not coped with the chosen step sizes. This is despite the fact some of these
schemes are not symmetric in time for the uu
gx term and are thus counter-intuitive. Having said
this, the only scheme that appears to preserve the first CLaw exactly is the symmetric scheme,
which is the norm-preserving scheme in [11, 12]. Also of note, for the one soliton problem,
the scheme which is angled upwards (red scheme) travels slightly faster than the symmetric
scheme, which in turn is faster than the scheme that is angled downwards (green scheme). For
this problem, the scheme that is compact and very slightly angled upwards (the cyan scheme)
appears to be conserving the third CLaw the best. In the δ = 0.0222 problem it is the downwards
scheme that performs best, but in both cases the narrow box scheme is preserving the third CLaw
the best. For the very coarse δ = 0.0222 problem, M = 60 and N = 1500, the schemes don’t
blow up but their solution profiles merely resemble noise. Increasing the number of points to
M = 120 and N = 3000, shown in Figure 6.13, we can see that, even though the overall shape
of the solution is roughly correct, small sawtooth waves have developed. Thus the extra-spatial
step used in these schemes has allowed for the existence of these spurious waves which, with a
coarse grid, can cause the solution to eventually resemble noise.
The results of the other schemes are shown in Figure 6.12. This figure displays the schemes for
the parameter choices that were found to best preserve the second CLaw, with the chosen step
sizes, from the suggestions made in §5.3.1.2. Clearly these schemes do not preserve the second
CLaw exactly, and their performance is comparable to the eight-point schemes. Of interest is
the fact, for each of these cases, the best choice of parameters was for the scheme with had the
most compact and symmetric linearization. However, by playing with the parameters, schemes
can be found, for this problem, that preserve the second CLaw better than this choice; but there
seems to be little to be gained from this, due to the exact conservation of the second conserved
quantity attained by schemes from the first family.
The tenth and eleventh schemes’ discretized characteristic is not very compact and neither
scheme seems to preserve the second CLaw particularly well.
Overall, using the wider stencil enables us to find methods that preserve the second Claw exactly,
and so give us nonlinear stability, which is desirable. However the cost of the extra spatial step is
that it allows for the existence of spurious waves which, as the schemes are conservative, persist
and can break up the solution profile. Thus unless exact conservation can be attained, as in the
first family of schemes, it is preferable to use more compact schemes, as comparable conservation
properties can be attained.
134
Chapter 6: Analysis and Numerical Experimentation
135
Figure 6.11: Single soliton solution of KdV (c = 8) with the first ten-point scheme that preserve the 1st and 2nd CLaws in characteristic form; M = 400,
N = 100, t = 2
Chapter 6: Analysis and Numerical Experimentation
136
Figure 6.12: Single soliton solution of KdV (c = 8) with ten-point schemes that preserve the 1st and 2nd CLaws in characteristic form; M = 400, N = 100,
t=2
Chapter 6: Analysis and Numerical Experimentation
137
Figure 6.13: KdV (δ = 0.0222 ) with the first ten-point scheme that preserve the 1st and 2nd CLaws in characteristic form; M = 120, N = 3000, t = 10
Chapter 6: Analysis and Numerical Experimentation
The most curious schemes discovered in §5.3 were those that that preserve the first and third
CLaws in characteristic form (Table 5.3) which had skewed discretizations of ut . In Figures 6.14
and 6.15 the results of the first scheme are shown in green, this is the only symmetric scheme, the
schemes which are angled downwards are the second (cyan) and fourth schemes (red), and the
schemes which are angled upwards are the third (magenta) and the fifth (blue). The skewness
of the time discretizations imposes a step-size restriction. If ν = λµr , for consistency we require
r < 1, so the time steps need to be of a similar size or larger than the spatial steps. Changing
the value of the parameters in the second and third schemes had little effect on the solution
profile, though the parameter values that best preserved the third CLaw are shown in Figure
6.14. It is clear from this that the schemes where uet is angled downwards have the soliton
traveling too slowly and the schemes which are angled upwards have the soliton traveling too
fast. Additionally, the schemes which are angled downwards are preserving the CLaws better
than the upwards, and even the symmetric, schemes.
The step-size restrictions means that we are unable to directly compare these schemes with the
others for the coarse δ = 0.0222 example, so instead we have halved the space step and doubled
the time step so that M = 120 and N = 750. For this example the fifth scheme doesn’t cope
and neither does β = 0 in the third scheme; however, other values of β such as 1/48 do cope
and the narrow box scheme also copes. Does this suggest that a downwards angle is preferable
in this situation? To alleviate concerns about whether these schemes converge, we have refined
the step sizes in Figure 6.15 to show that all the schemes are behaving well despite the strange
discretizations. We also show the second and third schemes with the best values of β we found
at preserving the third CLaw in this situation. However, some values are much worse than those
shown, for example β = 0 in the second scheme and β = 1/96 in the third scheme have an error
in the third CLaw of around 3 × 10−3 , whereas the error in the multisymplectic scheme is about
1 × 10−3 .
138
Chapter 6: Analysis and Numerical Experimentation
139
Figure 6.14: Single soliton solution of KdV (c = 8) with eight-point schemes that preserve the 1st and 3rd CLaws in characteristic form; M = 1600,
N = 80, t = 2
Chapter 6: Analysis and Numerical Experimentation
140
Figure 6.15: KdV (δ = 0.0222 ) with eight-point schemes that preserve the 1st and 3rd CLaws in characteristic form; M = 800, N = 4000, t = 10
Chapter 6: Analysis and Numerical Experimentation
We now examine the two ten-point schemes that preserve the third CLaw in characteristic form,
(see Table 5.4). Schemes with the same linearization are depicted with the same line-styles in the
following figures. The solid line is the choice that makes the schemes symmetric and compact.
The dotted line is the case where the central terms have been removed, the dashed scheme is
the case where we have sought to give equal weight to the outer and inner terms, and finally
the dash-dotted case was chosen by trying to improve the preservation of the third conserved
quantity for the first scheme, for the soliton problem.
Figure 6.16 shows the results from the single soliton problem. Clearly, of the parameter choices
not found by experimentation, the symmetric and compact choice preserves the 3rd CLaw the
best for each family. The second scheme with this parameter choice is Furihata’s scheme which
preserves the 3rd Claw exactly. However, by experimenting, we have found a parameter choice
(dash-dotted) for the first scheme that slightly outperforms the symmetric case. Not surprisingly,
the least compact schemes (the dotted ones) are doing a poor job at preserving the CLaws.
Though, for this choice of step sizes and soliton speed, their solution profiles are closest to the
position of the actual soliton. Apart from Furihata’s scheme, the narrow box scheme preserves
both the second and third CLaws in a similar manner to the best of the other schemes, even
though it is not constructed to do so.
For the δ = 0.0222 problem, with M = 60 and N = 1500 the symmetric and the dash-dotted
schemes cannot cope with the coarseness of the discretization, however, for the schemes that
do cope the resulting solutions are no longer smooth. If we refine the mesh to M = 120 and
N = 3000 shown in Figure 6.17 all the schemes cope. It is interesting to note that the schemes
which have the same linearization but are from different families, and so preserve the third CLaw
differently, have very similar behaviour. There is one major difference: the schemes from the
second family (blue) are very smooth, whereas some of the first family’s solutions have developed
small sawtooth waves. This is similar to the ten-point schemes that preserve the first and second
CLaws and it is what we expect from the wider stencil, so it is remarkable that the second family’s
solutions remain so smooth. The good performance of the second family compared with the first
family for this problem could be because schemes from this family have underestimated the
second conserved quantity, thus maybe introducing some dissipation, whereas schemes from the
first family have increased the second conserved quantity slightly.
By increasing the number of points in the discretizations, we have been able to find a scheme
that preserves the 3rd CLaw much better than any of the eight-point schemes. However, the
cost of this is that the resulting schemes do not cope with the very coarse discretization where
the nonlinear term is large.
The final scheme we need to study is the three-step explicit scheme that preserves the first and
third CLaws. As it is explicit, we will use more time steps than the implicit schemes so as
to ensure stability. We found that we needed ν = 14 µ3 for the three-step scheme to be stable,
which is similar to the necessary condition we found (6.6); this is similar to the condition for
the three-parameter family, where we found using ν = 13 µ3 produced stable results. With the
increased number of time steps used for Figure 6.18, the explicit scheme takes approximately
the same amount of time to solve as the implicit schemes, which use ν = 15 µ. We also include,
from Chapter 2, the three-parameter explicit scheme with (0, 1/6, 0) that preserves the first two
CLaws and does a good job preserving the third CLaw. We compare these two schemes with
the symmetric ten-point implicit schemes that preserve the first and third CLaws. From Figure
6.18 we can see the three-step explicit scheme does a better job at preserving the third CLaw
141
Chapter 6: Analysis and Numerical Experimentation
than the three-parameter scheme and the first implicit scheme. In fact, the performance of the
three-step explicit scheme also exceeds that of the narrow box scheme (not shown in this figure).
It is also of note that the three-parameter scheme is actually doing a better job of preserving the
third CLaw than the first implicit ten-point scheme, even though it was not designed to do this.
When attempting to use the explicit schemes to solve the δ = 0.0222 problem we encounter a
problem: we have to use a very large number of time steps. Because the explicit schemes both
use five points in space they allow the parasitic waves to form, so to get a smooth solution we
need a small spatial step and hence a very small time step. Using ν = 1/4µ3 to ensure stability
means that when M = 80 in Figure 6.19 we need 256000 time steps for the explicit schemes,
whereas we can chose to use N = 150 for the implicit schemes. The explicit schemes take far
longer than the implicit schemes and the resulting profile is still far from smooth. Thus the
explicit schemes are not practical for this problem.
142
Chapter 6: Analysis and Numerical Experimentation
143
Figure 6.16: Single soliton solution of KdV (c = 8) with ten-point schemes that preserve the 1st and 3rd CLaws in characteristic form; M = 400, N = 100,
t=2
Chapter 6: Analysis and Numerical Experimentation
144
Figure 6.17: KdV (δ = 0.0222 ) with ten-point schemes that preserve the 1st and 3rd CLaws in characteristic form; M = 120, N = 3000, t = 10
Chapter 6: Analysis and Numerical Experimentation
145
Figure 6.18: The three-step explicit scheme that preserves the 1st and 3rd CLaws compared with the ten-point implicit schemes that preserve the same
CLaws, and with the explicit three-parameter family that preserves the first and second schemes. The implicit schemes used M = 400, N = 100 and the
explicit schemes used M = 400, N = 8000.
Chapter 6: Analysis and Numerical Experimentation
146
Figure 6.19: The three-step explicit scheme that preserves the 1st and 3rd CLaws (for δ = 0.0222 ) compared with the ten-point implicit schemes that
preserve the same CLaws, and with the explicit three-parameter family that preserves the first and second schemes. The blue and magenta profiles are
very close to one another. The implicit schemes used M = 80, N = 150 and the explicit schemes used M = 80, N = 256000.
Chapter 6: Analysis and Numerical Experimentation
6.4
Numerical Dispersion Relations
In order to try to better understand the behaviour of the schemes we have numerically investigated, we linearize them about the constant solution u(x, t) = ρ and study their numerical
dispersion relations. The linearizations and their dispersion relations are shown in Appendix A
in Tables A.1, A.2, A.3 and A.4. The numerical dispersion relation is calculated by using the
trial solution vmn = ei(kµm+ωνn) in the linearized schemes. The trial solution is 2π-periodic so
we only need to consider
−π ≤ µk ≤ π,
−π ≤ νω ≤ π.
However, before plotting the dispersion relations for the the linearized KdV equation, to understand the effects of the skew derivative, let us consider the one-way wave equation ut − cux = 0.
Its dispersion relation is ω = ck and so its phase velocity is cp ≡ ωk = c and its group velocity is
cg ≡ dω
dk = c. We now consider the discretization
0=
1
2ν (Sn
− I)(u−10 + u00 ) −
cα
µ (u01
− u−10 ) −
c(1−α)
(u00
µ
− u−11 ),
so for α = 0 the u
fx term is angled downwards, for α = 12 the scheme is symmetric, and for α = 1
it is skewed upwards. We can now calculate the dispersion relation for this equation and see
the effects of changing α on the phase and group velocities. The numerical dispersion relation
is thus
kµ
kµ
c
cos
−
sin
cos ων
0 = ν1 − µc (2α − 1) sin ων
2
2
µ
2
2 .
If cos kµ
= 0 then cos
2
relation as
ων
2
= 0 so ω = k = 0. We can therefore consider the dispersion
ω=
2
ν
arctan
cν
µ−νc(2α−1)
tan
kµ
2
.
Hence the numerical phase velocity and group velocity are
γ(α)
kµ
2
2 sec2
arctan γ(α) tan kµ
cp = kν
, cg = µν ,
2
2
kµ
1+ γ(α) tan
(6.12)
γ(α) =
cν
µ−νc(2α−1) .
2
Now because tan and arctan are monotonic functions, the dispersion relation will have the correct
shape provided that µν > c(2α − 1) and if this case the sign of the group and phase velocities
will be preserved. Note that the symmetric case, α = 21 , will preserve the correct shape for all
values of c. In addition
dcp
2γ(α)2
kµ
2
2 tan
=
> 0,
dα
kν
2
kµ
1+ γ(α) tan
2
because sgn(1/k) = sgn(tan(2k/2)). Therefore the more the scheme is angled upwards the
greater the phase velocity of the numerical wave form which we can see demonstrated in Figure
6.20(c). However this is not true for the group velocity which can be seen in Figure 6.20(b).
We will now examine the numerical dispersion relations for the linearized KdV equation
0 = ut + ρux + δuxxx ,
which has the dispersion relation ω = δk 3 − ρk. We plot figures for two different cases: firstly
ρ = δ = 1 with µ = 2/15 and ν = 13 µ3 , and secondly ρ = 1, δ = 0.05, µ = 2/15 and ν = µ.
147
Chapter 6: Analysis and Numerical Experimentation
(a) Dispersion Relation
(b) Group Velocity
(c) Phase Velocity
Figure 6.20: Dispersion relations of numerical schemes with different values of α for the one-way
wave equation (µ = 0.1, ν = 0.075)
The first dispersion relation we examine is for the first three schemes that preserve the first and
second CLaws (5.20, 5.22, 5.24) and the effect of different discretizations of ut . The first thing
to note is that the different schemes are indistinguishable for the same value of ψ so Figure 6.21
only depicts the case of the first scheme (5.20) with = 0. We note that ψ = 1/8 yields the
uet term from the multisymplectic scheme, and ψ = 0 yields the narrow box scheme. For both
cases ψ = −1/24, 0, and 1/8 all retain the correct qualitative shape of the dispersion relation,
with the ψ = 0 case following the exact relation most accurately. For ψ = 1/6 and ψ = 1/4,
the dispersion relation no longer has the correct shape; for large k there exist two values of ω
that satisfy the dispersion relation, and the additional high frequency term is referred to as a
parasitic wave. For Figure 6.21(b), if δ is increased the the parasitic waves disappear for the
ψ = 1/6 case, but they persist for the ψ = 1/4 case at least until δ = 10. This shows that
compactness appears to be more important than preserving the exact structure of the second
CLaw, and explains the superior performance of the ψ = 0 and ψ = −1/24 schemes when close
to the Burger’s equation in Figure 6.2. Additionally, Figure 6.21 shows that as ψ increases the
numerical dispersion relation steepens, hence the phase speed and group velocity increase with
ψ. This explains why increasing ψ resulted in faster solitons.
We then examined the dispersion relations for the rest of the eight-point schemes that we numerically examined that preserve the first and second CLaws. All these schemes had the most
2
compact and symmetric ut (ψ = 0) and uxxx discretizations. For the ρ = δ = 1, µ = 15
and
1 3
ν = 3 µ case the schemes were indistinguishable, showing the same behaviour as the ψ = 0
2
scheme in Figure 6.21(a). However in the ρ = 1, δ = 0.05, µ = 15
and ν = µ case, the schemes
are qualitatively the same, however they now all approximately lie between the ψ = −1/24 and
the ψ = 1/8 (multisymplectic case) curves in Figure 6.21(b). Therefore the skewness of the u
fx
term appears to be having little effect in these cases; the compactness of the discretizations for
ut and uxxx seems to be the most important factor.
For the ten-point schemes shown in Figure 6.22 there is, as in the eight-point case, no qualitative
difference between the behaviour of the different schemes. However one can see that the most
upwardly skewed scheme (1/8, −1/24) from the first family will have a greater phase speed for
a given wavenumber than the other schemes. The major thing of note is that the shape of the
dispersion relation is no longer correct. For low frequencies there exist additional waves with
large values of |k|. As ω → 0 and |k| → πµ, these will resemble sawtooth waves in space. We can
148
Chapter 6: Analysis and Numerical Experimentation
(a) The ρ = δ = 1, µ =
case.
2
15
and ν =
1 3
µ
3
(b) The ρ = 1, δ = 0.05, µ =
case.
2
15
and ν = µ
Figure 6.21: Dispersion relations of the eight-point schemes that preserve the 1st and 2nd CLaws
with different values of ψ, for k ∈ [− πµ , πµ ] and ω ∈ [− πν , πν ]
(a) The ρ = δ = 1, µ =
2
15
and ν =
1 3
µ
3
case.
(b) The ρ = 1, δ = 0.05, µ =
ν = µ case.
2
15
and
Figure 6.22: Dispersion relations of the 10-point schemes that preserve the 1st and 2nd CLaws,
for k from − πµ to πµ and ω from − πν to πν
149
Chapter 6: Analysis and Numerical Experimentation
see these waves occurring in the solution profile in Figure 6.13. Having a non-compact spatial
stencil allows for the existence of spurious waves.
Figure 6.23 shows the dispersion relations for the schemes which preserve the first and third
CLaw with only eight-points. Not surprisingly, when ν = 31 µ3 only the symmetric scheme
bears any real resemblance to the actual dispersion relation, and is closer to the actual relation
than the multisymplectic case. The reason for this is because with this choice of step sizes the
schemes are inconsistent. As schemes with different linearizations, but from the same family, are
behaving very similarly, it is the skewed discretization of the ut term that is causing the observed
disparity. However, when µ = ν in Figure 6.23(b), the first three schemes are qualitatively
correct. The fourth and fifth schemes have the correct shape for larger values of δ. The more
skewed the schemes, the further they stray from the actual dispersion relation. The upwardly
skewed schemes (third and fifth) have too great a phase velocity and the downwardly skewed
schemes (second and fourth) are too slow. The effects of this in the nonlinear scheme can be
seen in the markedly different speeds the solitons travel at in Figure 6.15.
(a) The ρ = δ = 1, µ =
2
15
and ν =
1 3
µ
3
case.
(b) The ρ = 1, δ = 0.05, µ =
ν = µ case.
2
15
and
Figure 6.23: Dispersion relations of the eight-point schemes that preserve the 1st and 3rd CLaws
used in the numerics, for k from − πµ to πµ and ω from − πν to πν
Figure 6.24 depicts dispersion relations for the schemes that preserve the first and third CLaws
using ten points. Qualitatively, these dispersion relations are the same as those of the ten-point
schemes that preserve the first and second CLaws (see Figure 6.22), allowing for spurious waves
with low frequencies and high wave numbers. For ν = 31 µ3 , the different schemes give the
same result. However, for the µ = ν case (Figure 6.24(b)) differences can now be seen between
the different schemes which will affect the phase speed of the waves. However, what is most
interesting is that at low wave numbers the dispersion relation matches the actual relation very
closely, unlike any of the ten-point schemes that preserve the first and second CLaws.
Finally, Figure 6.25 shows the dispersion relation for the three-step explicit scheme that preserves
the third CLaw. Clearly, when µ = ν, the dispersion relation is very poor and this choice is not
allowed by the linear stability analysis. For ν = 1/3µ3 and ν = µ4 , there exist high-frequency
parasitic waves for all wave-numbers. Unsurprisingly, not being compact in space or (in this case
especially) time allows for the existence of parasitic waves.
Overall, the dispersion relations for the linearized schemes have confirmed the importance of
150
Chapter 6: Analysis and Numerical Experimentation
(a) The ρ = δ = 1, µ =
2
15
and ν =
1 3
µ
3
case.
(b) The ρ = 1, δ = 0.05, µ =
ν = µ case.
2
15
and
Figure 6.24: Dispersion relations of the ten-point schemes that preserve the 1st and 3rd CLaws
used in the numerics, for k from − πµ to πµ and ω from − πν to πν
(a) The ρ = δ = 1, µ =
ν = 13 µ3 case.
2
15
and (b) The ρ = 1, δ = 0.05, µ =
and ν = µ case.
2
15
(c) The ρ = 1, δ = 1, µ =
and ν = µ4 case.
2
15
Figure 6.25: Dispersion relation for the explicit three-step scheme that preserve the 1st and 3rd
CLaws, for k from − πµ to πµ and ω from − πν to πν
151
Chapter 6: Analysis and Numerical Experimentation
having as compact a discretization as possible in space and time, to prevent parasitic waves
from occurring. By choosing ψ = 0 for the eight-point schemes that preserve the first two
CLaws, we can match the dispersion relation in the same manner as the narrow box scheme
and better than the multisymplectic scheme. The dispersion relations indicate, along with our
investigation of the one-way wave equation, that skewing the u
fx term upwards results in a faster
phase speed, but does not appear to greatly affect the shape of the dispersion relation. It is also
of interest that the ten-point schemes that preserved the third CLaw had a closer match to the
actual dispersion relation than those ten-point schemes that preserved the second CLaw. This
leads to the question: when it is not possible to preserve all the CLaws of an equation, which
one is the most important to preserve?
6.5
Summary
Our numerics have shown that, for given parameter choices, the CLaw preserving schemes can
outperform a multisymplectic and a volume preserving scheme, at preserving the CLaw it is
designed to preserve. In particular by widening the stencil to five spatial points, schemes that
exactly preserve the second and third CLaws can be found. However this comes at a cost;
we have clearly demonstrated that compact schemes are to be preferred over wider stencils to
prevent the formation of parasitic waves. Overall the volume preserving method seems to do the
best job at approximately preserving all three physical CLaws.
The most interesting schemes that we have discovered are the three eight-point schemes that,
when we form their averages, preserve the second CLaw exactly. Hence we have the exact
conservation property along with compactness. Additionally, for a periodic domain, a restriction
on the number of spatial steps will ensure unconditional stability for these schemes. Also of
note was the good performance, for the soliton problem, of the three-step explicit scheme that
preserves the third CLaw.
A major unanswered question is, if there is a free parameter in the scheme, how should it be
chosen? In this chapter we were using a trial and error approach which is not very satisfying.
Insisting on compactness, either of the scheme or its linearization, is probably a reasonable
approach.
We also showed that when the uu
gx term was skewed, the schemes behaved reasonably well and
in some cases better than the symmetric case. Therefore further study needs to be done to
ascertain the effects of such discretizations and when such a discretization might be desirable.
The work in this chapter indicates that the more a scheme is skewed upwards, the faster the
phase speed is.
Finally, as it appears difficult to find a method that preserves multiple CLaws, for a given
problem, which is the most important CLaw to preserve?
152
Chapter 7
Conclusions and Further Study
7.1
The Best-Performing Methods
The motivation for studying CLaws of difference equations was to develop numerical methods for
a given PDE that locally preserves its CLaws. Therefore, to conclude, I present what I believe
to be the most significant methods for discretizing KdV that we have found. These schemes are
displayed in Figures 7.1 and 7.2, where we have used ν = 1/5µ and ν = µ respectively for the
implicit schemes.
Firstly, we found the three-parameter family of explicit schemes in Table 2.1 which preserve
the first second CLaws. This family included the Zabusky-Kruskal (Z-K) scheme; however, the
parameter choice (0, 1/6, 0) was found to be the best performing scheme, significantly outperforming the Z-K scheme at tracking the position of the single soliton and preserving the third
CLaw (see Figures 2.4 and 7.1). Its good behaviour is not understood. It uses a wider stencil
than the Z-K scheme to discretize the uux term that is not symmetric about its centre. The
advantage of being explicit means the scheme is cheap to compute; however, as the method uses
two time steps, the step size restriction ν = 13 µ3 means that for problems that require a fine
spatial mesh, it is not practical (see Figure 6.19).
The next notable schemes are the eight-point one-step schemes that preserve the first and second
CLaws. The best of these is the first scheme (5.21). If the average of the scheme is considered, so
that u(xm − 1/2µ, tn ) ≈ 12 (umn + u(m−1)n ), then the mass and momentum are exactly conserved.
This scheme has one parameter in the non-linear term. It is not clear how to choose this; however,
I believe there are two sensible a priori choices. The choice = 0 (the blue scheme in Figures 7.1
and 7.2) makes the nonlinear term as compact as possible. This scheme can cope well with very
coarse meshes; however, the numerical soliton travels slower than the actual solution and the
third CLaw is not preserved very well. The alternative is = −1/24 (the red scheme in Figures
7.1 and 7.2), which causes the linearization to be as compact as possible. This scheme’s soliton
travels faster, closer to the actual position, and the third CLaw is better preserved. The cost
of this is that the scheme cannot cope with very coarse discretizations. An alternative method
is to tune the parameter to best preserve the third CLaw, or some other property, for a given
problem and mesh size of interest.
With eight points we were also able to preserve the first and third schemes. Some of these
schemes were highly skewed, which resulted in the numerical soliton traveling either too fast
153
Chapter 7: Conclusions and Further Study
or too slow. Therefore, the most noteworthy scheme is the sole symmetric one (5.33) (the
cyan scheme in Figures 7.1 and 7.2). This scheme does not preserve the energy particularly
well (considering it preserves the third CLaw in characteristic form); however, it is interesting
simply because it shows that it is possible to preserve the third CLaw using a compact stencil
and so obtain better behaviour on a coarse grid. In fact, it has the same linearization as the
multisymplectic scheme against which we compared it. Also, for all these schemes that preserve
the first and third CLaws, as the time step increases compared to the spatial step, they preserve
the third CLaw better compared to other schemes. (The multisymplectic scheme’s performance
also improves especially compared to the narrow box scheme.)
The final scheme that deserves mention is the three-step explicit scheme (the magenta scheme in
Figure 7.1) that preserves the first and third CLaws (5.37). This suffers, like the three-parameter
family, from requiring a very large number of time steps if the spatial mesh needs to be fine.
However, when this stability requirement is satisfied, it preserves the third CLaw very well.
154
Chapter 7: Conclusions and Further Study
155
Figure 7.1: The most notable schemes. The dashed lines show the average given by u(xm − 1/2µ, tn ) ≈ 21 (umn + u(m−1)n ) for the eight-point schemes.
The implicit schemes used M = 400, N = 100 and the explicit schemes used M = 400, N = 8000.
Chapter 7: Conclusions and Further Study
156
Figure 7.2: The most notable schemes with ν = µ. The dashed lines show the average given by u(xm − 1/2µ, tn ) ≈ 12 (umn + u(m−1)n ) for the eight-point
schemes, M = 1600 and N = 80.
Chapter 7: Conclusions and Further Study
7.2
Summary and Some Open Questions
The main result of this thesis has been to form a method for characterizing CLaws of difference
equations. By showing a one-to-one correspondence between seeds of characteristics and CLaws
we can then prove the converse of Noether’s theorem for difference equations: that there is a
one-to-one correspondence between variational symmetries and CLaws for difference equations
that can be put into Kovalevskaya form.
We have then shown that the seeds of CLaws are in the kernel of the ALSC and have used this
to form a new method for constructing CLaws of quad-graph equations. This method requires
us being able to construct the CLaws from the seeds, which is a non-trivial procedure. In order
to do this, we require the difference equation to be in Kovalevskaya form. So an open problem
is how to the construct CLaws from seeds for quad-graphs, without needing to transform to
Kovalevskaya form. Another interesting question is: what else is in the kernel of the ALSC
other than seeds of CLaws? In particular, can an additional constraint on the solutions of the
ALSC be found, which will ensure that they are seeds of CLaws, thereby escaping the need to
reconstruct the characteristic?
Finally, a larger problem: can a characteristic for CLaws of some class of implicit difference
equations be found? This is an interesting question because evolution equations (which are in
Kovalevskaya form) can be discretized by fully implicit methods, as we do in this thesis.
The second focus of this thesis was on developing a method to integrate a partial difference
equation numerically so as to locally preserve as many CLaws as possible. The method developed is a brute force approach that is applicable to PDEs in two dimensions with polynomial
nonlinearities on a fixed mesh. The method requires solving a large overdetermined system of
quadratic equations; this is its main limiting factor. We currently calculate the Groebner basis
to solve the system, which is very expensive in time and memory. Thus, an important area for
future research is to find a more efficient method for solving this highly-structured system of
equations.
The method is practical for searching for one-step methods, for which we were able to find new
and known methods that discretize KdV to preserve the first and second CLaws together and
the first and third CLaws together. It is an open problem as to whether the second and third
CLaws of KdV can be preserved together. We suspect that if it is possible, this will require at
least two time steps; this has the potential to allow parasitic waves to form. However, the extra
constraint on the behaviour, that comes from preserving the additional CLaw may prevent them
from occurring. Since we are unable, at least currently, to preserve all the CLaws we desire, this
raises the question: what is the most important CLaw to preserve?
The methods that we have found behave comparably to a multisymplectic scheme, and can
outperform it for conserving a given CLaw. This suggests that if no multisymplectic structure
is known for an equation that has CLaws, this brute force approach may yield a good method.
However, apart from very coarse discretizations, a compact scheme found by a volume preserving
technique performed the best. Nevertheless, some of the new eight-point schemes we have found
can be considered to preserve the second CLaw exactly whilst having a compact stencil, which
is a very desirable property.
Another area for further research is whether the method can be extended to cope with complexvalued equations, such as the nonlinear Schrödinger equation (NLS), and with systems of equations. Also, can we adapt the method for equations with a non-polynomial nonlinearity such as
157
Chapter 7: Conclusions and Further Study
the sine-Gordon equation (sG)? However it should be noted, that unlike KdV [5], the NLS and
sG equations are integrable equations that have homoclinic structures. This presents a greater
numerical challenge, as for solutions close to the homoclinic solution, round off errors can lead
to numerically induced chaos [5, 4, 2, 3]. Other possible questions to investigate are: can the
aim of preserving CLaws be combined with an adaptive mesh method? And finally, can some
form of backwards error analysis be used on the methods that we have found?
158
Appendix A
Linearizations of the Schemes
To linearize the different schemes for KdV, about the constant solution, we take the Gâteaux
derivative (see §4.2 for a definition) of the scheme acting on vmn and then substitute uij = ρ.
The linearization of the multisymplectic scheme is
1
(Sn − I)(v−20 + 3v−10 + 3v00 + v10 ) +
0 = 8ν
+
ρ
8µ
1
2µ3 (Sn
+ I)(−v−20 + 3v−10 − 3v00 + v10 )+
(−v−20 + v11 + v10 − v−21 + v00 − v−11 + v01 − v−10 )
and its numerical dispersion relation is
0 = − 83 ν −1 + 18 µρ − 32 µ−3 sin 12 kµ − 21 ων + 38 ν −1 + 18 µρ − 23 µ−3 sin 12 kµ + 12 ων +
+ − 18 ν −1 + 18 µρ + 12 µ−3 sin 32 kµ − 12 ων + 18 ν −1 + 18 µρ + 21 µ−3 sin 32 kµ + 12 ων .
The narrow box scheme’s linearization is
ρ
1
(Sn −I)(v−10 +v00 )+ 2µ
(v01 +v00 −v−11 −v−10 )+ 2µ1 3 (Sn +I)(−v−20 +3v−10 −3v00 +v10 ),
0 = 2ν
and its numerical dispersion relation is
0 = − 21 ν −1 + 12 µρ − 23 µ−3 sin 12 kµ − 12 ων + 12 ν −1 +
1
3
1
1 sin 3/2kµ − 2 ων
1 sin 2 kµ + 2 ων
+2
+2
.
µ3
µ3
1ρ
2µ
− 32 µ−3 sin
1
2 kµ
+ 12 ων +
The linearization of the three-step explicit scheme is given in (6.3) and its numerical dispersion
relation is
1
0 = 3ν
sin
+
1
2
3νω
2
sin
ων
2
+
1
2µ
cos
− kµ −
sin (kµ) + µ13 21 + 2η sin ων
−
2
1
ων
ων
.
2 sin 2 + kµ − η sin 2 + 2kµ
ων
2
159
1
2
+ η sin
ων
2
− 2kµ +
(A.1)
1
+ (−2!0 − 41 + 1/3) v01 − 21 v−20 + − 61 − 20 v−21 + (41 + 20 − 1/3) v−10 + − 61 + 21 + 4
!0 v−11 + 20 + 6 v10 + 21 v11
1
1
ρ
ρ 6 +20
ρ(1/3−20 −41 )
3
3
1
sin 12 kµ + 12 ων +
− 2µ
sin 12 kµ − 12 ων + 2ν
+
− 2µ
+ 2µ
sin 32 kµ − 12 ων + 2 µ1 + 12 µ−3 sin 32 kµ + 12 ων
3
3
3
µ
µ
1
1
1
1
1
1
1
µf
ux2 = −81 +42 + 41 v00 + −4
1 − 12
2 v−10 + −42 +8
1 v10 + 4!
2 + 12 v11
!2 −41 + 4 v01 + −42 − 12 v−20 + −4
!+42 v−21 + 41 − 4 +4
!1 − 4 v−11 + 12 −42 +4
1
1
1
1
−42−41
ρ −81 +42 + 4
ρ
ρ −42+41 + 12
ρ 12+42
1
3
3
1
1
1
sin 12 kµ− 12 ων + 2ν
sin 12 kµ+ 12 ων +
sin 32 kµ− 21 ων +
sin 32 kµ+ 21 ων
0=
− 2ν
− 2µ
+ 4 µ
− 2µ
+ 2µ
+ 2µ
3
3
3
3
µ
µ
µ
µf
ux3 = (−41 − 22 + 1/3) v00 + −42 − 21 + 61 v01 + − 16 − 22 v−20 − 21 v−21 +! − 61 + 42 + 21 v−10 + (−1/3 + 22 + 41 ) v−11 + 21 v10 + 61 + 22 v11 !
1
1
ρ 6 +22
ρ 6 −42 −21
ρ(−41 −22 +1/3)
ρ
1
3
1
3
0 = − 2ν
+
− 2µ
sin 12 kµ − 12 ων + 2ν
+
− 2µ
+ 12 µ−3 sin 32 kµ + 12 ων
sin 12 kµ + 12 ων + 2 µ1 + 12 µ−3 sin 32 kµ − 12 ων +
3
3
µ
µ
µ
5
1
5
1
µf
ux4 = (−121 + 182 + 2/9) v00 + 18
− 61 v01 + − 18
− 62 v−20 + 122 − 61 − 91 v−21 + − 18
+ 61 v−10 + (−182 − 2/9 + 121 ) v−11 + 91 + 61 − 122 v!10 + 62 + 18
v11
1
+
6
ρ
2
ρ(−121 +182 +2/9)
ρ(−122+61 +1/9)
18
−1
1
3 1
1
3
1
3
1
1
5
1
1
1
1
3
1
0=
− 2ν − 2µ3 sin 2 kµ− 2 ων + 2ν +ρ 18 −61 µ − 2 µ3 sin 2 kµ+ 2 ων +
+ 2µ3 sin 2 kµ− 2 ων +
+ 2µ3 sin 2 kµ+ 2 ων
µ
µ
µ
µf
ux8 = (−21 + 1/3) v00 + 61 + 21 v01 + − 16 + 21 v−20 − 21 v−21 + − 16!− 21 v−10 + (21 − 1/3) v−11 + 21 v10 + 16 − 21 v11
!
1
1
ρ 6 +21
ρ 6 −21
ρ(−21 +1/3)
ρ1
1
3
1
1
3
1
1
1
3
1
1
0 = − 2ν
+
−
sin
kµ
−
ων
+
+
−
sin
kµ
+
ων
+
2
+
sin
kµ
−
1/2ων
+
+
sin 32 kµ + 1/2ων
µ
2
2
2ν
µ
2
2
µ
2
µ
2µ3
2µ3
2µ3
2µ3
µf
ux9 = 32 κv00 + − 32 κ + 14 v01 − 32 κv−20 + − 14 + 32 κ v−21 + − 14 +!32 κ v−10 − 32 κv−11 + − 32 κ + 14 v10 +!23 v11 κ
1
3
3
1
ρ 4−2κ
ρ 4−2κ
3 −3
1
3
1
1
1
+ 32 ρκ
sin 21 kµ − 12 ων + 2ν
+
− 2µ
+ 2µ
0 = − 2ν
sin 32 kµ + 12 ων
sin 12 kµ + 12 ων +
sin 32 kµ − 21 ων + 32 ρκ
3
3
µ − 2µ
µ
µ
µ + 2µ3
160
1
µf
ux10
= − 6 v−20−
0=
1
− 2ν
−
3
2µ3
1
1
1
v11
6 v−21 + 6 v10 + 6
1
1
1
sin 2 kµ − 2 ων + 2ν
−
3
2µ3
sin
1
2 kµ
+
1
2 ων
+
1 ρ
6 µ
+
1
2µ3
sin
3
2 kµ
−
1
2 ων
+
1 ρ
6 µ
+
1 −3
2µ
sin
3
2 kµ
+
1
2 ων
Appendix A: Linearizations of the Schemes
Table A.1: Linearized eight-point schemes that preserve the first and second CLaws and their dispersion relations
µf
ux1 = −21 + 16 − 40 v00
1
ρ −40 −21 + 6
1
0 = − 2ν
+
µ
1
+ 23 − 16 − 22 v−10 + (22 − 1/3 + 23 ) v−11
! + (1/3 − 22 − 23 ) v10 + 22 − 23+ 6 v11 + 23 v21
1
1 1
1
ρ 6+22−23
−
η
sin
2kµ
−
ων
ρ
2
1
1
sin kµ− 12 ων + 16
+ 1/3 µ3 + 21 µη3 sin 2kµ+ 21 ων
− 2µ
− 2µ
sin kµ+ 12 ων + 2 2
3
3
µ
µ3
1
1
1
1
1
1
µf
ux2 = − 12
+ 83 − 44 v00 +
84 v−10 + − 14!+ 43 v−11 + 14 − 43 v10 + 43 + 12
− 84 v11 + 44 + 12
4 − !
12 v−20 + −43 − 12 +
! 44 + 12 − 83 v01 + −4
! v21
1
1
1
1
1
1
1
− η sin(2kµ−1/2ων)
ρ 44−83+12
ρ −43+4
ρ 43−84+12
ρ 44+12
−η
1
1
0 = ν1 + 16
sin 12 ων + 16
sin kµ− 12 ων + 16
sin kµ+ 12 ων + 2 2
+ 16
+ 2µ3
− 2µ
− 2µ
+ 21 µη3 sin 2kµ+ 12 ων
3
3
µ
µ
µ
µ
µ3
1
1
1
µf
ux3 = (24 + 43 ) v00 + (−24 − 43 ) v01 + −24 − 61 v−20 + (−2
4 + 23 ) v11 + 6 + 24 v21!
! 3 + 24 ) v−10 + − 6 + 24 + 23 v−11 + 6 − 24 − 23 v10+ (−2
1
1
1 1
1
1
ρ −24−23+6
ρ 24+6
− η sin 2kµ−2 ων
−η
ρ(−24−23 )
ρ(23−24 )
1
0 = ν1 + 16
+ 2µ3
− 12 µ13 sin kµ− 12 ων + 16
− 2µ
sin kµ+ 12 ων + 2 2
+ 16
+ 12 µη3 sin 2kµ+ 21 ων
sin 12 ων + 16
3
µ
µ
µ
µ
µ3
1
1
1
5
5
1
1
µf
ux4 = −184 − 19 +123 v00 +
+ −64 +63 − 18
! 9 −123 +184 v01 + −18 −64 v−20 + −9 +184 −63 v−10
! v−11 + −63 + 18+64 v10+ 63 + 9 −184 v11 + 64 + 18
! v21
1
1
1
1 1
1
1
1
ρ 63−184+9
ρ 64+18
ρ 184+9−123
− η sin 2kµ−2 ων
−η
η
5
1
1
1
0 = ν1 +
+ 2µ3 sin 12 ων + ρ
− 2µ
+
+ 2µ
sin kµ+ 12 ων + 2 2
sin 2kµ+ 12 ων
3
3
6µ
6 64 −63 + 18
µ − 2µ3 sin kµ− 2 ων +
6µ
6µ
µ3
µf
ux10 = 32 κv00 − 32 κv01 − 32 κv−20 + 32 κ − 14 v−10 + 32 κ −!14 v−11 + − 32 κ + 14 v10 + − 32 κ + 41 !v11 + 32 v21 κ
3
3
1
1
1
1
1
ρ
−
κ
ρ
−
κ
1/2− 2 η sin 2kµ− 2 ων
4
2
1
1 η
2 −η
0 = ν1 − 14 ρκ
sin 12 ων + 1/6 4 µ 2
− 21 µ−3 sin kµ − 12 ων + 16
− 2µ
sin kµ + 12 ων +
+ 14 ρκ
sin 2kµ + 12 ων
3
3
µ + µ3
µ
µ + 2 µ3
µ
161
µf
ux11 = − 61 v−20 − 16 v−11 + 16 v10 + 61 wm+2,n+1
1
−η
ρ
0 = ν1 + 2µ3
sin 12 ων + − 16 µ
− 1/2 µ13 sin kµ −
1
2 ων
−
1
2
1
sin kµ+ 2 ων
µ3
+
1
1
1
2 − 2 η sin 2kµ− 2 ων
µ3
+
1 ρ
6 µ
+
1 η
2 µ3
sin 2kµ +
1
2 ων
Table A.3: Linearized eight-point schemes that preserve the first and third CLaws and their dispersion relations
0=
0=
0=
0=
0=
0=
0=
0=
0=
0=
1
1
1
1
1
1
1
1
1
3
1
1
1
3
1
1
1
1
3
3
− 8 v−10 + 8 v01 − 8 v−11 + 8 v00 − 8 v−21 + 8 v10 + 8 v11 − 8 v−20 θ
− 2 v−10 − 2 v00 + 2 v−11 + 2 v01
v
− v − v
+ v + v − v
− v + v
+
+ 2 −10 2 01 2 −21 2 10µ32 11 2 −20 2 00 2 −11
ν
µ
ρ
ρ
ρ
ρ
1
3
3
1
1
− 2ν
sin 12 kµ − 12 ων + 12 ν1 + 18 µ
sin 12 kµ + 21 ων + 18 µ
sin 32 kµ − 21 ων + 18 µ
sin 32 kµ + 21 ων
+ 18 µ
− 2µ
− 2µ
+ 2µ
+ 2µ
3
3
3
3
1
1
1
1
1
1
1
1
3
3
1
1
1
1
3
3
v00 + 6 v11 + 12 v01 − 6 v−20 − 12 v−21 − 12 v−10 − 6 v−11 + 12 v10 θ
v−11 −v00
4(v01 +v−20 −v−21 −v−10 +v−11 +v10 −v00 −v11 )θβ
v
− v − v
+ v + v − v
− v + v
+
+ 6
+ 2 −10 2 01 2 −21 2 10µ32 11 2 −20 2 00 2 −11
ν
µ
µ
1 ρ
3
1 ρ
3
1 ρ
1
1 ρ
1
− ν1 − 4 ρβ
sin 12 kµ − 21 ων + 4 ρβ
sin 12 kµ + 12 ων + 4 ρβ
sin 32 kµ − 12 ων + −4 ρβ
sin 32 kµ + 12 ων
µ + 6 µ − 2µ3
µ + 12 µ − 2µ3
µ + 12 µ + 2µ3
µ + 6 µ + 2µ3
1
1
1
1
3
3
1
1
1
1
3
3
v00 + 4 v11 − 4 v−20 − 4 v−11 θ
v01 −v−10
v
− v − v
+ v + v − v
− v + v
(−8v00 +8v−20 −8v−21 +8v01 +8v−11 +8v10 −8v−10 −8v11 )θβ
+
+ 4
+ 2 −10 2 01 2 −21 2 10µ32 11 2 −20 2 00 2 −11
ν
µ
µ
1 ρ
1
1
3
1
1 ρβ
1 ρ
3
1
− 18 ρβ
sin 12 kµ − 21 ων + 18 ρβ
sin 12 kµ + 12 ων + 21 µ13 + 18 ρβ
sin 32 kµ − 21 ων + 2µ
3 − 8 µ + 4 µ sin 2 kµ + 2 ων
µ + 4 µ − 3/2 µ3
µ + ν − 2µ3
µ
1
1
1
1
1
1
1
1
3
3
1
1
1
1
3
3
− 6 v−10 + 6 v01 − 12 v−11 + 12 v00 − 6 v−21 + 6 v10 + 12 v11 − 12 v−20 θ
v−21 −v10
v
− v − v
+ v + v − v
− v + v
+
+ 2 −10 2 01 2 −21 2 10µ32 11 2 −20 2 00 2 −11
ν
µ
ρ
ρ
1 ρ
1 ρ
3
3
1
1
sin 12 kµ − 21 ων + 16 µ
− 2µ
sin 12 kµ + 12 ων + − ν1 + 16 µ
+ 2µ
sin 32 kµ − 12 ων + 12
sin 32 kµ + 1/2ων
3
3
12 µ − 2µ3
µ + 2µ3
1
1
1
1
1
1
1
1
3
3
1
1
1
1
3
3
v00 + 6 v11 + 12 v01 − 6 v−20 − 12 v−21 − 12 v−10 − 6 v−11 + 12 v10 θ
v11 −v−20
v
− v − v
+ v + v − v
− wm,n + 2 v−11
+ 6
+ 2 −10 2 01 2 −21 2 10 µ23 11 2 −20 2
ν
µ ρ
1 ρ
3
1 ρ
3
1 ρ
1
1
sin 12 kµ − 21 ων + 12
sin 12 kµ + 12 ων + 12
sin 32 kµ − 12 ων + ν1 + 16 µ
+ 2µ
sin 32 kµ + 1/2ων
3
6 µ − 2µ3
µ − 2µ3
µ + 2µ3
Appendix A: Linearizations of the Schemes
Table A.2: Selection of the linearized ten-point schemes that preserve the first and second CLaws
µf
ux1 = 23 + 42 − 16 v00 + !
−42 − 23 + 61 v01 − 23 v−20
1
1
ρ(−22+1/3−23 )
ρ 6−23−42
−η
0 = ν1 + 16
+ 2µ3
sin 12 ων + 16
µ
µ
1
1
1
1
1
1
1
1
v00 + 6 v21 − 6 v01 − 6 v−20 − 12 v−10 − 12 v−11 + 12 v10 + 12 v11 θ
(−4v01 −4v−20 +4v−10 +4v−11 −4v10 −4v11 +4v00 +4v21 )θ5
−v00 +v01
+
+ 6
+
ν
µ
µ
1
1
1
1
2αv11 −2αv−10 −2ηv11 +2ηv−10 +ηv21 −ηv−20 + 2 v00 − 2 v01 +4αv00 −4αv01 −2ηv00 +2ηv01 +ηv−21 −ηv20 − 2 v−21 + 2 v20 −2ηv−11 +2ηv10 +2αv−11 −2αv10 +v−11 −v10
+
3
µ
1
sin 2kµ− 2 ων
ρ2
ρ2
ρ25
ρ2
ρ
ρ
1 ρ
1 −3
1
1 ρ
1
1
1
sin
kµ
+
0 = ν1 − 2 µ 5 − 16 µ
sin 12 ων + −2 µ 5 + 12
−
µ
sin
kµ
−
ων
+
−2
+
−
ων
+
+ 2 µ 5 + 61 µ
3
3
µ
2
2
µ
12 µ
2
4
2µ
µ
1
1
1
1
1
1
1
1
(−2v01 −2v−20 +2v−10 +2v−11 −2v10 −2v11 +2v00 +2v21 )θ5
−v00 +v01
6 v00 + 6 v21 − 6 v01 − 6 v−20 − 12 v−10 − 12 v−11 + 12 v10 + 12 v11 θ
0=
162
0=
+
+
ν
µ
1
1
1
1
1
1
1
1
1
1
1
1
− 2 v00 +αv00 + 2 v01 −αv01 − 2 αv−20 − 2 v−21 + 2 v−21 α+ 2 v−10 + 2 v−11 − 2 v10 − 2 v11 + 2 v20 − 2 v20 α+ 2 αv21
+
µ3
0=
1
ν
−2
ρ5
µ
−
1 ρ
6 µ
sin
1
2 ων
ρ
+ −2 µ5 +
1 ρ
12 µ
−
1 −3
2µ
sin kµ −
1
2 ων
ρ
+ −2 µ5 +
1 ρ
12 µ
−
1
2µ3
1 1
4 µ3
sin 2kµ +
1
2 ων
+
µ
sin kµ +
+
1
2 ων
+
1
4
1
sin 2kµ− 2 ων
µ3
ρ
+ 2 µ5 +
1 ρ
6 µ
+
1 1
4 µ3
sin 2kµ +
1
2 ων
Appendix A: Linearizations of the Schemes
Table A.4: Linearized ten-point schemes that preserve the first and third CLaws and their dispersion relations
Appendix B
Maple Code for Solving the
ALSC
In this section we provide the maple code used to solve the ALSC for the potential Lotka–
Volterra equation (4.21) with a seed of the form (4.22).
We begin by loading the packages we require, solving the quad-graph equation in the four
different directions, providing the ansatz for the seed and the ALSC.
>restart;with(LREtools):with(DEtools):
>omega:=(m,n)->u[m,n]+u[m,n+1]*u[m,n]/u[m+1,n];
>Omega_L:=(m,n)->-(-u[m, n+1]+u[m-1, n])*u[m, n]/u[m-1, n];
>Omega_R:=(m,n)->-u[m, n]*u[m, n-1]/(-u[m+1, n]+u[m, n-1]);
>Omega:=(m,n)->u[m, n]*u[m, n-1]/(u[m, n-1]+u[m-1, n]);
>Q:=A1(m,n,u[m,n-1],u[m,n],u[m,n+1])/((diff(omega(m,n),u[m,n+1])))+
B1(m,n,u[m-1,n],u[m,n],u[m+1,n])/((diff(omega(m,n),u[m+1,n])))+
(A2(m,n,u[m,n],u[m,n+1],u[m,n+2])+B2(m,n,u[m,n],u[m+1,n],u[m+2,n]))/
(diff(omega(m,n),u[m,n]))+C(m,n,u[m,n],u[m+1,n],u[m,n+1]);
>ALSC:=shift(shift(diff(omega(m,n),u[m,n])*Q,m,1),n,1)+
shift(diff(omega(m,n),u[m+1,n])*Q,n,1)
+shift(diff(omega(m,n),u[m,n+1])*Q,m,1)-Q:
We now consider the initial conditions on the cross with centre (0, 0) and isolate the B1 term
by differentiating w.r.t. u−10 , following the reasoning of (4.14) to (4.15).
>ALSC1:=collect(subs(u[m-1,n+1]=Omega_L(m,n),ALSC),B1):
>eq1:=collect(diff(ALSC1,u[m-1,n]),B1);
>eq1a:=collect(combine(eq1*diff(omega(m,n),u[m+1,n])),B1);
>eq1a1:=subs(u[m+1,n+1]=omega(m,n),subs(u[m+2,n+1]=omega(m+1,n),eq1a));
>eq1a2:=numer(simplify(subs(u[m+1,n]=Omega_R(m,n+1),subs({u[m-1,n]=Omega(m,n+1)
,u[m+2,n]=Omega_R(m+1,n+1)},diff(eq1a1,u[m,n+1]))))):
>sys1a:={coeffs(collect(shift(eq1a2,n,-1),u[m,n-1]),u[m,n-1])}
To generate the next system of equations we define the necessary differential operators and
rearrange eq1 as described up to (4.16) in §4.3.
163
Appendix B: Maple Code for Solving the ALSC
>eq1b:=collect(eq1/(diff(Omega_L(m,n),u[m-1,n])),B1,simplify);
>L:=f->diff(Omega_L(m,n),u[m-1,n])*diff(f,u[m,n])diff(Omega_L(m,n),u[m,n])*diff(f,u[m-1,n]);
>L1:=f->L(omega(m,n))*diff(f,u[m+1,n])-diff(omega(m,n),u[m+1,n])*L(f);
>eq1b1:=collect(numer(simplify(L1(eq1b))),u[m,n+1]):
>sys1b:={coeffs(eq1b1,u[m,n+1])}:
In order to use the function pdsolve all the functions need to have scalar arguments. By making
the required substitutions we can then solve the system of PDEs and assign the solution. Going
back up the hierarchy of equations gives a linear difference equation to solve.
>subvar:={u[m,n-1]=a,u[m-1,n]=b,u[m,n]=c,u[m+1,n]=d,u[m+2,n]=e,
u[m,n+1]=f,u[m,n+2]=g};
>sys1:=subs(subvar,sys1a union sys1b):
>pdsolve(sys1);
>B1:=(m, n, b, c, d) -> _F1(m, n, c, d)+_F3(m, n)/(b+d);
>simplify(subs(u[m+1,n+1]=omega(m,n),eq1b));
>_F3:=(m,n)->_f3(m);
We make the further simplification
_F1:=(m, n, c, d)->0;
because it, along with its coefficient in the ALSC, have the same arguments as the undetermined
function C.
We can now isolate the function A1 by differentiating w.r.t. u0−1 . We then follow the same
procedures as for B1.
>ALSC2:=collect(subs(u[m+1,n-1]=Omega_R(m,n),ALSC),A1):
>eq2:=collect(diff(ALSC2,u[m,n-1]),A1);
>eq2a:=collect(combine(eq2*diff(omega(m,n),u[m,n+1])),A1);
>eq2a1:=subs(u[m+1,n+1]=omega(m,n),subs(u[m+1,n+2]=omega(m,n+1),eq2a)):
>eq2a2:=factor(numer(simplify(diff(eq2a1,u[m+1,n])))):
>eq2a2a:=subs( u[m,n+1]=Omega_L(m+1,n),subs({u[m,n-1]=Omega(m+1,n),
u[m,n+2]=Omega_L(m+1,n+1)},eq2a2));
>eq2a2b:=shift(collect(factor(numer(simplify(eq2a2a))),A1),m,-1);
>sys2a:={coeffs(collect(eq2a2b,u[m-1,n]),u[m-1,n])};
>eq2b:=collect(eq2/diff(Omega_R(m,n),u[m,n-1]),A1,simplify);
>L1:=f->diff(Omega_R(m,n),u[m,n-1])*diff(f,u[m,n])diff(Omega_R(m,n),u[m,n])*diff(f,u[m,n-1]);
>L2:=f->L1(omega(m,n))*diff(f,u[m,n+1])-diff(omega(m,n),u[m,n+1])*L1(f);
>eq2b1:=collect(numer(simplify(L2(eq2b))),u[m+1,n]);
>sys2b:={coeffs(eq2b1,u[m+1,n])};
>sys2:=subs(subvar,sys2a union sys2b):
>pdsolve(sys2);
>A1:=(m, n, a, c, f) ->_F2(m, n, c, f)+_F4(m, n)/a;
>simplify(eq2b);
164
Appendix B: Maple Code for Solving the ALSC
>_F4:=(m,n)->_f4(n);
>_F2:=(m,n,c,f)->0;
We now consider the initial conditions on the cross centred at (1, 0). Following the reasoning of
(4.17) to (4.18) we can isolate B2.
>ALSC3:=subs(u[m+3,n+1]=omega(m+2,n),ALSC):
>eq3:=diff(ALSC3,u[m+3,n]);
>eq3a:=collect(shift(eq3,m,-1)*diff(omega(m,n),u[m,n])/
diff(omega(m,n),u[m,n+1]),B2,simplify);
>eq3a1:=diff(subs(u[m+1,n+1]=omega(m,n),eq3a),u[m,n+1]):
>eq3a1a:=shift(numer(simplify((subs(u[m+1,n]=Omega_R(m,n+1),
subs(u[m+2,n]=Omega_R(m+1,n+1),eq3a1))))),n,-1);
>sys3a:={coeffs(collect(eq3a1a,u[m,n-1]),u[m,n-1])}:
By defining suitable differential operators we can generate the system of equations resulting from
(4.19)
>eq3b:=collect(shift(eq3,m,-1)/shift(diff(omega(m,n),u[m+1,n]),m,1),
B2,simplify);
>L1:=f->diff(omega(m,n),u[m,n])*diff(f,u[m+1,n])diff(omega(m,n),u[m+1,n])*diff(f,u[m,n]);
>L2:=f->L1(omega(m+1,n))*diff(f,u[m+2,n])diff(omega(m+1,n),u[m+2,n])*L1(f);
>eq3b1:=(numer(simplify(subs(u[m+1,n+1]=omega(m,n),(L2(eq3b))))));
>sys3b:={coeffs(collect(eq3b1,u[m,n+1]),u[m,n+1])}
>sys3:=subs(subvar,sys3a union sys3b):
>pdsolve(sys3);
>B2:=(m, n, c, d, e) -> _F5(m, n, c, d)+_F7(m, n)/(e+c);
>simplify(subs(u[m+1,n+1]=omega(m,n),shift(eq3,m,-1)));
>_F7:=(m,n)->_f7(m);
>_F5:=(m,n,c,d)->0;
The last two statements are a result of reducing eq3 to a linear difference equation and that F 5
and its coefficient in the ALSC can be absorbed by C.
By shifting the centre of our cross of initial conditions to (0, 1) we can perform similar operations
on A2.
>ALSC4:=subs(u[m+1,n+3]=omega(m,n+2),ALSC):
>eq4:=shift(diff(ALSC4,u[m,n+3]),n,-1);
>eq4a:=subs(u[m+1,n+1]=omega(m,n),collect(eq4*diff(omega(m,n),u[m,n])/
(diff(omega(m,n),u[m+1,n])),A2,simplify));
>eq4a1:=numer(simplify(subs(u[m-1,n+1]=Omega_L(m,n),
subs(u[m-1,n+2]=Omega_L(m,n+1) ,shift(diff(eq4a,u[m+1,n]),m,-1)))));
>sys4a:={coeffs(collect(eq4a1,u[m-1,n]),u[m-1,n])};
For the potential Lotka–Volterra equation, we don’t need the additional system of equations but
we can generate it using the following code.
165
Appendix B: Maple Code for Solving the ALSC
>eq4b:=collect(eq4/shift(diff(omega(m,n),u[m,n+1]),n,1),A2,simplify);
>L1:=f->diff(omega(m,n),u[m,n])*diff(f,u[m,n+1])
-diff(omega(m,n),u[m,n+1])*diff(f,u[m,n]);
>L2:=f->L1(omega(m,n+1))*diff(f,u[m,n+2])-diff(omega(m,n+1),u[m,n+2])*L1(f);
>eq4b1:=(numer(simplify(subs(u[m+1,n+1]=omega(m,n),(L2(eq4b))))));
>sys4b:={coeffs(collect(eq4b1,u[m+1,n]),u[m+1,n])};
All that remains is to solve the system of equations.
>sys4:=subs(subvar,sys4b union sys4a):
>pdsolve(sys4);
>A2:=(m, n, c, f, g) -> _F9(m, n)*g/c^2+_F8(m, n, c, f);
>simplify(eq4a);
>_F9:=(m,n)->_f9(n);
>_F8:=(m,n,a,b)->0;
We now simplify the seed.
>Q:=collect(Q,{_f4,_f3,_f9,_f7,C},simplify);
We now centre the cross of initial conditions at (1, 1).
>ALSC5:=subs({u[m+2,n]=Omega_R(m+1,n+1),u[m,n+2]=Omega_L(m+1,n+1)}
,subs({u[m+3,n]=Omega_R(m+2,n+1),u[m,n+3]=Omega_L(m+1,n+2)},ALSC)):
>ALSC6:=(subs(u[m,n]=Omega(m+1,n+1),subs({u[m-1,n]=Omega(m,n+1),
u[m,n-1]=Omega(m+1,n)},ALSC5))):
>ALSC7:=collect(ALSC6,{_f4,_f3,_f9,_f7,C},simplify);
We can obtain a difference equation by differentiating w.r.t. opposite points on the cross.
>eq5:=factor(numer(simplify(diff(ALSC7,u[m+2,n+1],u[m,n+1]))));
>_f7:=(m)->_f3(m+1);
Differentiating w.r.t. to the other pair of points doesn’t in this case result in an additional
difference equation.
We now form a system of equations for C.
>eq7:=collect(shift(shift((diff(ALSC7,u[m+1,n+2],u[m+2,n+1])),m,-1),n,-1),C,
simplify):
>eq8:=collect(subs(u[m+1,n+1]=omega(m,n),(shift((diff(ALSC7,u[m+1,n+2],u[m,n+1]
)),n,-1))),C,simplify):
>eq9:=collect(simplify(subs(u[m+1,n+1]=omega(m,n),(diff(ALSC7,u[m+1,n],u[m,n+1]
)))),C,simplify):
>eq10:=collect(subs(u[m+1,n+1]=omega(m,n),shift((diff(ALSC7,u[m+1,n],u[m+2,n+1]
)),m,-1)),C,simplify):
>bigsys:=subs(subvar,{eq7,eq8,eq9,eq10}):
This system of equations is overdetermined. In order to find the additional constraints we need
to use the rif simp function from the DEtools package. This requires that the functions all have
the same arguments. So we need to perform a substitution to remove shifts of functions.
166
Appendix B: Maple Code for Solving the ALSC
>subfunc:={_f3(m)=f3m,_f3(m+1)=f3m1,_f4(n)=f4n,_f4(n+1)=f4n1,_f9(n)=f9n,
_f9(n-1)=f9n_1};
>bigsys2:=subs(subfunc,bigsys):
>rifsys:=rifsimp(convert(bigsys2,diff));
This yields additional difference equations to solve.
>diffeqn1:=_f4(n+1)+_f9(n-1);
>diffeqn2:=_f4(n)-(_f9(n)-2*_f9(n-1));
>_f9:=n->k[1]*n+k[2];
>_f4:=n->-_f9(n-2);
The system of differential equations for C can now be solved.
>bigsys1:=simplify(bigsys):
>pdsolve(bigsys1,C);
>C:=(m, n, c, d, f)->(-f*_f3(m+1)*d*c+c^2*(d+f)*_F12(m,n)+
c^2*d*f*(d+f)*_F14(m, n)+d*_F13(m, n)+f*_F15(m, n)*d^2+c*(d+f)*(_f3(m)*d+
((n-1)*k[1]+k[2])*f))/(c^2*f*(d+f));
Finally we generate a system of linear difference equations in the arbitrary functions.
>ALSC8:=collect(numer(simplify(ALSC7)),{u[m, n+1], u[m+1, n],u[m+1, n+1],
u[m+1, n+2],u[m+2, n+1]},’distributed’):
>diffsys:={coeffs(ALSC8,{u[m,n+1],u[m+1,n],u[m+1,n+1],u[m+1,n+2],u[m+2,n+1]})}:
Solving this system of difference equations gives the solution of the ALSC. The coefficient of
each arbitrary constant corresponds to an independent solution.
167
Appendix C
Maple Code for Finding
Discretizations
We include in this section the maple worksheet used to find discretizations of KdV that preserve
some of its first three CLaws.
Begin by loading the packages that are needed.
>restart;with(LREtools):with(Groebner):
The following is a procedure for creating a set, called var, which contains all of the uij s needed
for the scheme and the discrete Euler operator.
>variables:=proc(M,N)
local i,j;
global var:={u[m,n]};
for i from -M to M do;
for j from -N to N do;
var:= var union {u[m+i,n+j]};
od;
od;
end proc;
The discrete Euler operator is then given by this piece of code.
>eulerop:=proc(f,M,N)
local i,j,E;
E:=0;
for i from -M to M do;
for j from -N to N do;
E:=E+shift(shift(diff(f,u[m+i,n+j]),m,-i),n,-j);
od;
od;
E;
end proc;
169
Appendix C: Maple Code for Finding Discretizations
f2 (5.4).
The following procedure is for calculating the quadratic terms such as uu
gx (2.28) and u
>squareterm:=proc(A,B,C,E,epsilon)
local i,j,k,l;
global SQ;
SQ:=sum(sum(sum(epsilon[i,j,l,j]*u[m+i,n+j]*u[m+l,n+j],l=i..B),i=A..B)+
sum(sum(sum(epsilon[i,j,l,k]*u[m+i,n+j]*u[m+l,n+k],l=A..B),i=A..B),
k=(j+1)..E),j=C..E);
end proc;
The following procedure is for calculating the conditions, such as (2.29) and (5.5), that the
coefficients need to satisfy for the quadratic terms to be suitable discretizations .
>coeffsquare:=proc(f,A,B,C,E)
local i,j,k,l;
global eqf0:=0,eqfmu:=0, eqfnu:=0,eqfmu2:=0;
for j from C to E do
for i from A to B do
#square terms
eqf0:=eqf0+coeff(f,u[m+i,n+j]^2);
eqfmu:=eqfmu+coeff(f,u[m+i,n+j]^2)*(i+i );
eqfnu:=eqfnu+coeff(f,u[m+i,n+j]^2)*(j+j );
eqfmu2:=eqfmu2+coeff(f,u[m+i,n+j]^2)*(i^2+i^2 );
for l from j to j do
for k from i+1 to B do
eqf0:=eqf0+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1);
eqfmu:=eqfmu+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1)*(i+k);
eqfnu:=eqfnu+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1)*(j+j);
eqfmu2:=eqfmu2+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1)*(i^2+k^2);
od;
od;
for l from j+1 to E do
for k from A to B do
eqf0:=eqf0+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1);
eqfmu:=eqfmu+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1)*(i+k);
eqfnu:=eqfnu+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1)*(j+l);
eqfmu2:=eqfmu2+coeff(coeff(f,u[m+i,n+j],1),u[m+k,n+l],1)*(i^2+k^2);
od;od;
od;od;
end proc;
The following procedure takes a system of equations and splits it into a further system of equations according the coefficients of µ and ν.
170
Appendix C: Maple Code for Finding Discretizations
>RemoveMuNu:=proc(sys)
local i;
global bigsys:={};
for i from 1 to nops(sys) do
bigsys:=bigsys union {coeffs(collect(sys[i],{mu,nu},’distributed’),{mu,nu})};
od;
bigsys:
end proc;
The above procedures are all that are necessary to search for discretizations. However we would
also like to impose symmetry assumptions on the discretizations. The procedures need to be able
to deal with whether there are even or odd number of points in the spatial and time directions.
The first procedure is for imposing 180◦ symmetry on the linear expressions.
>sym180ansatz:=proc(iota,A,B,C,E)
local i,j;
if
type(E-C,odd)=true
then
for j from 0 to (E-C-1)/2 do
for i from 0 to (B-A) do
iota[A+i,C+j]:=iota[B-i,E-j];
od;od;
else
for j from 0 to (E-C)/2-1 do
for i from 0 to B-A do
iota[A+i,C+j]:=iota[B-i,E-j];
od;od;
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
iota[A+i,(E+C)/2]:=iota[B-i,(E+C)/2];
od;
else
for i from 0 to (B-A)/2-1 do
iota[A+i,(E+C)/2]:=iota[B-i,(E+C)/2];
od;
end if;
end if;
end proc;
The next procedure is for imposing 180◦ antisymmetry on the linear expressions.
>antisym180ansatz:=proc(iota,A,B,C,E)
local i,j;
if
type(E-C,odd)=true
then
171
Appendix C: Maple Code for Finding Discretizations
for j from 0 to (E-C-1)/2 do
for i from 0 to (B-A) do
iota[A+i,C+j]:=-iota[B-i,E-j];
od;od;
else
for j from 0 to (E-C)/2-1 do
for i from 0 to B-A do
iota[A+i,C+j]:=-iota[B-i,E-j];
od;od;
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
iota[A+i,(E+C)/2]:=-iota[B-i,(E+C)/2];
od;
else
for i from 0 to (B-A)/2-1 do
iota[A+i,(E+C)/2]:=-iota[B-i,(E+C)/2];
od;
iota[(B+A)/2,(E+C)/2]:=0
end if;
end if;
end proc;
If we wish to impose stricter symmetry conditions then we need the following procedures. To
be symmetric in both space and time we need this procedure.
>symXsymTansatz:=proc(iota,A,B,C,E)
local i,j;
if
type(E-C,odd)=true
then
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
for j from 0 to (E-C-1)/2 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=iota[A+i,C+j];
od;od;
else
for j from 0 to (E-C-1)/2 do
for i from 0 to (B-A)/2-1 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=iota[A+i,C+j];
od;
iota[(A+B)/2,E-j]:=iota[(A+B)/2,C+j];
172
Appendix C: Maple Code for Finding Discretizations
od;
end if;
else
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
for j from 0 to (E-C)/2-1 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=iota[A+i,C+j];
od;
iota[B-i,(E+C)/2]:=iota[A+i,(E+C)/2];
od;
else
for j from 0 to (E-C)/2-1 do
for i from 0 to (B-A)/2-1 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=iota[A+i,C+j];
od;
iota[(A+B)/2,E-j]:=iota[(A+B)/2,C+j];
od;
for i from 0 to (B-A)/2-1 do
iota[B-i,(E+C)/2]:=iota[A+i,(E+C)/2];
od;
end if;
end if;
end proc;
The following procedure imposes symmetry in the time direction and antisymmetry in the spatial
direction.
>antisymXsymTansatz:=proc(iota,A,B,C,E)
local i,j;
if
type(E-C,odd)=true
then
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
for j from 0 to (E-C-1)/2 do
iota[B-i,C+j]:=-iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;od;
else
for j from 0 to (E-C-1)/2 do
173
Appendix C: Maple Code for Finding Discretizations
for i from 0 to (B-A)/2-1 do
iota[B-i,C+j]:=-iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;
iota[(A+B)/2,E-j]:=0;iota[(A+B)/2,C+j]:=0;
od;
end if;
else
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
for j from 0 to (E-C)/2-1 do
iota[B-i,C+j]:=-iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;
iota[B-i,(E+C)/2]:=-iota[A+i,(E+C)/2];
od;
else
for j from 0 to (E-C)/2-1 do
for i from 0 to (B-A)/2-1 do
iota[B-i,C+j]:=-iota[A+i,C+j];
iota[A+i,E-j]:=iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;
iota[(A+B)/2,E-j]:=0;iota[(A+B)/2,C+j]:=0;
od;
for i from 0 to (B-A)/2-1 do
iota[B-i,(E+C)/2]:=-iota[A+i,(E+C)/2];
od;
iota[(B+A)/2,(E+C)/2]:=0;
end if;
end if;
end proc;
The last procedure we need imposes symmetry in the spatial direction and antisymmetry in the
time direction.
>symXantisymTansatz:=proc(iota,A,B,C,E)
local i,j;
if
type(E-C,odd)=true
then
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
174
Appendix C: Maple Code for Finding Discretizations
for j from 0 to (E-C-1)/2 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=-iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;od;
else
for j from 0 to (E-C-1)/2 do
for i from 0 to (B-A)/2-1 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=-iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;
iota[(A+B)/2,E-j]:=-iota[(A+B)/2,C+j];
od;
end if;
else
if type(B-A,odd)=true
then
for i from 0 to (B-A-1)/2 do
for j from 0 to (E-C)/2-1 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=-iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;
iota[B-i,(E+C)/2]:=0;iota[A+i,(E+C)/2]:=0;
od;
else
for j from 0 to (E-C)/2-1 do
for i from 0 to (B-A)/2-1 do
iota[B-i,C+j]:=iota[A+i,C+j];
iota[A+i,E-j]:=-iota[A+i,C+j];
iota[B-i,E-j]:=-iota[A+i,C+j];
od;
iota[(A+B)/2,E-j]:=-iota[(A+B)/2,C+j];
od;
for i from 0 to (B-A)/2-1 do
iota[B-i,(E+C)/2]:=0;iota[A+i,(E+C)/2]:=0;
od;
iota[(B+A)/2,(E+C)/2]:=0;
end if;
end if;
end proc;
We now need procedures to impose 180◦ antisymmetry and symmetry on the quadratic terms.
The following procedure imposes symmetry.
>SquareTerm180:=proc(A,B,C,E,b)
175
Appendix C: Maple Code for Finding Discretizations
local i,j,k,l,bcheck;
for i from 0 to (E-C)-1 do
for j from 0 to (E-C)-1-i do
for k from 0 to (B-A) do
for l from 0 to (B-A) do
b[A+k,C+i,A+l,C+j]:=b[B-l,E-j,B-k,E-i];
od;od;od;od;
for i from 0 to (E-C) do
for k from 0 to (B-A)-1 do
for l from 0 to (B-A)-1-k do
b[A+k,C+i,A+l,E-i]:=b[B-l,C+i,B-k,E-i]
od;od;od;
unassign(’i’,’j’,’k’,’l’);
bcheck:=sum(sum(sum(b[i,j,l,j],l=i..B),i=A..B)+
sum(sum(sum(b[i,j,l,k],l=A..B),i=A..B),k=(j+1)..E),j=C..E);
end proc;
This procedure imposes antisymmetry.
>SquareTermAnti180:=proc(A,B,C,E,b)
local i,j,k,l,bcheck;
for i from 0 to (E-C)-1 do
for j from 0 to (E-C)-1-i do
for k from 0 to (B-A) do
for l from 0 to (B-A) do
b[A+k,C+i,A+l,C+j]:=-b[B-l,E-j,B-k,E-i];
od;od;od;od;
for i from 0 to (E-C) do
for k from 0 to (B-A)-1 do
for l from 0 to (B-A)-1-k do
b[A+k,C+i,A+l,E-i]:=-b[B-l,C+i,B-k,E-i]
od;od;
for k from 0 to (B-A) do
b[A+k,C+i,B-k,E-i]:=0;
od;
od;
unassign(’i’,’j’,’k’,’l’);
bcheck:=sum(sum(sum(b[i,j,l,j],l=i..B),i=A..B)+
sum(sum(sum(b[i,j,l,k],l=A..B),i=A..B),k=(j+1)..E),j=C..E);
end proc;
Having created all the necessary procedures we are now in a position to search for discretizations.
We start by choosing the size of our stencil. In this example we choose all the terms to depend
on the same points and then form the discretizations for the various terms with our chosen
symmetry assumptions.
>A:=-2;B:=2;C:=0;E:=1;
176
Appendix C: Maple Code for Finding Discretizations
>unassign(’kappa’);
>sym180ansatz(kappa,A,B,C,E):
>#symXsymTansatz(kappa,A,B,C,E):
>Q2:=sum(sum(u[m+j,n+i]*kappa[j,i],j=A..B),i=C..E):
>Q2_0:=sum(sum(kappa[j,i],j=A..B),i=C..E):
>solve(Q2_0-1):assign(%);
>unassign(’eta’);
>antisym180ansatz(eta,A,B,C,E):
>#antisymXsymTansatz(eta,A,B,C,E):
>u_xxx:=sum(sum(eta[j,i]*u[m+j,n+i],j=A..B),i=C..E):
>equ_xxx0:=sum(sum(eta[j,i],j=A..B),i=C..E):
>equ_xxx1:=sum(sum(eta[j,i]*j,j=A..B),i=C..E):
>equ_xxx2:=sum(sum(eta[j,i]*j^2,j=A..B),i=C..E);
>equ_xxx3:=sum(sum(eta[j,i]*j^3,j=A..B),i=C..E);
>solve({equ_xxx0,equ_xxx1,equ_xxx2,equ_xxx3-3!}):assign(%);
>unassign(’c’);
>antisym180ansatz(c,A,B,C,E):
>symXantisymTansatz(c,A,B,C,E):
>u_t:=sum(sum( c[j,i]*u[m+j,n+i],j=A..B),i=C..E):
>eqc0:=sum(sum( c[j,i],j=A..B),i=C..E):
>eqc1t:=sum(sum( i*c[j,i],j=A..B),i=C..E):
>u_tsys:={eqc0,eqc1t-1}:
>solve(u_tsys);assign(%);
>unassign(’epsilon’);
>SquareTermAnti180(A,B,C,E,epsilon):
>squareterm(A,B,C,E, epsilon):
>uu_x:=SQ:
>coeffsquare(uu_x,A,B,C,E);
>equu_x0:=eqf0: equu_xmu:=eqfmu: equu_xnu:=eqfnu: equu_xmu2:=eqfmu2:
>uu_xsys:={equu_x0,equu_xmu-1}:
>solve(uu_xsys):assign(%);
>unassign(’alpha’,’i’,’j’);
>sym180ansatz(alpha,A,B,C,E);
>#symXsymTansatz(alpha,A,B,C,E)
>u_xx:=sum(sum(alpha[j,i]*u[m+j,n+i],j=A..B),i=C..E):
>eqalpha0:=sum(sum(alpha[j,i],j=A..B),i=C..E);
>eqalpha1:=sum(sum(alpha[j,i]*j,j=A..B),i=C..E);
>eqalpha2:=sum(sum(alpha[j,i]*j^2,j=A..B),i=C..E);
>u_xxsys:={eqalpha0,eqalpha1,eqalpha2-2!};
>solve(u_xxsys);assign(%);
>unassign(’beta’,’j’,’i’);
177
Appendix C: Maple Code for Finding Discretizations
>SquareTerm180(A,B,C,E,beta):
>squareterm(A,B,C,E,beta):
>usquare:=SQ:
>coeffsquare(usquare,A,B,C,E);
>equsquare0:=eqf0:
>solve(equsquare0-1):assign(%):
>KdV:=u_t/nu+uu_x/mu+u_xxx/(mu^3):
>Q3:=usquare+2*u_xx/(mu^2):
To impose that the discretization satisfies the first CLaw in characteristic form we use the
following piece of code.
>eq1:=simplify(expand(eulerop(uu_x,5,5))):
>c1:=collect(eq1,var,’distributed’):
>sys1:={coeffs(c1,var)}:nops(sys1);
>solve(sys1):assign(%):
To impose the discretization satisfies the second and third CLaws we generate the following
systems of equations.
>eq2:=simplify(eulerop(Q2*KdV,5,5)):
>c2:=collect(eq2,var,’distributed’):
>sys2:={coeffs(c2,var)}:
>eq3:=simplify(eulerop(Q3*KdV,5,5)):
>c3:=collect(eq3,var,’distributed’):
>sys3:={coeffs(c3,var)}:
>sys23:=sys2 union sys3
This system still contains µ and ν so we split this into a larger system using
>RemoveMuNu(sys23):
>big23:=bigsys:
The algorithm for calculating the Groebner Basis is most efficient if it works with a polynomial
ideal, so we convert our system to this form.
>with(PolynomialIdeals):
>big23alt:=convert(bigsys2,PolynomialIdeal):
We use maple’s heuristic algorithm for deciding the ordering of the variables. We then calculate
the Groebner basis using the total degree ordering (graded reverse lexicographic ordering).
>V23:=SuggestVariableOrder(big23alt):nops({V23alt}):
>B23:=Basis(big23alt,tdeg(V23)):B23[1];
If B231 = 1 then there is no solution to the system of equations. If there is a solution we convert
the Basis to one with the lexicographic ordering, to enable us to solve it.
178
Appendix C: Maple Code for Finding Discretizations
>L23:=Walk(B23,tdeg(V23),plex(V23)):
>S23:=Solve(L23):K:=nops(S23alt);
Finally S23 contains any solution families that have been found. We now need to check that the
resulting discretizations are consistent. We do this by substituting the solution into KdV and
then applying the following procedure with different values of r.
>v:=(m,n)->mtaylor(f(x+m*mu,t+n*nu),[mu=0,nu=0],6);
>taylorcheck:=proc(f,M,N,r)
local i,j,F,alpha;
F:=f;
for i from -M to M do
for j from -N to N do
F:=subs(u[m+i,n+j]=v(i,j),F);
od;od;
F:=subs(nu=lambda*mu^r,F);
F:=(limit(F,mu=0));
end proc;
This procedure replaces the discretization with the appropriate Taylor series and then lets the
step sizes tend to zero using ν = λµr .
179
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