a
TEL-AVIV UNIVERSITY
The Raymond and Beverly Sackler Faculty of Exact Sciences
School of Mathematical Sciences
Bounded-Error Finite Dierence Schemes
for Initial Boundary Value Problems
on Complex Domains
Thesis submitted for the degree "Doctor of Philosophy"
by
Adi Ditkowski
Submitted to the Senate of Tel-Aviv University
June 1997
This work was carried out under the supervision of
Professor Saul S. Abarbanel
This work is dedicated to the memory of my late father
Haim Ditkowski
I wish to express my gratitude and deepest appreciation to Professor Saul S. Abarbanel for his guidance, counseling, encouragement and for his friendship.
I would like to thank my wife Sigal, my mother Yona, and my sister Idit for standing
by me during the past years.
I wish to thank my colleagues in TAU. In particular, I would like to thank Doron
Levy for the fruitful discussions, and for the time we spent together.
I would also like to thank the Institute for Computer Applications in Science and
Engineering (ICASE) , NASA Langley Research Center, Hampton, VA, for its warm
hospitality.
Finally, I would like to thank the Roberto Nemirovsky doctoral fellowship and the
Bauer-Neuman Chair in applied mathematics and theoretical mechanics for supporting
this research.
Contents
Introduction
1
1 General theory and description of the method
7
1.1 The one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 The multi dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 The diusion equation
2.1
2.2
2.3
2.4
Construction of the scheme .
Numerical example . . . . .
Conclusions . . . . . . . . .
Appendix . . . . . . . . . .
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3 The advection diusion equation
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3.1 Construction of the scheme . . . . . . . . . .
3.2 Numerical examples . . . . . . . . . . . . . .
3.2.1 One dimensional case . . . . . . . . .
3.2.2 A steady state two dimensional case .
3.2.3 A 2-D time dependent example . . .
3.3 Conclusions . . . . . . . . . . . . . . . . . .
3.4 Appendix, The case a < 0 . . . . . . . . . .
4 Mixed derivatives and parabolic systems
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17
17
23
27
28
37
38
44
44
46
51
53
54
57
4.1 The scalar mixed derivatives problem . . . . . . . . . . . . . . . . . . . . 57
4.2 Second order scheme for the scalar mixed derivatives problem . . . . . . 63
4.3 Parabolic systems the diusion part of the Navier-Stokes equations . . . 69
i
5 Bounded error schemes for the wave equation
76
Summary
86
Bibliography
88
5.1 Description of the method . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Construction of the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 81
ii
Introduction
This thesis is concerned with the construction of bounded-error nite-dierence numerical schemes, for initial boundary value problems (IBVP), on complex, multi-dimensional
shapes. Applications where such problems arise are, among others, heat transfer, acoustic and electro-magnetic wave propagation and uid dynamics. In constructing numerical schemes for these problems some diculties may arise. Some of them are:
1. Imposing stable boundary conditions: Close to the boundary one-sided approximations have to be used. These approximations have to be stable, fulll the boundary
conditions and maintain the global accuracy of the scheme.
2. Irregular domains: In multi-dimensional domains the boundary may not necessarily coincide with the nodes of the mesh.
3. Long time integration: The error may grow rapidly in time even when the scheme
is stable in the classical sense.
4. Low viscosity: For some schemes large numerical oscillations may occur when the
boundary-layers are not resolved.
5. Lack of 'modularity': Suppose we have two dierentiation matrices Dx and Dy in
the x and y directions respectively, all the eigenvalues of which have non positive
real part (i.e. they are strictly stable, see the denition below). The sum of these
matrices will not necessarily preserve this virtue. This is an example of lack of
modularity.
In this thesis a method is presented that allows one to construct nite-dierence schemes
which resolve some of these diculties.
1
The standard way to construct nite-dierence schemes that numerically solve a
PDE, or a system of PDE's, on a given mesh is to nd a consistent approximation to
the given problem and then prove that the approximation is stable. Lax's equivalence
theorem insures that the scheme converges, i.e. that for a xed time T the numerical
solution to the scheme converges to the analytic solution of the dierential problem as
the mesh size h ! 0, see for example 33], 13] . Note that even when the scheme is stable
the error may grow exponentially in time, see 5]. Additional types of stability are strong
stability which means that an estimate of the solution at any given time is obtained in
terms of the forcing function, initial data and boundary data and strict stability which
means that the energy dissipation introduced by the boundaries is essentially preserved
by the numerical scheme. In the case of semi-discrete schemes strict stability implies
that all the eigenvalues of the coecient matrix of the corresponding ODE system have
non positive real part.
The stability analysis for fully-discrete algorithms can be a dicult task. The analysis is somewhat simpler when carried out for semi-discrete schemes. If one converts a
semi-discrete scheme to a fully-discrete one by using Runge-Kutta or other multi-step
methods, then stability is assured under conditions given in Kreiss and Wu 22] or Levy
and Tadmor 23].
There are essentially two methods to analyze stability of nite-dierence approximations of PDE's with non-periodic boundary conditions: the energy method and the
Laplace transform method. The rst paper in the area of the Laplace transform method
was written by Goganov and Ryabenkii 14] in 1963. This gave necessary normal-mode
conditions for stability, analogous to the Von Neumann necessary condition for pure initial value problem. The sucient condition was given by Kreiss 18] and the complete
theory for dissipative schemes was presented in 19]. Extentions of Kreiss results, including some nondissipative and implicit methods were in Osher 30]. A general stability
theory, for hyperbolic IBVP, based on the Laplace transform method, for the fullydiscrete case, was presented by Gustafsson, Kreiss and Sundstrom (G.K.S.) 12]. This
was a breakthrough in the study of the stability analysis of nite-dierence schemes.
Later, the theory was generalized to the semi-discrete case by Strikwerda 37]. In 13]
Gustafsson, Kreiss and Oliger prove that, under mild assumptions, G.K.S. stability
2
leads to strong stability. It should be realized however that the above theories are
one-dimensional in nature. In the d-dimensional case they are applicable only if d ; 1
boundary conditions are periodic. Even then the application of this theory in several
space dimensions is complicated. In fact, to my knowledge, there is only one example of
this type of stability analysis carried out for two-dimensional 2 2 hyperbolic systems,
see 4].
Energy methods for stability analysis were introduced in the 50's, see 33]. In the 70's
Kreiss and Scherer 20] 21] proved the existence of dierence operators approximating
a hyperbolic PDE using a summation by parts formula and weighted norms. They also
used a projection operator to impose the boundary conditions on semi-discrete schemes.
Lately this technique was generalized by Olsson 28] 29] and Strand 35]36] who proved
strict stability for a larger class of operators and orders of accuracy. Another approach
to impose boundary conditions and to insure strict stability was presented by Carpenter,
Gottlieb and Abarbanel 8] who used simultaneous approximation terms (SAT) to treat
boundary conditions. Using this technique high order implicit schemes were constructed.
Many problems that arise from applications are concerned with solving PDE.'s in
complex multidimensional geometries. Typical problems are uid ow and electromagnetic wave propagation. Some of the methods which attack this problems are unstructured grids, body-tted coordinates, and overlapping meshes. These methods are
successfully applied in CFD. However, they suer from two major drawbacks - the length
of time it takes to generate the grid, measured in weeks for typical large scale CFD computation, and the complexity added to the computation due to the fact that the PDE's
have to be transformed to the new coordinate system. Another approach is to use a
cartesian grid. This method uses a 'simple' cartesian grid, but requires a complex treatment at boundaries which do not necessarily coincide with grid nodes or cell-surfaces.
Papers describing the use of cartesian grids using nite dierence algorithms and nite
volume methods were published in the past twenty years, see for example 32] 31] 9]
11]. Lately there is a growing interest in cartesian grid methods combined with mesh
renement. One of the goals of current research is to automate grid generation, see for
example 27] 6] 24] 25] and also some of the home-pages devoted to cartesian grids,
e.g., Dr. John Melton :http://oldwww.nas.nasa.gov/melton/cartesian.html and Capt.
3
Michael Aftosmis: http://george.arc.nasa.gov/aftosmis/. Currently there are several
industrial CFD codes that use cartesian grids, for example TRANAIR and MGAERO.
The Finite-Dierence Time-Domain (FDTD) method, was rst applied by Yee in 1966
38] to the problem of solving numerically Maxwell's equations. Yee used two cartesian
grids, an electric eld E-grid and a magnetic eld H-grid which are oset from each
other both spatially and temporally. A leap-frog scheme was utilized to advance the
elds in time. When FDTD methods are applied to problems on complex geometries,
\staircased" or \lego-type" approximations are used to represent the boundaries. These
approximations to the geometry may, in certain cases, lead to signicant errors 7] 15].
Mesh renement, as well as other approaches, are used to overcome this diculty and
to resolve small-scale structures, such as a narrow slot see 26] 17]. See also the FDTD
survey paper 34].
One of the diculties that may arise in the analysis of nite-dierence schemes is that
some of the properties of the discrete operators are not necessarily preserved when the
operators are added. For example, even if we have strictly stable dierentiation matrices
Dx and Dy (i.e. all their eigenvalues have non positive real part), it is not assured that
the representation of @=@x + @=@y by Dx + Dy will possess only eigenvalues with non
positive real part. This implies that it is the matrix Dx + Dy that should be checked
for stability. This analysis may be extremely complicated on complex multidimensional
geometries. Thus arises the motivation to construct 'modular' schemes, in the sense
that the properties of the operators that are essential for stability, or convergence, are
preserved when the operators are added.
In this work a methodology for constructing nite-dierence semi-discrete schemes,
for initial boundary value problems (IBVP), on complex, multi-dimensional shapes is
presented. The implementation of this methodology for constructing nite-dierence
approximations for various operators and orders of accuracy is discussed and selected
numerical verications are presented.
Unlike the 'standard approach' where convergence is proven by using stability, here
we prove convergence directly, by deriving an equation for the error and bounding the
error norm. The standard requirement of stability is replaced by error boundness, which
means that the error norm is bounded by a function of the time t, the mesh size h and
4
the exact solution to the dierential problem u (typically a Sobolev norm of u), i.e.
k k< F (u h t) F < 1 8 t < 1 F ! 0 as h ! 0. Throughout this work we
use the error boundness in a stricter sense. We require that k k, the L2 norm of ,
be bounded by a \constant" proportional to hm (m being the spatial order of accuracy)
for all t < 1, or at most grow linearly in time, the time coecient being proportional
to hm. By Lax's equivalence theorem, a scheme which possesses an error bounded by
a linear growth in time is strictly stable. Additionally, the use of error-boundedness
analysis enables us not only to prove convergence directly, but also to get an estimate
on the actual error generated by the scheme.
The method presented here for building up semi-discrete schemes on complex, multidimensional shapes has as its starting point the construction of one dimensional schemes
on a uniform grid with boundary points that do not necessarily coincide with the extremal nodes of the mesh. The boundary conditions are imposed using simultaneous
approximation terms (SAT) which are a generalization of the penalty method presented
in 8]. The 1-D schemes are built in a way that the coecient matrix of the corresponding ODE system which represents the error evolution in time is negative denite (N.D.)
and bounded away from 0 by a constant independent of the size of the matrix, or is at
least non-positive denite (N.P.D.). These properties, N.D. or N.P.D., enable us to prove
that the scheme is error-bounded by a \constant", or error-bounded by linear growth
in time, respectively. Since a sum of two negative (non-positive) denite matrices is
a negative (non-positive) denite matrix, a multidimensional scheme can be built by
adding dierentiation operators each of which is negative (non-positive) denite. This
is the sense in which such schemes are 'modular'.
The error boundness proof and details of constructing multidimensional schemes on
complex shapes are given in chapter 1.
In chapter 2 a 4th-order accurate scheme is developed for solving the diusion equation in one or more dimensions, on irregular domains. The scheme is constructed on
a rectangular grid using the method presented in chapter 1. Numerical examples in
2-D show that the method is eective even where standard schemes, stable by traditional denitions, fail. The general theory and the results presented in this chapter were
published in 2].
5
In chapter 3 the methodology presented in chapter 1 is used to develop second
order accurate schemes which solve multi-dimensional linear hyperbolic and diusion
equations on complex shapes. These algorithms are used to solve the linear advectiondiusion equation, including the low viscosity case. Numerical examples show that
the method can give a good approximation to the solution outside the boundary-layer
even when the viscosity is low, and the grid is coarse with respect to the boundary
layer thickness. Standard schemes converge much slower and generate oscillations. The
material presented in this chapter was published in 3].
In chapter 4 the methodology presented in chapter 1 is adopted to construct schemes
for parabolic equations and systems containing mixed-derivatives. A second order
accurate bounded-error scheme is constructed to represent a scalar mixed-derivatives
parabolic operator. This scheme is then generalized to solve the diusion part of the
Navier-Stokes equations in two and three space dimensions. This generalization may
also be applicable to other parabolic systems.
In chapter 5 the same method is adopted to solve the wave equation. The diculties
that arise from attempts to apply the method presented in chapter 1 naively are discussed, and a way to resolve them and to solve the problem is proposed. Consequently
we have a second-order accurate bounded-error scheme to solve the wave equation on
complex shapes.
6
Chapter 1
General theory and description of
the method
1.1 The one dimensional case
We consider the following problem
@u = L(u) + f (x t) ; x ; t 0
L
R
@t
u(x 0) = u0(x)
(1:1:1b)
BL (u(;L t)) = gL(t)
(1:1:1c)
BR(u(;R t)) = gR(t)
(1:1:1d)
(1:1:1a)
Where L, BL, BR are linear dierential operators. For example: in the inhomoge@ 2u +f (x t) k > 0,
neous diusion equation with Dirichlet boundary condition L(u) = k @x
2
BL(u(;L t)) = u(;L t), and BR(u(;R t)) = u(;R t). In the inhomogeneous hyperbolic equation with Dirichlet boundary condition L(u) = a @u
@x + f (x t) for a < 0,
BL(u(;L t)) = u(;L t), and no boundary condition is given on the right side.
Let us spatially discretize (1.1.1a) on the following uniform grid:
∆x=h
γ h
L
ΓL
x1
x2
x3
xj-1
xj
γ h
R
xj+i
Figure 1.1: One dimensional grid.
7
xN-2
xN-1
ΓR
xN
Note that the boundary points do not necessarily coincide with x1 and xN . Set xj+1 ;
xj = h, 1 j N ; 1 x1 ; ;L = L h 0 L < 1 ;R ; xN = Rh 0 R < 1.
The projection of the exact solution u(x t) to (1.1.1) onto the above grid is uj (t) =
u(xj t) =4 u(t), similarly fj (t) = f (xj t) =4 f (t) is the projection of the inhomogeneous
term. Let D~ be a matrix representing numerical approximation to the dierential operator L at internal points, without specifying yet how it is being built. Then we may
write
d u(t) = D~ u(t) + B + f + T
(1.1.2)
dt
where T is the truncation error due to the numerical dierentiation. The boundary
vector B has entries whose values depend on gL gR, L R in such a way that D~ +B
represents the dierential operator L everywhere to the desired accuracy. The standard
way of nding a numerical approximate solution to (1.1.1) is to omit T from (1.1.2) and
solve
d v(t) = D~ v(t) + B + f
(1.1.3)
dt
where v(t) is the numerical approximation to the projection u(t). An equation for the
solution error vector, (t) = u(t) ; v(t), can be found by subtracting (1.1.3) from
(1.1.2):
d = D~ (t) + T(t)
(1.1.4)
dt
Unlike the 'standard approach' where convergence is proven by using stability, here
we prove convergence directly, by deriving an equation for the error and bounding the
error norm. The standard requirement of stability is replaced by error boundness which
means that the error norm is bounded by a function of the time t, the mesh size h and
the exact solution to the dierential problem u (typically a Sobolev norm of u), i.e.
k k< F (u h t) F < 1 8 t < 1 F ! 0 as h ! 0. Throughout this work, our
requirement for error boundness is that either k k, the L2 norm of , be bounded by
a \constant" proportional to hm (m being the spatial order of accuracy) for all t < 1,
or at most grow linearly in time, the time coecient being proportional to hm . Note
that this requirement is more severe than strong stability, which allows for exponential
temporal growth of the error.
In many cases it can be shown that if D~ is constructed using central dierencing in
8
a standard manner, i.e., away from the boundaries the numerical second derivative is
symmetric and the numerical rst derivative is antisymmetric, (and near the boundaries
one uses \non-symmetric" dierentiation), then there are ranges of R and L for which
D~ is not negative denite. Since in the multi-dimensional case one may encounter all
values of 0 L R 1, this is unacceptable.
The \common practice" is to use D~ v + B to approximate D~ u + B. Here, however, the
basic idea is to use a dierentiation matrix D to approximate the dierential operator
L everywhere and use a penalty-like term in the numerical algorithm to represent the
boundary conditions. This way of imposing the boundary conditions gives us more
degrees of freedom in the construction of the scheme. Special properties of the scheme,
like non-positive deniteness in L2 norm, can now be achieved using these extra degrees
of freedom. This idea of using penalty-like term was presented in the SAT procedure of
ref 8] here, however, it will be modied and applied in a dierent manner.
Note rst that the solution projection uj (t) satises, besides (1.1.2), the following
dierential equation:
du = Du + f + T
(1.1.5)
e
dt
where now D is indeed a representation of L, that does not use the boundary values, and
therefore Te 6= T but it too is a truncation error due to dierentiation.
Next let the semi-discrete problem for v(t) be, instead of (1.1.3),
dv = Dv ; (A v ; g ) ; (A v ; g )] + f
(1.1.6)
L L
L
R R
R
dt
where gL = (1 : : : 1)T gL(t) gR = (1 : : : 1)T gR (t), are vectors created from the left
and right boundary values as shown. The matrices AL and AR are dened by the
relations:
ALu = gL ; TL ARu = gR ; TR
(1.1.7)
i.e., each row in AL(AR) is composed of the coecients extrapolating u to its boundary
value gL(gR), at ;L(;R ) to within the desired order of accuracy. ( The error is then
TL(TR) ).
The diagonal matrices L and R are given by
L = diag (L1 L2 : : : LN )
R = diag (R1 R2 : : : RN ) :
9
(1.1.8)
Though in principle the penalty terms (ALv ; gL) and (ARv ; gR) are added to each
point, in practice they are added just near the boundaries, i.e. most of the Lj s and Rj s
are zero. Subtracting (1.1.6) from (1.1.5) we get
d = D ; A ; A ] + T
(1.1.9)
L L
R R
1
dt
where
T1 = Te ; LTL ; RTR
Taking the scalar product of with (1.1.9) one gets:
1 d k k2 = ( (D ; A ; A ) ) + ( T )
L L
R R
1
2 dt
= ( M ) + ( T1 )
(1.1.10)
We notice that ( M ) is ( (M + M T ) )=2, where
M = D ; LAL ; RAR:
(1.1.11)
If M + M T can be made negative denite then
( (M + M T ) )=2 ;c0 k
k2
(c0 > 0)
(1.1.12)
where ;c0 is an upper bound on the largest eigenvalue of M + M T . Equation (1.1.10)
then becomes
1 d k k2 ;c k k2 +( T )
0
1
2 dt
and using Schwarz's inequality we get after dividing by k
d k k ;c k k + k T k
0
1
dt
and therefore (using the fact that v(0) = u(0))
k
k k k Tc0 kM (1 ; e;c t)
\constant" k T kM = max0 t k T ( ) k. This
1
0
(1.1.13)
where the
\constant" is function of
1
1
the exact solution u and its derivatives.
If we indeed succeed in constructing M such that M + M T is negative denite,
with c0 > 0 independent of the size of the matrix M as it increases, then it follows
10
from (1.1.13) that the norm of the error will be bounded for all t by a constant which
is O(hm ) where m is the order of the spatial accuracy of the nite dierence scheme
(1.1.6). The algorithm is then a bounded error scheme, and as h ! 0, v(t) converges to
u(t).
When c0 = 0, as in the case of hyperbolic equations, the dierential inequality is
dk
dt
leading to
k k T1 k
(1.1.14)
k k k T1 kM t (1.1.15)
i.e. a linear growth in time, a result typical of hyperbolic systems convergence, however,
is unaected.
1.2 The multi dimensional case
In this section we show how to use a one-dimensional scheme, whose properties were
described in the previous section, as a building block for multi-dimensional schemes
Pd (xr )
d
of the type @u
@t = r=1 L (u) + f (x t) x = (x1 : : : xd ) 2 R . For the sake of
simplicity we describe in detail the construction in the two-dimensional case (d = 2). A
brief explanation on how to construct a multi-dimensional schemes in d dimensions is
given at the end of this section. The more general case, where the evolution operator
contains mixed derivatives, will be examined in chapter 4.
We consider the following inhomogeneous linear dierential equation, with constant
coecients, in a domain . To begin with we shall assume that is convex and has
a boundary curve @ 2 C 2. The convexity restriction is for the sake of simplicity in
presenting the basic idea it will be removed later. We thus have
@u = L(x)(u) + L(y)(u) + f (x y t) x y 2 t 0
@t
u(x y 0) = u0(x y)
B (u(x y t))j@ = uB (t)
We shall refer to the following grid representation:
11
(1:2:1a)
(1:2:1b)
(1:2:1c)
γ ∆y
T
y
k=MR
γ ∆x
L
γ ∆x
R
Ω
k
k=2
k=1
j=1 j=2 j=3
γB ∆y
j
j=MC
x
Figure 1.2: Two dimensional grid.
We have MR rows and MC columns inside . Each row and each column has a discretized
structure as in the one 1-D case, see Figure 1.2, with !x not necessarily being equal
to !y. Let the number of grid points in the kth row be denoted by Rk and similarly
let the number of grid points in the j th column be Cj . Let the solution projection be
designated by Ujk (t). By U(t) we mean, by analogy to the 1-D case,
U(t) = (u11 u21 : : : uR 1 u12 u22 : : : uR 2 : : : u1MR u2MR : : :uRMR MR )
(u1 u2 : : : uMR ):
(1.2.2)
1
2
One should note that the x-position of j = 1 is dierent on dierent rows ( i.e.
dierent k's).
Thus, we have arranged the solution projection array in vectors according to rows,
starting from the bottom of . The projection of the forcing function, F(t), was also
arranged in the same manner.
If we arrange this array by columns (instead of rows) we will have the following
structure
U(C)(t) = (u11 u12 : : : u1C u21 u22 : : : u2C : : : uMC 1 uMC 2 : : : uMC CMC )
(u(1C) u(2C) : : : u(MCC) )
(1.2.3)
1
2
12
Since U(C)(t) is just a permutation of U(t), there must exist an orthogonal matrix
P such that
U(C)(t) = P U:
(1.2.4)
If the length of U(t) is `, then P is an ` ` matrix whose each row contains ` ; 1 zeros
and a single 1 somewhere.
The dierential operator L(x) in (1.2.1a) is represented on the kth row by the dierentiation matrix Dk(x), whose structure is the same as that of D in (1.1.5). Similarly let
L(y) be given on the j th column by Dj(y) , whose structure is also given by (1.1.5). With
this notation the projection of L(x) + L(y) is:
;L(x) + L(y) u (t) = D(x)U + D(y) U(C) + T(x) + T(y) ij
e
e ij
(1.2.5)
where D(x) and D(y) are (` `) matrices with a block structures shown in (1.2.6).
2 D(x)
1
6
D2(x)
D(x) = 664
...
DM(xR)
2 D (y )
3
77 (y) 66 1 D2(y)
75 D = 64
...
DM(y)C
3
77
75
(1.2.6)
T(ex) and T(ey) are the truncation errors associated with D(x) and D(y) , respectively.
We now call attention to the fact that D(x) and D(y) do not operate on the same vector.
This is xed using (1.2.4):
;L(x) + L(y) u (t) = ;L(x) + L(y) U = (D(x) + P T D(y) P )U + T(x) + P T T(y)
ij
e
e
Thus (1.2.1a) becomes, by analogy to (1.1.5),
dU = (D(x) + P T D(y) P )U + (T(x) + P T T(y)) + F
e
e
dt
Before proceeding to the semi-discrete problem let us dene:
Mk(x) = Dk(x) ; Lk ALk ; Rk ARk
(1.2.7)
(1.2.8)
(1.2.9)
where Lk ALk are the L and AL dened in section 1.1, appropriate to the kth row
similarly for Rk and ARk . In the same way, dene
Mj(y) = Dj(y) ; Bj ABj ; Tj ATj
13
(1.2.10)
where the subscripts B and T stand for bottom and top respectively.
We can now write the semi-discrete problem by analogy to (1.1.6)
dV = (M(x) + P T M(y)P )V + G(x) + P T G(y) + F y
dt
where V is the numerical approximation to U
2 M (x)
1
6
M2(x)
6
(
x
)
M = 64
and
...
MM(xR)
3
2 M (y)
77 (y) 66 1 M2(y)
75 M = 64
...
h
MM(yC)
(1.2.11)
3
77
75 (1.2.12)
i
G(x) = (L gL + R gR ) : : : (Lk gLk + Rk gRk ) : : : (LMR gLMR + RMR gRMR ) h
i
G(y) = (B gB + T gT ) : : : (Bj gBj + Tj gTj ) : : : (BMC gBMC + TMC gTMC ) :
1
1
1
1
1
1
1
1
(1.2.13)
Subtracting (1.2.11) from (1.2.8) we get in a fashion similar to the derivation of (1.1.9):
dE = (M(x) + P T M(y)P )E + T
(1.2.14)
2
dt
where E = U ; V is the two dimensional array of the errors, ij , arranged by rows as a
vector. T2 is proportional to the truncation error.
The time rate of change of k E k2 is given by
1 d k E k2= (E (M(x) + P T M(y)P )E) + (E T )
(1.2.15)
2
2 dt
The symmetric part of M(x) + P T M(y)P is given by
1 (M(x) + M(x)T ) + P T (M(y) + M(y)T )P ]
(1.2.16)
2
Note that when this scheme is used in practice, the 1-D algorithem (1.1.6) is implemented on each
row, to compute the numerical approximation to (X ) , and on each column, to compute the numerical
approximation to (y) . Therefor the scheme may be written as:
v = h (x) V + G(x) + (y) V(C ) + G(y) + Fi
y
L
L
d
ij
dt
M
M
ij
:
Writing the scheme in the manner of equation (1.2.11), enables us to use standard linear-algebra
manipulations, and thus make the proof easier.
14
Clearly M(x) + M(x)T and M(y) + M(y)T are block-diagonal matrices with typical
T
T
blocks given by Mk(x) + Mk(x) and Mj(y) + Mj(y) . We have already shown in the one
dimensional case that each one of those blocks is either negative denite and bounded
away from zero by c0, or is non-positive denite. Therefore the operator (1.2.16) is also
negative denite and bounded away from zero, or non-positive denite. The rest of the
proof follows the one dimensional case and thus the norm of the error, k E k, is bounded
by a constant, or, in the hyperbolic case, at most grows linearly with t.
If the domain is not convex or simply connected then either rows or columns,
or both, may be \interrupted" by @ . In that case the values of the solution on each
\internal" interval (see Figure 1.3 below) are taken as separate vectors.
y
Uj
(c),(2)
Ω
(1)
Uk
k
Uj
Ω
(2)
Uk
(c),(1)
x
j
Figure 1.3: Two dimensional grid, non convex domain.
Decomposing \interrupted" vectors in this fashion leaves the previous analysis unchanged. The length of U (or U(C)) is again `, where ` is the number of grid nodes
inside . The dierentiation and permutation matrices remain ` `. Note that adding
more \holes" inside @ does not change the general approach.
If the domain 2 Rd , in practice algorithm (1.1.6) is implemented on each direction
xr r = 1 : : : d, in the same way it was implemented for each row, to compute in the
15
two-dimensional case, see footnote to equation (1.2.11). In order to prove the convergence of this algorithm we write the scheme as in equation (1.2.11), using the proper
permutation matrices. Then we get a bound on on the error norm as was done in the
two-dimensional case.
The following chapters describe how to construct M , see (1.1.11), for various operators and orders of accuracy, so that the basic assumption of the theory, i.e. that
M is either negative denite or non-positive denite is fullled. Selected numerical
verications are also presented.
16
Chapter 2
The diusion equation
This chapter is concerned with 4th-order approximations to the the diusion equation
in one and two dimensions, on irregular domains, using the method presented in chapter
1.
In section 2.1 we develop the one-dimensional semi-discrete algorithm that approximate second derivative to 4th-order accuracy. This scheme satises the requirement
presented in chapter 1, therefore the scheme is bounded by a \constant" and can be
generalized to the multi-dimensional case, i.e. the Laplace operator.
Section 2.2 presents numerical results in two space dimensions demonstrating the
error boundeness of the scheme, constructed in section 2.1. These results show that the
method is eective even where standard schemes, stable by traditional denitions, fail.
2.1 Construction of the scheme
We consider the following problem
@u = @ 2u + f (x t) ; x ; t 0 k > 0
L
R
@t @x2
u(x 0) = u0(x)
(2:1:1b)
u(;L t) = gL (t)
(2:1:1c)
u(;R t) = gR (t)
(2:1:1d)
(2:1:1a)
and f (x t) 2 C 4.
As in chapter 1 let us discretize (2.1.1a) spatially on the following uniform grid:
17
∆x=h
γ h
L
ΓL
x1
x2
x3
xj-1
xj
γ h
R
xj+i
xN-2
xN-1
ΓR
xN
Figure 2.1: One dimensional grid.
Note that the boundary points do not necessarily coincide with x1 and xN . Set xj+1 ;
xj = h, 1 j N ; 1 x1 ; ;L = L h 0 L < 1 ;R ; xN = Rh 0 R < 1.
The projection of the exact solution u(x t) to (2.1.1) onto the above grid is uj (t) =
u(xj t) =4 u(t), similarly fj (t) = f (xj t) =4 f (t) is the projection of the inhomogeneous
term. Let v(t) be the numerical approximation to the projection u(t), given by equation
(1.1.6).
dv = Dv ; (A v ; g ) ; (A v ; g )] + f (t)
L L
L
R R
R
dt
= M v + LgL + RgR + f (t)
(2.1.2)
where
du = Du + f (t) + T :
e
dt
gL = (1 : : : 1)T gL (t) gR = (1 : : : 1)T gR(t), are vectors created from the left and
right boundary values as shown. The matrices AL and AR are dened by the relations:
ALu = gL ; TL
ARu = gR ; TR
i.e., each row in AL(AR) is composed of the coecients extrapolating u to its boundary
value gL(gR), at ;L(;R ) to within the desired order of accuracy. ( The error is then
TL(TR) ).
The diagonal matrices L and R are given by
L = diag (L1 L2 : : : LN )
R = diag (R1 R2 : : : RN )
The numerical solution error vector is given by,
(t) = u(t) ; v(t)
18
where
and
d = Mv + T
1
dt
T1 = Te ; LTL ; RTR:
The rest of this section is concerned with the case of m = 4, i.e, a spatially fourth
order accurate nite dierence algorithm.
Let the n n dierentiation matrix, D, be given by
2 45 ;154 214 ;156
66 10 ;15 ;4 14
66 ;1 16 ;30 16
66
;1 16 ;30
66
;1 16
1 6
...
12h2 666
66
66
64
61
;6
;1
16
;30
...
;1
;10
1
;1
3
77
77
77
77
77
77
;1 777
16 ;1 77
;15 10 5
;1
... ...
;30 16 ;1
16 ;30 16
;1 16 ;30
1 ;6
14 ;4
;10 61 ;156 214 ;154
16
...
16
;1
45
(2.1.3)
The upper two rows and the lower two rows represent non-symmetric fourth order
accurate approximation to the second derivative without using boundary values. The
internal rows are symmetric and represent central dierencing approximation to uxx to
the same order. Note that D is not negative denite, and neither is the symmetric part
of 21 (D + DT ) which is given by:
19
3
2 90 ;144 213 ;156 61 ;10
77
66 ;144 ;30 12 13 ;6 1
77
66 213 12 ;60 32 ;2 0
77
66 ;156 13 32 ;60 32 ;2
77
66 61 ;6 ;2 32 ;60 32 ;2
77
66 ;10 1 0 ;2 32 ;60 32 ;2
77
66
1 6
77
... ... ... ... ...
6
2
77
6
24h 6
77
66
2
32
;
60
32
;
2
0
1
;
10
;
66
;2 32 ;60 32 ;2 ;6 61 777
66
;2 32 ;60 32 13 ;156 77
66
0 ;2
32 ;60
12 213 77
64
1 ;6
13 12 ;30 ;144 5
;10
;156
61
213
;144
In order to construct M we need to specify AL, AR, L and R. We construct AL as
follows:
AL = A(L) + cLA(eL)
(2.1.4)
2 1 2 3 4 5
6
A(L) = 664 ..1 2 3 4 5
.
where
0 ::: 0
0 ::: 0
1 2 3 4 5 0 : : : 0
3
77
75 cL = diag ;201=71 0 : : : 0]
2
;1
6
;1
A(eL) = 664 ..
.
The 's are given by
;1
5
5
5
;10
;10
;10
10
10
10
;5
;5
;5
(2.1.6)
1 0 ::: 0
1 0 ::: 0
1 0 ::: 0
25 + 35 2 + 5 3 + 1 4
1 = 1 + 12
L
24 L 12 L 24 L
2 =
;
2 + 3 3 + 1 4
4L + 13
L
3
2 L 6 L
20
(2.1.5)
3
77
75 :
(2.1.7)
90
2 + 2 3 + 1 4
3 = 3L + 19
L
L 4 L
4
4 =
;
4
7 2 7 3 1 4
L + L + L + L
3
3
6
6
(2.1.8)
2 + 1 3 + 1 4
5 = 14 L + 11
L
24
4 L 24 L
Note that A(L)v gives a vector whose components are the extrapolated value of v at x =
;L (i.e., v;L (t)), to fth order accuracy while A(eL)v gives a vector whose components
represents (@ 5v1=@x5)h5. Since cL (see 2.1.6) is of order unity, then ALv = (A(L) +
cLA(eL))v represents an extrapolation of v to v;L to fth order.
Before using AL in (2.1.2) we dene L:
L = 121h2 diag1 2 3 4 5 0 : : : 0]
(2.1.9)
where
1
2
3
4
5
=
=
=
=
=
71=21
(;94 ; 21)=1
(113 ; 31)=1
(;56 ; 41)=1
(11 ; 51)=1
(2.1.10)
The right boundary treatment is constructed in a similar fashion, and the formulae
corresponding to (2.1.5) - (2.1.11) become:
2 0 ::: :::
66 0 : : : : : :
(
R
)
A = 64 ..
.
AR = A(R) + cRAe(R)
3
0 0 N ;4 N ;3 N ;2 N ;1 N
0 0 N ;4 N ;3 N ;2 N ;1 N 77
75 0 : : : : : : 0 0 N ;4 N ;3 N ;2 N ;1 N
cR = diag0 0 : : : 0
21
;20N =71]
(2.1.11)
(2.1.12)
(2.1.13)
20
66 0
(
R
)
Ae = 64 ..
.
The 's here are:
0 ::: 0 1
0 ::: 0 1
0 0 ::: 0 1
;5
;5
;5
10
10
10
;10
;10
;10
;1 37
;1 77
5
;1
5
5
5
(2.1.14)
25 + 35 2 + 5 3 + 1 4
N = 1 + 12
R
24 R 12 R 24 R
N ;1 =
;
3 + 3 3 + 1 4
4R + 13
R
3
2 R 6 R
2 + 2 3 + 1 4
N ;2 = 3R + 19
R
R 4 R
4
N ;3 =
;
4
7 2 7 3 1 4
R + R + R + R
3
3
6
6
(2.1.15)
2 + 1 3 + 1 4 N ;4 = 14 R + 11
R
24
4 R 24 R
R = 121h2 diag0 : : : N ;4 N ;3 N ;2 N ;1 N ]
N =
N ;1 =
N ;2 =
N ;3 =
N ;4 =
71=2N
(;94 ; N ;1N )=N
(113 ; N ;2N )=N
(;56 ; N ;3N )=N
(11 ; N ;4N )=N
(2.1.16)
(2.1.17)
Note that the extrapolating errors TL and TR are only O(h5). Therefore the scheme
near the boundaries is third order accurate.
In order to show that the error estimate (1.1.13) is valid, i.e. the error is bounded
by a \constant00, it is sucient to prove that 12 (M + M T ) is negative denite, and its
eigenvalues are bounded away from zero. The proof is done by writing 12 (M + M T ) as
a sum of ve matrices. One matrix is negative denite and the other are non-positive
22
ones. The details are given in the appendix to this chapter, section 2.4, where it is
proven that 21 (M + M T ) is indeed negative denite, and its eigenvalues are bounded
away from zero by (;2=24), even as N ! 1.
In the 2-D case the mesh size h becomes !x or !y respectively, when used in
conjunction with D(x) or D(y) .
2.2 Numerical example
In this section we describe numerical results for the following problem:
@u = (u + u ) + f (x y t)
(x y) 2 t > 0
(2.2.1)
xx
yy
@t
where is the region contained between a circle of radius ro = 1=2 and inner circle of
radius ri 0:1. The inner circle is not concentric with the outer one. Specically is
described by
(x ; :5)2 + (y ; :5)2 1=4 \ (x ; :6)2 + (y ; :5)2 (:1 ; )2 0 :1
(2.2.2)
The geometry thus looks as follows:
y
1
ro =.5
ri =.1- δ
0.5
Ω
y=0
0.5 0.6
x=0
1
x
Figure 2.2:
The Cartesian grid in which is embedded spans 0 x y 1. We took !x = !y,
and ran several cases with !x = 1/50, 1/75, 1/100.
23
The source function f (x y t) was chosen dierent from zero so that we could assign
an exact analytic solution to (2.2.1). This enables one to compute the error Eij =
Uij ; Vij \exactly" (to machine accuracy). We chose = 1 and
u(x y t) = 1 + cos(10t ; 10x2 ; 10y2)
(2.2.3)
f (x y t) = 400(x2 + y2) cos(10t ; 10x2 ; 10y2)
; 50 sin(10t ; 10x2 ; 10y2)
(2.2.4)
This leads to
From the expression for u(x y t) one obtains the boundary and initial conditions.
The problem (2.2.1), (2.2.2), (2.2.4) was solved using both a \standard" fourth order
algorithm and the new \SAT", or \bounded error", see footnote to equation (1.2.11).
The temporal advance was via a fourth order Runge-Kutta.
The standard algorithm was run for !x = 1=50 and a range of 0 < :01 (:09 <
ri :1). We found that for :0017323, the runs were stable and the error bounded for
\long" times (105 time steps, or equivalently t = 2). For 0 < :0017233 the results
began to diverge exponentially from the analytic solution. The \point of departure"
depended on . A discussion of these results is deferred to the next section. Figures 2.3
to 2.5 show the L2-norm of the error vs. time for dierent radii of the inner \hole".
The same congurations were also run using the \bounded error" algorithm, and
the results are shown in gures 2.6 to 2.9. It is seen that for 's for which the standard
methods fails, the new algorithm still has a bounded error, as predicted by the theory.
To check on the order of accuracy, the \SAT" runs (with = 0) were repeated
for !x = !y = 1=75 and 1/100. Figure 2.10, 2.11, and 2.12 show the logarithmic
slope of the L2 L1 and L1 errors to be less than ;4 i.e., we indeed have a 4th order
method. That the slopes are larger in magnitude than 4.5 is attributed to the fact
that as !x = !y decreases the percentage of \internal" points increases (the boundary
points have formally only 3rd oder accuracy). It is therefore possible that if the number
of grid points was increased much further, the slope would tend to ;4. Lack of computer
resources prevented checking this point further. (For !x = 0:01, running 20,000 time
steps, t = 1:0, the cpu time on a CRAY YMP is about 5 hours). It should also be noted
24
that the \bounded-error" algorithm was run with a time step, !t, twice as large as the
one used in the standard scheme. At this larger !t the standard scheme \explodes"
immediately.
err
err
0.0003
0.0003
0.00025
0.00025
0.0002
0.0002
0.00015
0.00015
0.0001
0.0001
0.00005
0.00005
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
t
Figure 2.3: = 0:0017325, Standard
scheme
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
t
Figure 2.4: = 0:0017323, Standard
scheme
err
err
140000
0.0003
120000
0.00025
100000
0.0002
80000
0.00015
60000
0.0001
40000
0.00005
20000
0.0002
0.0004
0.0006
0.0008
t
0.001
Figure 2.5: = 0:0015, Standard
scheme
0
0.5
1
1.5
2
2.5
3
Figure 2.6: = 0, SAT scheme
25
3.5
4
t
err
err
0.0003
0.0003
0.00025
0.00025
0.0002
0.0002
0.00015
0.00015
0.0001
0.0001
0.00005
0.00005
0
0.5
1
1.5
2
2.5
3
3.5
4
t
Figure 2.7: = 0:0015, SAT scheme
0
err
1.5
2
2.5
3
3.5
4
t
Log[L1]
Fit: 4.12882 - 4.64381 Log[N]
-2.5
0.00025
-3
0.0002
-3.5
0.00015
-4
0.0001
-4.5
0.00005
-5
0.5
1
1.5
2
2.5
3
3.5
4
1.7
1.8
1.9
2
Log[N]
2.1
t -5.5
Figure 2.9: = 0:0017325, SAT scheme
Figure 2.10: Order of accuracy L1
Log[L2]
Log[max]
Fit: 4.41253 - 4.67035 Log[N]
Fit: 4.88824 - 4.57438 Log[N]
-2.5
-2.5
-3
-3
-3.5
-3.5
-4
-4
-4.5
-4.5
-5
1
Figure 2.8: = 0:0017323, SAT scheme
0.0003
0
0.5
1.7
1.8
1.9
2
2.1
Log[N]
2.2
-5.5
-5
1.7
1.8
1.9
2
Log[N]
2.1
-5.5
Figure 2.11: Order of accuracy L2
Figure 2.12: Order of accuracy L1
A study of the eect of the size of !t shows that the instabilities exhibited above by
the \standard" scheme are due to the time-step being near the C.F.L.-limit. The reason
for this strong dependence on the geometry is that the \standard" dierentiation matrix
has entries which are O (1=(h2 )). In the \SAT" dierentiation matrix the entries are
O (1=(h2(1 + ))). This instability problem could probably be solved also by using an
26
implicit method for the time integration.
2.3 Conclusions
(i) The theoretical results show that one has to be very careful when using an algorithm whose dierentiation matrix, or rather its symmetric part, is not negative
denite. For some problems, such \standard" schemes will give good answers (i.e.,
bounded errors) and for others instability will set in. Thus, for example, the \standard" scheme for the 1-D case has a matrix which, for all 0 < L R < 1, though
not negative denite has eigenvalues with negative real parts. This assures, in the
1-D case, the error boundness. In the 2-D case, even though each of the block
sub-matrices of the ` ` x-and-y dierentiation matrices has only negative (realpart) eigenvalues, it is not assured that the sum of the two ` ` matrices will have
this property. This depends, among other things, on the shape of the domain and
the mesh size (because the mesh size determines, for a given geometry, the L and
R's along the boundaries).
Thus we might have the \paradoxical" situation, that for a given domain shape,
successive mesh renement could lead to instability due to the occurrence of destabilizing 's. This cannot happen if one constructs, as was done here, a scheme
whose dierentiation matrices have symmetric parts that are negative denite.
It is also interesting to note that if one uses explicit standard method then the
allowable C.F.L. may decrease extremely rapidly with change in the geometry that
causes decrease in the 's. This point is brought out in gures 2.3 to 2.5.
(ii) Note that the construction of the 2-D algorithm, and its analysis, which were
based on the 1-D case, can be extended in a similar (albeit more complex) fashion
to higher dimensions.
(iii) Also note that if the diusion coecient , in the equation
ut = r2u
is a function of the spatial coordinates, = (x y z), the previous analysis goes
27
through but the energy estimate for the error is now for a dierent, but equivalent
norm.
2.4 Appendix
Using D given in (2.1.3) and AL L AR R given in equations (2.1.5) to (2.1.18) we are
now ready to construct
1 (M + M T ) = 1 D + DT
2
2
;
;
L(A(L) + cLA(eL)) + R(A(R) + cRA(eR))]
L(A(L) + cLA(eL)) + R(A(R) + cRA(eR))]T
(2.4.1)
Upon using equations (2.1.3)-(2.1.18) in (2.4.1) one gets:
M + MT =
22
1
24h2
66
66
66
66
66
66
66
66
66
66
64
0
(
L
)
W
0
;2
32
0 :::
0 ;2 32 ;60
;2 32
;2
0
;2
32
;60
32
...
0
;2
32 ;2
;60 32 ;2
... ... ...
;2 32 ;60
;2 32
;2
0
0
where W (L) and W (R) are 6 6 blocks given by:
0
...
32
;2
0
W (R)
3
77
77
77
77
77
77
77
7
: : : 0 77
77
77
5
(2.4.2)
W (L) = W1(L) + W2(L)
(2.4.3)
W (R) = W1(R) + W2(R)
(2.4.4)
28
9
8 0
i = 1 or j = 1 =
<
W1(ijL) = :
;(ij + j i) i j 6= 1 80
9
i
= N or j = N =
<
W1(ijL) = :
;(N ;iN ;j + N ;j N ;i) 1 < i j < 5
0 N ; i N ; j 4
2 ;1 0 0 0 0 0 3
66 0 ;30 12 13 ;6 1 77
6
;60 32 ;2 0 777
W2(L) = 66 00 12
60 32 ;2 7
64 0 ;136 ;322 ;32
;60 32 5
0
1
;2
0
32
2 ;60 32 ;2 0 1
66 32 ;60 32 ;2 ;6
6
32 13
W2(R) = 66 ;20 ;322 ;60
32 ;60 12
64 1 ;6 13
12 ;30
0
0
0
0
;60
(2.4.5)
(2.4.6)
(2.4.7)
3
0
0 77
0 77
0 77
05
0 ;1
(2.4.8)
The next task is to show that M~ = 21 (M + M T ) is negative denite. We write the
symmetric matrix M~ as a sum of ve symmetric matrices,
h
i
M~ = 241h2 0M~ 1 + 2M~ 2 + (24 ; 0)M~ 3 + M~ 4 + M~ 5 :
(2.4.9)
We shall show that M~ 1 is negative denite, and that M~ j (j = 2 : : : 5) are non-positive
denite. The M~ 's are given by
3
2;1 0 0
20
77
66 0 ;2 1 0 0
77
66 0 1 ;2 1 0
77
66 0 0 1 ;2 1
77 = M1L + M^ 1 + M1R (2.4.10)
6
M~ 1 = 66 0 0 0 1 ;2 1
77
... ... ...
66
7
66
1 ;2 1
0 77
4
1 ;2
05
1
0 0 ; 20
29
where
2 ;1=2
0
6
0
M1L = 664
0
0
...
0
3
2
77 R 66 0
75 M2 = 64
...
and M^ 1 is the remaining (N ; 2) (N ; 2) middle block.
20
66 0
66 0
66 0
66 0
66 0
66
M~ 2 = 666
66
66
66
66
66
4
20
66 0
66 0
66 0
66 0
66 0
66 0
66
66
~
M3 = 66
66
66
66
66
66
66
64
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;1
2
;1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;1
1
0
0
0
0
2 ;1
;5 4 ;1
4 ;6 4 ;1
... ... ... ... ...
;1 4 ;6 4
;1 4 ;5
;1 2
0
0
0
0
0
0
0
;1=20
3
77
77
77
77
77
77
77
77
7
0 77
0 77
0 77
0 77
05
0
;1
2
;1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
;2
1
... ... ...
1
0
30
;2
1
1 ;1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(2.4.11)
0
0
0
0
0
0 0
0
1
;2
1
3
77
75
0
0
0
0
0
0
3
77
77
77
77
77
77
77
77
77
77
77
7
0 77
0 77
0 77
0 77
05
0
0
0
0
0
0 0
(2.4.12)
2 ;1=2
0
0
0
0
66
66
;30 + 2
12 ; 13
;6
66 0
66
;222 ;(23 + 32) ;(24 + 42) ;(25 + 52)
66
66
12 ; ;60 + 2
32 ; ;2
66 0
66
;(23 + 32) ;233 ;(34 + 43) ;(35 + 53)
~
M4 = 66
66
13
32 ; ;60 + 2
32 ; 66 0
66
;(24 + 42) ;(34 + 43) ;244 ;(45 + 54)
66
66
;6
;2
32 ; ;58 + 66 0
66
;(25 + 52) ;(35 + 53) ;(45 + 54) ;255
4
0
1
;2
0
28 ; 3
77
77
77
1
77
77
77
77
0
77
77
77
;2 777
77
77
7
28 ; 77
77
5
0
;26 + (2.4.13)
31
2 ;26 + 66
66
66
66 28 ; 66
66
66
66 ;2
66
~
M5 = 66
66
66 0
66
66
66
66 1
66
66
4
0
28 ; ;2
0
3
77
77
77
77
77
77
77
77
77
77 :
77
77
77
77
77
77
77
77
5
1
0
;58 + 2 32 ; ;2
;6
;2N ;4N ;4 ;(N ;3N ;4 ;(N ;2N ;4 ;(N ;1N ;4
0
32 ; ;60 + 2 32 ; 13
;(N ;3N ;4 ;2N ;3N ;3 ;(N ;2N ;3 ;(N ;1N ;3
0
+N ;4N ;3) +N ;4N ;2) +N ;4N ;1)
+N ;4N ;3)
;2
32 ; ;(N ;2N ;4 ;(N ;2N ;3
+N ;4N ;2) +N ;3N ;2)
+N ;3N ;2) +N ;3N ;1)
;60 + 2 12 ; ;2N ;2N ;2 ;(N ;1N ;2
0
;6
13
12 ; ;30 + 2
;(N ;1N ;4 ;(N ;1N ;3 ;(N ;1N ;2 ;2N ;1N ;1
0
+N ;2N ;1)
+N ;4N ;1) +N ;3N ;1) +N ;2N ;1)
0
0
0
;1=2
0
(2.4.14)
Let us consider M^ 1 - see (2.4.10) it may be decomposed as follows:
2 1 ;1
66 . . .
6
M^ 1 = ; 66
64
...
... ...
...
;1
1
32 1
77 66 ;1
77 66
77 66
54
...
... ...
... ...
;1 1
3 2
77 6 0
77 + 66
77 66
5 4
...
0
...
;1
3
77
77
75
(2.4.15)
The last matrix in non-positive denite. The rst term is a product of a regular matrix
with its transpose, hence its negative is a negative denite matrix. Thus we established
that M^ 1 is negative denite for any nite dimension N . All its eigenvalues are negative.
It remains to show that the eigenvalues of M~ 1=h2 (see (2.4.9) ) are bounded away from
zero by a constant as h ! 0 (N ! 1).
32
Consider a symmetric tridiagonal matrix S with, like M^ 1, constant diagonals:
2b a 0
66 a b a
66 0 a b a
S = 66
... ... ...
64
a b
3
77
77
77 :
7
a5
(2.4.16)
a b
Designate
b a by Dj the determinant of the upper-left j j sub-matrix. Thus D1 = b D2 =
det a b , etc.
We have then D1 = b, D2 = b2 ; a2 and in general
Dj = bDj;1 ; a2Dj;2
(3 j N )
It can be shown that the solution to the recursion relation (2.4.17) is
A B
1
Dj = ; a2 j + j
1 2
where
h p 2 2i
1
1 = 2a2 b + b ; 4a
(2.4.17)
(2.4.18)
(2.4.19)
h p 2 2i
1
2 = 2a2 b ; b ; 4a
(2.4.20)
A = ;1 (D2 ; bD1) 1 + D1]
1
2
(2.4.21)
(2.4.22)
B = ;1 (D2 ; bD1) 2 + D1]
1
2
We have already shown that M~ 1 is negative denite. The eigenvalues of M^ 1 are found
from
1 1 ~
^
det(M1 ; I) = ; ; det(M1 ; I ) ; ; = 0
(2.4.23)
20
20
thus either = ;1=20 < 0 (because 0 will be taken positive) or = eigenvalue
of M^ 1 < 0. We would like to investigate the behavior of the eigenvalues of 24h0 2 M~ 1. In
particular we would like to show that these eigenvalues (which are negative) are bounded
33
away from zero. To show this we analyze the behavior of M^ 1 ; I as N increases. We
now take S = M^ 1 ; I . Its determinant is given by DN ;2. Substituting (2.4.19)-(2.4.22)
into (2.4.18) with j = N ; 2 we get after some elementary manipulations
N ;2
DN ;2 = r2 N ;3 sin(N ; 1)
(2.4.24)
where
p
= 4 ; b2 b = ;2 ; a = 1
p
r = b2 + 2 = 2
= tan;1(=b)
(2.4.25)
DN ;2 = 0
(2.4.26)
From (2.4.23) we require
This is equivalent, see (2.4.24), to requiring
= Nk
k = 1 : : : N ; 2:
; 1
From the denition of and (2.4.25) we obtain
k p;( + 4)
tan N ; 1 = ; 2 + ( < 0):
(2.4.27)
(2.4.28)
Squaring (2.4.28) we get a quadratic equation for , the solution of which is
=
" k ;1=2#
;2 1 1 + tan2 N ; 1
k ;2 1 cos N ; 1 :
For any xed N , the smallest values of jj is given by (2.4.28) for k = 1,
max = ; min
jj = ;2 1 ; cos N ; 1 :
k
=
As N increases, we have
max
(2.4.29)
(2.4.30)
2
! ;2 1 ; 1 ; 2(N; 1)2 + O N14
2
= ; (N ; 1)2 ;2h2:
34
(2.4.31)
2
2
Thus the eigenvalues
2 of M^ 1=24h (and hence of M~ 1=24h ) are bounded away from zero
by the value ; 24 .
We now consider M~ 2. One can verify that
M~ 2 = ;M^ 2M^ 2T
(2.4.32)
where
20
66 0
66 0
66 0
66 0
66 0
66
66
M^ 2 = 666
66
66
66
66
66
66
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
;2
1
0
0
1
;2
1
0
1
;2 1
... ... ...
1
0
0
;2
1
1 ;2
0 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
77
77
77
77
77
77
77
77
77
77
77
77
77
77
75
(2.4.33)
Therefore M~ 2 is non-positive denite. In a similar fashion M~ 3 is non-positive denite
because
(2.4.34)
M3 = ;M^ 3M^ 3T
35
with
20
66 0
66 0
66 0
66 0
66 0
66
M^ 3 = 666
66
66
66
66
66
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;1
1
;1
... ...
1
;1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
77
77
77
77
77
77
77 :
77
7
0 77
0 77
0 77
0 77
05
(2.4.35)
0
0
0
0
0
0 0
The matrices M~ 4 and M~ 5 are N N matrices with zero entries except for 6 6 upper-left
(lower-right) blocks. It is sucient to show that these blocks are negative denite. This
was done symbolically using the Mathematica software and plotted for 0 L R < 1
and 0 = 1. M~ 4 and M~ 5 are indeed negative denite for, 0 R L < 1. Thus we
have shown that M~ = 12 (M + M T ) is indeed negative denite, and its eigenvalues are
bounded away from zero by (;2=24), even as N ! 1, and the error estimate (1.1.13)
is valid.
36
Chapter 3
The advection diusion equation
This chapter considers 2nd-order accurate approximations to model linear advectiondiusion equations in one and more dimensions, on domains which may be irregular.
In section 3.1 we treat a model \shock-layer" equation (linearized Burger's equation),
ut + aux = R1 uxx t 0 0 < x < 1 R 1:
We develop there a 2nd-order one dimensional semi-discrete scheme using the methodology presented in chapter 1. By constructing the scheme this way we can be sure that
the energy norm of the error to be temporally bounded for all t > 0 by a \constant"
proportional to the norm of the truncation error and that this property is also valid for
the multi-dimensional scheme.
Section 3.2 presents numerical results. Subsection 3.2.1 deals with the steady state
solution to the \shock-layer" equation for a large range of the \Reynolds number", R.
Oscillations that appear in the numerical solution when using a standard central nitedierencing, are eliminated (or dramatically reduced) when the bounded-error algorithm
is used.
Subsection 3.2.2 considers steady-state solution to a two dimensional scalar model
to the boundary layer equations,
ut + aux + buy = R1 uyy R 1 b < 0
both for rectangular and trapezoidal domains. Again, the bounded-error algorithm
out-performs the standard scheme in ways described therein.
37
Subsection 3.2.3 presents a time dependent example, modeling a boundary-layer
being excited sinusoidally,
ut + aux + buy = R1 uyy + b sink(x ; at)]:
Here, aside from the usual performance criteria, such as error-norms and quality of the
velocity proles, we see that the error-bounded algorithm also has a signicantly smaller
phase error.
3.1 Construction of the scheme
Consider the scalar advection-diusion problem
@u = a @u + 1 @ 2u + f (x t) ; x ; t 0 a > 0 y
L
R
@t @x R @x2
u(x 0) = u0(x)
u(;L t) = gL (t)
(3:1:1a)
(3:1:1b)
(3:1:1c)
u(;R t) = gR (t)
and f (x t) 2 C 2.
Let us discretize (3.1.1) spatially on the same grid as in chapters 1 and 2, and using
the same notation for the numerical approximation and for the error vector. The scheme
gets the form:
dv = M v + g + g + f (t)
L L
R R
dt
where
(3:1:2a)
M = R1 MP + aMH = R1 (DP ; LP ALP ; RP ARP )+ a(DH ; LH ALH ; RH ARH ) (3:1:2b)
This section is devoted to the task of constructing M in the case of m = 2, i.e., a
second order accurate nite dierence algorithm.
The results for the case 0 are found by an analysis anologus to the one presented in this section,
and are presented in the appendix to this chapter, section 3.4.
y
a <
38
We shall deal separately with the hyperbolic and parabolic parts of the r.h.s. of
(3.1.2b). The parabolic terms are given by:
2 1 ;2 1 0
66 1 ;2 1 0
66 0 1 ;2 1
66 0 0 1 ;2
1
...
DP = h2 666
66
66
4
1
...
1
3
77
77
77
77 ...
;2 1 0 0 777
1 ;2 1 0 77
1 ;2 1 5
1
;2
1
(3.1.3)
h (P )
i 1
1
4
LP = h2 diag L1 0 : : : 0 = h2 diag (2 + )(1
0 : : : 0 L + L )
h
i 1
1
4
(P )
RP = h2 diag 0 0 : : : RN = h2 diag 0 0 : : : (2 + )(1
R + R )
21
66 2 (2 + L.)(1 + L ) ;L(2. + L)
..
..
ALP = 64
1
1 ( + 2 )
2 L. L
..
1
2
(2
+
L )(1 + L ) ;L (2 + L ) (L + L )
2
2
2
66 0. : : :
ARP = 64 ..
0
...
0 ::: 0
1 ( + 2 )
2 R. R
..
1 ( + 2 )
2 R R
The hyperbolic terms are given by:
0 :::
...
0 :::
(3.1.4)
(3.1.5)
3
07
... 77 5
0
(3.1.6)
1 (2 + )(1 + ) 3
R
R 7
2
77 :
...
...
5
1
;R(2 + R) 2 (2 + R)(1 + R)
(3.1.7)
;R(2 + R)
82
;2 2
>
>
6
>
66 ;1 0
>
>
<66 ;1
1
DH = 2h >66
66
>
>
>
>
:4
1
0 1
... ... ...
;1 0
;1
39
3
77
77
77
7
0 77
15
1
0
;2 2
2c
66 1 c2
66
c3
6
+ 66
66
4
...
cN ; 2
20
66 1
66 1
+2hc~ 666
66
4
where
and
3 2 ;1 2 ;1
3
77 66 0 ;1 2 ;1
7
7
77
77 66 1 ;2 0 2 ;1
77
77 66
... ... ... ... ...
7
77 66
1 ;2 0 2 ;1 77
7
6
54
cN ;1
1 ;2 1 0 5
cN
1 ;2 1
39
;1 1
>
>
7
;1 ;1 1
77>
>
>
;1 0 ;1 1
7
77=
... ... ... ... ...
(3.1.8)
7
>
77>
1 ;1 0 ;1 1
>
1 ;1 ;1 1 5>
>
1 ;1 0 ck = N 1; 1 (cN ; c1)k + (Nc1 ; cN )]
(3.1.9)
c~ = 12 (c1 ; cN ):
(3.1.10)
We note that the rst term in (3.1.8) is the usual second order dierentiation matrix,
The additional matrices may be viewed as connectivity terms that allow the non-positive
denite property to be maintained as we go from one corner to the other. In the parabolic
case the need did not arise because there are penalty terms available at both corners.
These terms are O(h2), thus maintaining accuracy. One eect of using these connectivity
terms is that the 'core' of DH is not Toeplitz any more.
For a > 0 in (3.1.1a), the left boundary is, for the hyperbolic part, an \outow"
boundary on which we do not prescribe a \hyperbolic boundary condition", therefore,
in this case LH = 0. When a < 0, then RH = 0 { see the Appendix for details.
Here, with a > 0,
RH = diag 0 0 : : : R(HN);1 R(HN)]
(3.1.11)
and
2
66 0 . . .
ARH = 666
4 0
0
3
77
0
77
;R 1 + R 75
;R
40
1 + R
(3.1.12)
Next we shall show that the parabolic part of M is negative denite. The symmetric
part of MP M~ P = 12 (MP + MPT ), is found using equations (3.1.3) to (3.1.7), to be
2
3L ; 1 2 ; L
;
2
66
L + 1 2 + L
66
66 3L ; 1 ;4
2
66 L + 1
66
66 2 ; L
2
;4 2
66 2 + L
66
2
;4
66
6
...
M~ P = 2h1 2 666
66
66
66
66
0
66
66
66
66
64
2
...
2
3
77
77
77
0
77
77
77
77
77
77
77
...
77
77
77
;4 2
77
2 ; R 77
2
;4
2
2 + R 77
77
3
R;1 7
2
;4 R + 1 77
77
2 ; R 3R ; 1
;2 5
2 + R R + 1
(3.1.13)
We now decompose M~ P as follows:
8
>> 2
>> ;4 2
>> 66 2 ;4 2
>< 66
2 ;4
1
6
...
~
MP = 2h2 > 66
>> 66
>> 4
>:
2
...
2
20
3
66 0
66 0
77
66
77
7
...
77 + (1 ; ) 666
66
;4 2 77
66
5
2 ;4 2
4
2 ;4
41
0
0
0
0
0
;2
2
2
;4 2
... ... ...
2 ;4
0 2
2
;2
0
0
3
77
77
77
77
77
7
0 77
05
0
0
0
0 0
2
3L ; 1 ; 2 2 ; L
;
2(1
;
2
)
66
L + 1
2 + L
66
66 3L ; 1 ; 2 ;4(1 ; ) 2(1 ; )
66 L + 1
66 1 ; L
66
2(1 ; ) ;2(1 ; )
2
+
L
66
6
+ 66
66
66
66
66
66
0
66
64
0
39>
77>>
77>>>
77>
0
77>>
77>>
77>>
77>>
77=
77>
77>>
2
;
R
7>
;2(1 ; ) 2(1 ; )
2 + R 77>
>
77>>
>
2(1 ; ) ;4(1 ; ) 3R+;11 ; 2 777>
>
R
77>>>
2 ; R 3R ; 1 ; 2 ;2(1 ; 2) 5>
2 + R R + 1
(3.1.14)
We look for 1 > > 0 such that the second and third matrices in (3.1.14) are nonpositive denite. The rst matrix in (3.1.14) is already negative denite by the argument
leading to eq. (2.4.31), in the appendix to chapter 2. By the same argument it immediately follows that its largest eigenvalue is smaller than ;2. For 0 < < 1, the
second matrix in (3.1.14) is non-positive denite, see eq. (2.4.34) and (2.4.35) in that
appendix. The third matrix in (3.1.14) has two square 3 3 corners which are negative
for 0 < < :275. This completes the proof that M~ P is indeed negative denite ( for a
certain range of ).
Next we would like to show that M~ H = 21 (MH + MHT ) is non-positive denite. Using
equations (3.1.8){(3.1.12) we have
42
2 ;4 ; 2c1 1 + 2c1
66 1 + 2c1 ;2c1
66
66 0
0
1
M~ H = 4h 666
66
66
0
4
0
0
0
0
...
2cN + 2R N(H;)1
; 1 ; 2cN ; (1 + R)N(H;)1 + R N(H )
;1 ; 2cN ; (1 + R)N(H;)1 + RN(H ) 4 + 2cN ; 2(1 + R)N(H)
(3.1.15)
We now write M~ H as the sum of three \corner-matrices",
M~ H = 41h mH1 + mH2 + mH3 ]
where
2 ;4 ; 2c 1 + 2c
1
1
66
0
66 1 + 2c1 ;2c1
mH = 66
0
64
0
1
20
66
66
mH = 66
64 0
2
...
(3.1.16)
0
3
77
77
77 75
3
77
0
0
...
77
2RN(H;)1
;1 ; (1 + R)N(H;)1 + RN(H ) 777 5
(H )
(H )
(H )
;1 ; (1 + R2) N ;1 + RN 4 ;32(1 + R)N
0
66 0
77
77 :
...
mH = cN 666
7
4
2 ;2 5
3
;2
2
(3.1.17)
Clearly mH3 is N.P.D (non-positive denite) for 8cN 0. Also, mH1 is N.P.D for
c1 1=4. A simple computation shows that mH2 is N.P.D if N ;1 and N satisfy
N(H ) = 12++ ( 0)
(3.1.18)
R
43
3
77
77
77
77
77
77
75
N(H;)1 =
; 1 ;(1+R(1R;)2 )
(3.1.19)
Thus we have proved that M~ H is indeed non-positive denite, and therefore M~ = R1 M~ P +
aM~ H is negative denite for 8 R1 a > 0, with its eigenvalues bounded away below zero by
;2=R, 0 < < :275, and the (H)'s satisfying (3.1.18) and (3.1.19).
3.2 Numerical examples
3.2.1 One dimensional case
Here we consider the problem
@u + u = 1 u
@t x R xx
t 0 0 x 1
(3.2.1)
u(0 t) = 1
u(1 t) = 0
u(x 0) = u0(x)
The steady state solution to (3.2.1) is:
u(x) = 1 ;1 ;e e;R
;R(1;x)
(3.2.2)
Note that R (= 1=
) plays the role of Reynolds number in this model for a \linear shock
layer".
Eq. (3.2.1) was solved numerically by two methods. In one ,referred to as \standard",
we use central dierencing for the spatial dierentiation, and 4th-order Runge-Kutta in
time. In this \standard" case, there is no need for special treatment at the boundaries.
The numerical steady state approximation v, ( @@tv = 0 ), in this \standard" case,
satises, for (L = R = 1), the following nite dierence equation:
1 (v ; v ) ; 1 (v ; 2v + v ) = 0 (1 j N ; 1)
(3.2.3)
j
j ;1
2h j+1 j;1 Rh2 j+1
with v0 = 1 and vN = 0. The solution to (3.2.3) is:
j
2N ;j
+ hR :
vj = 1;;2N = 22 ;
(3.2.4)
hR
44
Notice that if the \cell Reynolds number", RC = hR > 2, then < 0 and the
numerical solution, vj , will be oscillatory. If RC < 2 then we resolve the \shock layer"
(or \boundary layer") and the solution will be smooth.
Numerical steady-state solutions of (3.2.1) using the \standard scheme", and using
the \bounded-error" algorithm, (3.1.2), described above are shown in Figures 3.1 to 3.6
for !x = 1=100 and various values of R. Both schemes were advanced to steady state
using 4th-order Runge-Kutta. It is clear that when RC < 2, both schemes give good
results. For RC = 10 (R = 1000) both show oscillations, but the new algorithm approximates the exact solution much better. When RC = 103 (R = 105 ), the \standard"
numerical solution is useless while the \bounded-error" scheme gives excellent results
in fact far better than for RC = 10.
u
2
1.75
1.5
u
2
1.75
Standard
Exact
1.5
1.25
1
1
0.75
0.5
0.5
0.25
0.25
0
0.2
0.4
0.6
1
0.8
x
0
Figure 3.1: Standard scheme, RC = 2
0.4
0.6
0.8
1
0.8
1
x
u
2
1.75
Standard
Exact
1.5
1.25
1.25
1
1
0.75
0.75
0.5
0.5
0.25
0.25
0
0.2
Figure 3.2: SAT, RC = 2
u
2
1.5
Exact
1.25
0.75
1.75
SAT
0.2
0.4
0.6
1
0.8
Figure 3.3: Standard scheme, RC = 10
x
0
SAT
Exact
0.2
0.4
0.6
Figure 3.4: SAT, RC = 10
45
x
u
7
6
u
7
Standard
Exact
6
5
5
4
4
3
3
2
2
1
1
0
0.2
0.4
0.6
1
0.8
Figure 3.5: Standard scheme, RC =
1000
x 0
SAT
Exact
0.2
0.4
0.6
0.8
Figure 3.6: SAT, RC = 1000
3.2.2 A steady state two dimensional case
Here we shall consider a linear steady-state problem, which models, in a way, the
2-D boundary layer equations. The formulation is as follows: (the time derivative is left
in the equation, since the approach to steady state will be via temporal advance.)
ut + aux + buy = R1 uyy t 0 0 x < 1 0 y 1
(3:2:5)
bRy
bRy
u(0 y t) = 11;;eebR + 101 bRe 2 sin y
(3:2:5a)
u(x 0 t) = 0
(3:2:5b)
u(x 1 t) = 1
(3:2:5c)
We also take a = 1, and in order to have a growing \boundary layer" on y = 0, we must
set b < 0.
The analytic solution to this problem is:
b2R2 x bRy
bRy
1
;
e
1
u(x y) = 1 ; ebR + 10 bRe 2 exp ; 4 ; 2 Ra sin y
(3.2.6)
Figure 3.7 is a 3-D rendition of u(x y) for R = 90 000. (This 3-D plot looks the same
p
to the eye for various ;1 < b < ;4= R = ;4=300.) Figure 3.8 is a plot of the \velocity
p
prole" inside the \boundary-layer" (0 < y < :04) at x = :1 :25 :9 and b = ;4= R.
The \bumps" at x = :1 and x = :25 may be considered as \emulating" results of uid
mechanics computation for an incompressible ow near the entrance to a channel, see
e.g. 1].
The numerical solution of (3.2.5) using a standard central dierencing scheme depends strongly on the value of b (at a given R). Figures 3.9 and 3.10 show the 3-D
46
1
x
p
plot of vjk with b = ;1 and b = ;4= R = ;4=300. Figs. 3.11 and 3.12 show the
4 , respectively. It should be emphasized
proles at x = :1 and x = :9 for b = ;1 and ; 300
that the \peak" in Figures 3.10 and 3.12 has nothing to do with the \bumps" in the
exact solution (see Figure 3.8). The \peak" occurs way outside the boundary layer, and
also the amplitude behavior with the x-coordinate is counter to that of Figure 3.12 The
\peak" is due to a purely numerical oscillation.
The same series of plots, but as computed by the new algorithm, is shown in Figures
3.13 to 3.16.
u
1
0.8
2
1.5
5
u
1
0.5
.5
0
50
0.6
x=0.1
0.4
x=0.25
0.2
x=0.9
40
30
20
40
*dx
30
*dy
10
20
0
0.01
0.02
0.03
10
Figure 3.7: Exact solution
Figure 3.8: Exact solution near the
boundary
47
0.04
y
2
2
1.5
5
u
1
0.5
.5
0
50
1.5
5
u
1
0.5
.5
0
50
40
30
*dx
20
40
40
30
30
*dy
*dx
20
40
30
10
20
*dy
10
Figure 3.9: Standard scheme, b = ;1
10
20
10
Figure 3.10: Standard scheme, b =
;4=300
u
u
2.5
1.4
x=0.1
x=0.9
2
1.2
1
1.5
0.8
0.6
1
x=0.1
0.4
0.5
x=0.9
0.2
0
0.2
0.4
0.6
1
0.8
Figure 3.11: Standard scheme, b = ;1
y
0
0.2
2
2
1.5
5
u
1
0.5
.5
0
50
40
30
20
*dx
40
30
20
30
10
20
1
0.8
40
30
*dy
0.6
Figure 3.12: Standard scheme, b =
;4=300
1.5
5
u
1
0.5
.5
0
50
40
0.4
*dy
10
Figure 3.13: SAT, b = ;1
10
20
10
Figure 3.14: SAT, b = ;4=300
48
*dx
y
u
u
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
x=0.1
0.4
x=0.1
0.4
x=0.9
x=0.9
0.2
0
0.2
0.2
0.4
0.6
1
0.8
y
Figure 3.15: SAT, b = ;1
0
0.2
0.4
0.6
0.8
Figure 3.16: SAT, b = ;4=300
It should be noted (see table 3.1) that the \bounded-error" algorithm converges to
steady state (residual L2 norm < 10;13 ) an order of magnitude faster than the standard
scheme when using the same !t, while cpu-time/iteration is about the same. The
standard scheme may be run at bigger !t ( by about a factor of 2 ) while the SAT
algorithm was already at its maximum CFL number. If we let each scheme run at its
own maximum !t then the non-dimensional times to get to a steady-state in both cases
are about equal, but the dierence in errors remains.
b = ;1
SAT
Standard
b = ;4=300
SAT
Standard
'time' to
L2
L1 norm
L2 norm L1 norm max error
'steady-state' residual of the error of the error of the error location
21.09
417
52.64
416
9.911e-14
9.987e-14
8.805e-05
0.485139
1.076e-04
0.674233
3.108e-04
-1.00423
45, 46
10, 4
9.943e-14 1.665e-04 1.142e-03
0.01220
9.967e-14 3.362e-03 2.447e-02
-0.2864
Table 3.1: Rectangular geometry results
50, 2
50, 2
We also ran the same equations for a non-strictly rectangular geometry, where the
upper boundary instead of being y = 1 is y = 1 ; (tan )x, where is the angle which
the upper boundary makes with the x-axis, see Figure 3.17.
49
1
y
b = ;4=300
SAT
Standard
'time' to
L2
L1 norm
L2 norm L1 norm max error
'steady-state' residual of the error of the error of the error location
52.56
401.11
9.984e-14 1.707e-04 1.156e-03
9.995e-14 3.448e-03 2.479e-02
Table 3.2: Trapezoid geometry results
0.01220
-0.2864
50, 2
50, 2
y
1
θ
Ω
0.5
y=0
x=0
0.5
1
x
Figure 3.17:
For many 's the results of the performance of the two schemes are unaected by the
change. However, there are some 's for which the standard scheme converges to steady
state much slower than before at its own maximum allowed !t, while the performance
of the bounded-error algorithm remains the same as before. For example, see table 3.2,
for the case of = 3:9. As in chapter 2, the point is that for non-rectangular geometry
the distance that a boundary is away from a computational mode, h, might become
extremely small and this causes the deterioration in the performance of the standard
scheme. Here it is reected in the fact that the standard scheme cannot \support" the
larger allowed !t that can be achieved for the case = 0. For more complex geometries
it is very dicult to predict a-priori what range the values of will take. The SAT
methods (the bounded error algorithm) are insensitive to the variations in caused by
the geometry of the domain.
50
3.2.3 A 2-D time dependent example
To check on the temporal \performance" of the bounded-error scheme, we considered
the following problem:
ut + aux + buy = R1 uyy + b sink(x ; at)] t 0 0 x < 1 0 y 1 (3:2:7a)
bRy
bRy ;( b2 R2 +2 ) x
2
4
Ra sin y + y sin kx
u(x y 0) = 11 ;;eebR + bR
e
10 e
bRy
bRy
2 sin y ; y sin kat
e
u(0 y t) = 11;;eebR + bR
10
u(x 0 t) = 0
u(x 1 t) = 1 + sink(x ; at)]
The exact solution of (3.2.7) is:
(3:2:7b)
(3:2:7c)
(3:2:7d)
(3:2:7e)
bRy
bRy ;( b2R2 +2 ) x
2
4
Ra sin y + y sink (x ; at)]
(3.2.8)
u(x y t) = 11;;eebR + bR
e
10 e
Again we take a = 1, R = 90 000, b = ;1, and ; p4R . The parameters and
k have certain constraints. If we want u > 0, we must take < 1. The number of
computational nodes, N , puts a lower bound of 2N on the wave-length, 1=k, i.e.,
1 < k < 2N . In the actual computations we used = 1=2 and k = 30. All the plots
for this time dependent case are shown for t = 10. Figure 3.18 shows a 3-D plot of
u(x y 10). As in the steady-state case, the plot looks the same to the eye for various
;1 < b < ;4=pR = ;4=300. Figures 3.19, and 3.20 show the 3-D plots of vjk for
the standard and bounded-error schemes respectively. Figure 3.21, shows a x-prole of
v at y = :2, for both schemes and the exact prole, for b = 1. Figure 3.22, gives the
same proles at y = :8. These plots bring out the dierences in the phase errors of
the numerical algorithms. Figures 3.23 to 3.26, repeat the same information as given in
p
Figures 3.19 to 3.22 , but for b = ;4 R = ;4=300. The ecacy of the bounded-error
p
algorithm is quite evident { even when b = ;4= R, where the norm-errors away from
the boundary layer are not dissimilar. The phase error of the right running waves is
quite a bit smaller in the case of the proposed present scheme.
51
2
2
1.5
5
u
1
0.5
.5
0
50
1.5
5
u
1
0.5
.5
0
50
40
30
20
40
*dx
40
30
20
40
30
*dy
*dx
30
10
20
*dy
10
10
20
10
Figure 3.19: Standard scheme, b = ;1.
Figure 3.18: Exact solution.
u
2.5
Exact
Standard
SAT
2
1.5
2
1.5
5
u
1
0.5
.5
0
50
1
40
0.5
30
20
40
*dx
0
0.2
0.4
0.6
0.8
30
*dy
-0.5
10
20
10
Figure 3.20: SAT, b = ;1.
Figure 3.21: b = ;1, y = 0:2 proles.
u
2.5
Exact
Standard
SAT
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
Figure 3.22: b = ;1, y = 0:8 proles.
52
x
1
x
2
2
1.5
5
u
1
0.5
.5
0
50
1.5
5
u
1
0.5
.5
0
50
40
30
20
40
*dx
40
30
30
*dy
*dx
30
10
20
*dy
10
10
20
10
Figure 3.24: SAT, b = ;4=300.
Figure 3.23: Standard scheme, b =
;4=300.
u
2.5
u
2.5
Exact
Standard
SAT
2
1.5
1
1
0.5
0.5
0.2
0.4
0.6
Exact
Standard
SAT
2
1.5
0
20
40
0.8
1
Figure 3.25: b = ;4=300, y = 0:2 proles.
x
0
0.2
0.4
0.6
0.8
Figure 3.26: b = ;4=300, y = 0:8 proles.
3.3 Conclusions
(i) A second order method has been developed which renders spatial second derivative
nite dierence operators negative denite. This is not surprising, since negative
deniteness was achieved for 4th order parabolic operators in chapter 2, see also
2].
(ii) A second order method has been developed which renders spatial rst derivative
nite dierence operators non-positive denite. For the case when boundary points
do not coincide with grid nodes ( 6= 1), this is a new result.
(iii) The results (i) and (ii) allow us to construct a solution operator for the advectiondiusion problem (and, of course, the diusion equation) which is negative denite,
53
1
x
thereby ensuring that this scheme is indeed error bounded.
(iv) The construction of these operators allows an immediate simple generalization to
multi-dimensional problems, on complex domains which are covered by rectangular
meshes. The proofs of the boundness of the error-norms carry over rigorously to
the (linear) multi-dimensional cases.
(v) Numerous numerical examples demonstrate the ecacy of this methodology.
3.4 Appendix, The case a < 0
As in the a > 0 case the hyperbolic terms are given by:
2c
66 1 c2
66
c3
6
+ 66
66
4
82
>
;2 2
>
66 ;1 0
>
>
>
<666 ;1
1
DH = 2h >66
>
66
>
>
>
:4
...
cN ;2
cN ;1
cN
1
0 1
... ... ...
;1 0
;1
3 2 ;1 2
77 66 0 ;1
77 66 1 ;2
77 66
...
77 66
75 64
2 0 ;1 1
66 1 ;1 ;1 1
66 1 ;1 0 ;1
+2hc~ 666 . . . . . . . . .
66
1 ;1
4
where
and
1
...
0
1
3
77
77
77
7
0 77
15
1
0
;2 2
;1
2 ;1
0 2 ;1
... ... ... ...
1 ;2 0 2
1 ;2 1
1 ;2
39
>
77>
>
77>
>
77=
...
;1 1 777>>>
;1 ;1 1 5>>
1 ;1 0
3
77
77
77
;1 777
05
1
(3.4.1)
ck = N 1; 1 (cN ; c1)k + (Nc1 ; cN )]
(3.4.2)
c~ = 12 (c1 ; cN ):
(3.4.3)
54
For a < 0 in (3.1.1a), the right boundary is, for the hyperbolic part, an \outow"
boundary on which we do not prescribe a \hyperbolic boundary condition", therefore,
in this case RH = 0, and
LH = diag L(H1 ) L(H2 ) 0 : : : 0 0]
2
1 + ;
66 1 + LL ;LL
ALH = 666
4
0
0
...
0
0
(3.4.4)
3
77
77
75
(3.4.5)
Since a < 0, and we want aM~ H to be non-positive denite, we need to show that
M~ H = 21 (MH + MHT ) is non-negative denite. Using equations (3.4.1){(3.4.5) we have
M~ H2 =
1
4h
;
4 ; 2c1 ; 2(1 + L )1(H ) 1 + 2c1 ; (1 + L)2(H ) + L1(H )
66 1 + 2c1 ; (1 + L)2(H ) + L1(H ) ;2c1 + 2L 2(H )
6
66
66
4
0
0
0
0
0
0
...
3
0 777
77
77
;1 ; 2cN 5
2cN
;1 ; 2cN 4 + 2cN
(3.4.6)
We now write M~ H as the sum of three \corner-matrices",
M~ H = 41h mH1 + mH2 + mH3 ]
where
2 ;2 2
66 2 ;2
mH = c1 666
4 0
1
55
0
0
...
0
3
77
77 75
(3.4.7)
2
;
4 ; 2(1 + L )1(H )
1 ; (1 + L)2(H ) + L1(H )
66 1 ; (1 + L )2(H ) + L 1(H )
+22(H )
66
0
0
mH = 66
64
0
2
20
66
mH = 666
4
0
3
3
77
77
;1 ; 2cN 75
...
0
0
0
0
0
...
0
3
77
77
77 75
(3.4.8)
2cN
;1 ; 2cN 4 + 2cN
Clearly mH1 is N.N.D (non-negative denite) for 8c1 0. Also, mH3 is N.N.D for
cN ;1=4. A simple computation shows that mH2 is N.N.D if 1 and 2 satisfy
1(H ) =
; 12++L
( 0)
2(H ) = 1 ;(1+L(1 ;)2 )
L
(3.4.9)
(3.4.10)
Thus we have proved that M~ H is indeed non-negative denite, and therefore M~ = R1 M~ P +
aM~ H is negative denite for 8 R1 > 0, with its eigenvalues bounded away from zero by
;2=R, 0 < < :275, and (H)'s satisfying (3.4.9) and (3.4.10), as in the a > 0 case
treated in the text.
56
Chapter 4
Mixed derivatives and parabolic
systems
In the previous chapters we rst developed a one-dimensional scheme, and then generalized it to the multi-dimensional case, following the outline presented in chapter 1.
However, since a mixed derivative problem is inherently multi-dimensional, a dierent
strategy should be adopted. In section 4.1 two methods to tackle the mix derivative
scalar problem are presented. In section 4.2 a 2nd order scheme is presented that solves
this problem. In section 4.3 parabolic systems are discussed, and a scheme is constructed that solves the diusion part of the Navier-Stokes equations in two and three
space dimensions.
4.1 The scalar mixed derivatives problem
Consider the scalar problem
d
@u = X
@ @ u
c
ij
@t ij=1 @xi @xj
(x1 : : : xd) 2
Rd t 0
u(x1 : : : xd 0) = u0(x1 : : : xd)
u(x1 : : : xd t)j@ = uB (t)
(4:1:1a)
(4:1:1b)
(4:1:1c)
It is assumed that this dierential problem is strictly-stable in the sense of Petrowski,
P
P
i.e. dij=1 cij i j > > 0 for all dj=1 j2 = 1, see for example 10], and that cij = cji
57
Note that equation (4.1.1) can also be written as:
0 1
@
@u = (@ : : : @ ) C B x.. C u
x
xd @ . A
@t
1
1
(4.1.2)
@xd
where the matrix C has cij as its entries. Using this notation, the matrix C is symmetric
and positive-denite.
One way to attack this mixed derivative problem problem is to convert equation
(4.1.1) to its canonical form, ut = r2u, using a change of variables. This standard
technique can be found in many P.D.E. textbooks, for example 16]. In the new variables
the problem was already solved in chapter 2.
In some cases this method can be very eective. Often however, especially when
parabolic systems have to be solved, a change of variables may not be possible since
a change of variables that diagonalizes one equation in a system may not diagonalize
the others. The rest of this section is devoted to the description of an alternative
approach, which will be demonstrated in two space dimensions. A second-order accuracy
scheme will be built in the next section. It should be emphasized that this procedure
might not always work. However, under certain conditions, bounded-error schemes for
scalar equations and systems can be generated by using method. These conditions are
discussed in the next section.
We consider the scalar problem
@u = a2(1 + 2)u + 2abu + b2(1 + 2)u (x y) 2 t 0
(4:1:3a)
xy
1 xx
2 yy
@t
u(x y 0) = u0(x y)
(4:1:3b)
u(x y t)j@ = uB (t)
(4:1:3c)
Let us use a multi-dimensional grid and the notation presented in section 1.2. Then
the projection of the exact solution u(t) satises:
dU = a2(1 + 2)D(xx)U + ab hD(x) P T D(y) P + P T D(y) P D(x)i U +
1
2
1
2
1
dt
2
2
T
(
yy
)
b (1 + 2)P D P U + T
(4.1.4)
58
where
2 D(xx)
1
6
D2(xx)
D(xx) = 664
...
DM(xxR)
3
2 D(yy)
77 (yy) 66 1 D2(yy)
75 D = 64
...
DM(yyC)
3
77
75
(4.1.5)
i.e. the matrix D(xx) is a diagonal-block matrix, the kth block in its diagonal, Dk(xx),
is a dierentiation sub-matrix which represents second x derivative when applied to the
kth row. Similarly the matrix D(yy) is also a diagonal-block matrix, and the j th block in
its diagonal, Dj(yy) , is a dierentiation sub-matrix which represents second y derivative
when applied to the j th column. The matrices
2 (x)
66 D1 D1(x)
D1(x) = 664
1
2
and
2 (y)
66 D1 D1(y)
D1(y) = 664
1
...
2
D1(xM)R
1
2
3
2 (x)
77
66 D2 D2(x)
77 D2(x) = 66
5
4
3
2 (y)
77
66 D2 D2(y)
77 D2(y) = 66
5
4
...
D2(xM)R
1
...
2
D1(yM)C
...
D2(yM)C
3
77
77
5
3
77
77
5
(4.1.6)
(4.1.7)
are dened in a similar way, with the dierence that the matrices D1(xk) , D2(xk) , D1(yj) and
D2(yj) represents rst derivative.
Now we can write the scheme:
dV = a2(1 + 2)M(xx)V +
dt
h (x)1 T (y)
i
(y )
( x)
T
ab M2 P M1 P + P M2 P M1 V +
b2(1 + 22)P T M(yy) P V + G(x) + P T G(y)
(4.1.8)
The M matrices have the same structure as the D matrices do, with the sub-matrices
dened in a manner similar to that of chapter 1:
59
Mk(xx)
M1(kx)
M2(kx)
Mj(yy)
M1(jy)
M2(jy)
and
=
=
=
=
=
=
Dk(xx) ; Lk ALk ; Rk ARk
D1(xk) ; Lk ALk ; Rk ARk
D2(xk)
Dj(yy) ; Bj ABj ; Tj ATj
D1(yj) ; Bj ABj ; Tj ATj
D2(yj)
h
(4.1.9)
i
G(x) = (L gL + R gR ) : : : (Lk gLk + Rk gRk ) : : : (LMR gLMR + RMR gRMR ) +
h
i
(L gL + R gR ) : : : (Lk gLk + Rk gRk ) : : : (LMR gLMR + RMR gRMR ) h
i
G(y) = (B gB + T gT ) : : : (Bj gBj + Tj gTj ) : : : (BMC gBMC + TMC gTMC ) +
h
i
(B gB + T gT ) : : : (Bj gBj + Tj gTj ) : : : (BMC gBMC + TMC gTMC ) :
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
(4.1.10)
After subtracting (4.1.8) from (4.1.4) we get:
dE = a2(1+2)M(xx)E+ab hM(x) P T M(y)P + P T M(y)P M(x)i+b2(1+2 )P T M(yy) P E+T
1
2
1
2
1
2
dt
(4.1.11)
where E = U ; V. The time rate of change of k E k2 is given by:
1 d k E k2 = a2(1 + 2) ;E M(xx)E + ab E M(x) P T M(y)P E +
2 dt
1T (y) (x) 2 22 ; T1 (yy) ab E P M2 P M1 E + b (1 + 2 ) E P M P E + (E T)
(4.1.12)
Equation (4.1.12) may be written as:
1 d k E k2 = a2(1 + 2) ;E M(xx)E + ab M(x)T E P T M(y)P E +
2
1
2 dt
T 1 (y)T
;
ab P M2 P E M(1x)E + b2(1 + 22) E P T M(yy)P E + (E T)
60
Now we may use the Schwarz inequality and get:
1 d k E k2 a2(1 + 2)(E M(xx)E) + ab k M(x)T E k k P T M(y)P E k +
1
2
1
2 dt
T
ab k P T M(2y) P E k k M(1x)E k +b2(1 + 22)(E P T M(yy) P E)+ k E k k T k
Then by using the inequality:
2
2
+2 (4.1.13)
we may write
1 d k E k2 a2 E (1 + 2)M(xx) + 1 M(x)T M(x) + M(x)M(x)T E +
1
1
1
2
2
2 dt
2
(y)T (y) (y) (y)T 1
2
T
2
(
yy
)
PE +
b E P (1 + 2)M + 2 M1 M1 + M2 M2
kEkkTk
(4.1.14)
If the matrices
M
(1 + 21)
and
(xx) +
1 M(x)T M(x) + M(x)M(x)T 1
1
2
2
2
(4.1.15)
M 21 M(1y)T M(1y) + M(2y)M(2y)T
(4.1.16)
are negative denite and bounded away from 0 by ;c0 c0 > 0 then the error norm
k E k is bounded, see chapter 1 for the details. Note that each of these matrices is a
diagonal-block-matrix, therefor it is sucient to prove that each of their blocks
(x)T (x) (x) (x)T (xx) 1
2
(1 + 1)Mk + 2 M1k M1k + M2k M2k
(4.1.17)
and
(y) T (y) (y) (y) T (yy) 1
2
(1 + 2)Mj + 2 M1j M1j + M2j M2j
(4.1.18)
(1 + 22)
(yy) +
are negative denite and bounded away from 0.
We can try to solve equation (4.1.1), in more than two space dimensions (d > 2),
using the same method. Writing the scheme, by analogy to equation (4.1.8) we get:
61
dV =
dt
d
X
cjj PjT M(xjxj )Pj V +
j =1
d j ;1
XX h
i
cij PiT M(2xi)Pi PjT M(1xj )Pj + PjT M(2xj )Pj PiT M(1xi)Pi V +
j =1 i=1
d
PjT G(xj )
j =1
X
(4.1.19)
where the Pj 's are the permutation matrices dened in section 1.2, (we take P1 = I )
and the M's and G's are dened in a way similar to equations (4.1.9) and (4.1.10)
(2)
respectively. Now we write cij = c(iji) c(ijj) i 6= j , ( thus for d = 2, cij = c12 = c(1)
1 2 c12 =
ab ), and after doing the same manipulations leading to equation (4.1.14) we get:
1 d k E k2
2 dt
d X
j =1
E PjT
"
cjj M
(xj xj ) +
#
!
1 X c(j)2 M(xj)T M(xj) + M(xj )M(xj)T P E +
j
1
1
2
2
2 i6=j ij
kEkkTk:
(4.1.20)
And as in the two-dimensional case we require that for all j j = 1 : : : d the
matrices
#
"
d
(xj)T (xj) (xj ) (xj)T X
1
(j ) 2
(
x
x
)
j
j
cjj M
+ 2 cij M1 M1 + M2 M2
i6=j
or, equivalently, their diagonal-blocks
"
d
1X
cjj Mk(xj xj ) + 2
i6=j
2
c(ijj)
T
T
M1(kxj ) M1(kxj ) + M2(kxj )M2(kxj )
#
(4.1.21)
(4.1.22)
be negative denite and bounded away from 0. It is shown in the next section that this
requirement is more dicult to satisfy than in the two-dimensional case.
62
4.2 Second order scheme for the scalar mixed derivatives problem
In this section we construct a second order accuracy scheme to the problem (4.1.3) using
the method presented in the previous section.
The matrices Mk(xx) = Dk(xx) ; Lk ALk ; Rk ARk are constructed from:
2 1 ;2 1 0
66 1 ;2 1 0
66 0 1 ;2 1
66 0 0 1 ;2
1
...
Dk(xx) = h2 666
66
66
4
For simplicity we take
1
...
1
3
77
77
77
77 ...
;2 1 0 0 777
1 ;2 1 0 77
1 ;2 1 5
1
;2
1
Lk = h12 diag 7 ;25L 0 : : : 0 0 (4.2.2)
Rk = h12 diag 0 0 : : : 0 7 ;25R 21
66 2 (2 + L.)(1 + L) ;L(2. + L)
..
..
ALk = 64
1
1 ( + 2 )
2 L. L
..
1
2
(2
+
L )(1 + L ) ;L (2 + L ) (L + L )
2
2
2
66 0. : : :
ARk = 64 ..
0 1 (R + R2 )
... 2 ...
0 : : : 0 1 (R + R2 )
2
(4.2.3)
3
::: 0 7
... 77 5
0
...
0 ::: 0
1 (2 + )(1 + ) 3
R
R 7
2
77 :
...
...
5
1
;R(2 + R) 2 (2 + R)(1 + R)
;R(2 + R)
The matrices M1(kx) = D1(xk) ; Lk ALk ; Rk ARk are composed of:
63
(4.2.1)
(4.2.4)
(4.2.5)
2 ;3 + c 4 ; 3c ;1 + 3c ;c
66 ;1 1 0 1 1 1 1
66
;1
0
1
66
;
1
0 1
1
6
(x)
... ... ...
D1k = 2h 66
66
;1 0
66
;1
4
cN
1
0
;1
1 ; 3cN
Where c1 = cN = 1.
0
1
0
0
1
;4 + 3cN 3 ; cN
(4.2.6)
Lk = h12 diag L1 L2 0 : : : 0 0] (4.2.7)
Rk = h12 diag 0 0 : : : 0 RN ;1 RN (4.2.8)
but we take L1 = L2 = RN ;1 = RN = 0, i.e. we don't use the penalty terms for
M1(kx).
Finally the matrices M2(kx) = D2(xk) are given by:
2 ;2 2
66 ;1 0
66 ;1
1
D2(xk) = 2h 666
66
4
1
0 1
... ... ...
;1 0
;1
3
77
77
77
7
0 77
15
(4.2.9)
1
0
;2 2
The denitions of the matrices Mj(yy) ,M1(jy) and M2(jy) are analogous.
After substituting (4.2.1) to (4.2.9) into (4.1.17) and denoting by M~ the symmetric
part of this matrix one gets,
64
3
77
77
77
77 77
77
7
05
2m
66 11
66 m12
66
66 m13
66
66 m14
66
66
66
1
M~ = 16h2 666
66
66
66
66
66
66
66
66
4
m12 m13 m14
0
m22 m23 m24
m23 m33 m34 m2
m24 m34 m44 m1 m2
m2 m1 m0 m1 m2
... ... ... ...
...
m2 m1 m0
m1
m2
m2 m1 mN ;3N ;3 mN ;2N ;3 mN ;1N ;3
m2 mN ;2N ;3 mN ;2N ;2 mN ;1N ;2
0
mN ;1N ;3 mN ;1N ;2 mN ;1N ;1
mNN ;3
mNN ;2
mNN ;1
(4.2.10)
where
m0 = 8 ; 32(1 + 21)
m1 = 16(1 + 21)
m2 = ;4
m11
m12
m13
m14
m22
= 26 ; 4(1 + 21)(10 + 11L ; 8L2 ; 5L3 )
= ;4(1 + 21)(2 ; 14L + 3L2 + 5L3 )
= ;14 + 2(1 + 21)(4 ; 7L ; 2L2 + 5L3 )
= 4
= 8 ; 32(1 + 21)
65
3
77
77
77
77
77
77
77
77
77
77 77
77
77
7
mNN ;3 77
77
mNN ;2 77
77
mNN ;1 77
5
mNN
m23
m24
m33
m34
m44
= 4 + 16(1 + 21)
= ;6
= 16 ; 32(1 + 21)
= ;4 + 16(1 + 21)
= 10 ; 32(1 + 21)
and
mNN =
mNN ;1 =
mNN ;2 =
mNN ;3 =
mN ;1N ;1 =
mN ;1N ;2 =
mN ;1N ;3 =
mN ;2N ;2 =
mN ;2N ;3 =
mN ;3N ;3 =
26 ; 4(1 + 22)(10 + 11R ; 8R2 ; 5R3 )
;4(1 + 22)(2 ; 14R + 3R2 + 5R3 )
;14 + 2(1 + 22)(4 ; 7R ; 2R2 + 5R3 )
4
8 ; 32(1 + 22)
4 + 16(1 + 22)
;6
16 ; 32(1 + 22)
;4 + 16(1 + 22)
10 ; 32(1 + 22)
We now decompose M~ as follows:
where:
h 2 ~
i
1
2
~
~
~
~
~
M = 16h2 161 M1 + 161 (1 ; )M2 + 4M3 + M4 + M5
2 ;2 1
66 1 ;2 1
66
1 ;2
6
...
~
M1 = 66
66
4
66
1
... ...
1 ;2
1
1
;2
1
3
77
77
77 77
7
15
;2
(4.2.11)
(4.2.12)
20 0 0 0
3
66 0 0 0 0
77
66 0 0 0 0
77
66 0 0 0 ;1 1
77
66
77
1 ;2 1
77 ... ... ...
M~ 2 = 666
77
66
1 ;2 1
7
66
0 1 ;1 0 0 0 77
66
0 0 0 0 77
4
0 0 0 05
0 0 0 0
20 0 0 0
3
66 0 0 0 0
0 7
77
66 0 0 ;1 2 ;1
77
66 0 0 2 ;5 4 ;1
77
66
;
1 4 ;6 4 ;1
77
6
.
.
.
.
.
~
.
.
.
.
.
M3 = 66
77 . . . . .
66
77
;1 4 ;6 4 ;1
66
;1 4 ;5 2 0 0 77
66 0
;1 2 ;1 0 0 577
4
0 0 0 0
2
66
66
66
66
66
66
1
M~ 4 = 16h2 666
66
66
66
66
66
4
0
0 0 0
+m11
3221 m12
;1621
m13
m14
m12
m22
+3221 ;1621
m23
m24
m13
m23
m33
2
;161 +3221 + 4
;1621
m14
m24
m34
;1621 ; 8
m34
m44
2
;161 ; 8 +16(1 + )21 + 20
0
(4.2.13)
(4.2.14)
3
77
77
77
77
0 777
77
77
77
77
77
77
77
05
(4.2.15)
and
67
2
66 0
66
66
66
66
66
1
M~ 5 = 16h2 666
66
66 0
66
66
66
4
mN ;3N ;3
+16(1 + )21 + 20
mN ;2N ;3
;1621 ; 8
mN ;1N ;3
mNN ;3
3
77
0
77
77
mN ;2N ;3 mN ;1N ;3 mNN ;3 777
77
;1621 ; 8
77
mN ;2N ;2 mN ;1N ;2 mNN ;2 77 :
77
+3221 + 4 ;1621 77
mN ;1N ;2 mN ;1N ;1 mNN ;1 77
;1621 +3221 ;1621 77
77
mNN ;2
mNN ;1 mNN 5
2
2
;161 +321 (4.2.16)
M~ 1 is negative denite by the argument leading to eq. (2.4.31), in the appendix
to chapter 2. By the same argument it immediately follows that its largest eigenvalue
is smaller than ;h22. M~ 2 and M~ 3 are non-positive denite, see equations (2.4.32) to
(2.4.35) in that appendix. The matrices M~ 4 and M~ 5 are non positive denite for all
0 L R 1, 0 max(1). For 21 2min 0:4342 there is no positive . If,
for example, we take 21 = 1=2 ranges from 0 to 0.02 . For 21 > 1=2 max increases
with 21 . These results were veried using the Mathematica software. We get the same
expressions for the dierentiation matrices in the y direction, the only dierence being
that in the y direction we have 2 instead of 1. Thus we have proven that if 21 22 1=2
then from (4.1.14), (4.2.11) and the fact that M~ 1 is bounded away from 0 by ;h22 it
follows that
1 d k E k2
2 dt
;0:02 a2(21 ; 1=2) + b2(22 ; 1=2) (E E) +
kEkkTk:
Then by using the denition:
c0 = 0:02 a2(21 ; 1=2) + b2(22 ; 1=2) 68
we get that
k E k k Tc0kM (1 ; e;c t)
\constant" k T kM = max0 t k T( ) k. This
0
(4.2.17)
where the
\constant" is a function of
the exact solution u and its derivatives.
The value of the maximal , ( max = 0:02 ), and the minimal value of 21 and 22,
( 1=2 ), could probably be improved by using dierent values for the coecients 's ,'s
and c's in (4.2.2), (4.2.3) and (4.2.6) to (4.2.8).
It should be noted that when we write equation (4.1.1) in the form (4.1.2), then in
order to prove the negative-deniteness of the discrete dierentiation operator in (4.1.8),
the diagonal of the matrix C should be very large with respect to the o-diagonal terms
( by a factor of 1.5 ). This problem limits the usefulness of this approach, especially in
the case of many space dimensions, d 3, where the ratio between the diagonal and
o-diagonal terms becomes larger. For example, if the matrix C has the form:
8 c c if i 6= j
< ij
cij = :
scj if i = j
then by equating eq. (4.1.22) to eq. (4.1.17) one can see that s should be at least
1:5(d ; 1).
In the general case, see equation (4.1.20), a ratiobetween the coecient of M(xjxj )
T
T
and the coecient of M(1xj ) M(1xj ) + M(2xj)M2(xj ) of least 3 will assure the negative
deniteness. When this ratio is grater the 3 we can bound the error-norm by a constant
using the same considerations leading to eq. (4.1.17).
4.3 Parabolic systems the diusion part of the NavierStokes equations
The analysis of general parabolic systems is much more complex than that for the
scalar equations. It can be carried out for some important parabolic systems, such as
diusion part of the Navier-Stokes equations, which will now be considered.
Consider the problem
@u(1) = 4 u(1) + u(1) + 1 u(2)
@t 3 xx yy 3 xy
69
(x y) 2 t 0
@u(2) = u(2) + 4 u(2) + 1 u(1) (x y) 2 t 0
xx 3 yy 3 xy
@t
u(1)(x y 0) = u(1)
(x y) 2
0 (x y )
u(2)(x y 0) = u(2)
0 (x y )
(x y) 2
u(1)(x y t)j@ = u(1)
B (t)
u(2)(x y t)j@ = u(2)
B (t)
(4.3.1)
where the Reynolds number, Re, has been absorbed in the temporal variable t.
Let us discretize (4.3.1) in the same way as we discretized the scalar equation (4.1.3).
dU(1) = 4 D(xx)U(1) + P T D(yy) P U(1) +
dt
3h
1 D(x) P T D(y) P + P T D(y) P D(x)i U(2) + T(1)
1
2
1
6 2
dU(2) = D(xx)U(2) + 4 P T D(yy) P U(2) +
dt
3
1 hD(x) P T D(y) P + P T D(y) P D(x)i U(1) + T(2)
1
2
1
6 2
(4.3.2)
The denitions for the D's and P are given in (4.1.5) to (4.1.7). We can now write
the scheme, analogous to equation (4.1.8).
dV(1) = 4 M(xx)V(1) + P T M(yy) P V(1) +
dt
3h
1 M(x) P T M(y)P + P T M(y)P M(x)i V(2) + G(x) + P T G(y) + G(x) + P T G(y)
1
2
1
11
11
12
12
6 2
dV(2) = M(xx)V(2) + 4 P T M(yy) P V(2) +
dt
3
1 hM(x) P T M(y)P + P T M(y)P M(x)i V(1) + G(x) + P T G(y) + G(x) + P T G(y)
1
2
1
21
21
22
22
6 2
(4.3.3)
where the M's are given in (4.1.9), the G's are analogous to those in equation
(4.1.10), and the rst number in the subscripts indicate in which equation the G's
are and the second one indicate whether the boundary conditions are of u(1) or u(2).
For example G(21x) reects the boundary values for u(1) in the x direction (rows) and it
appears in the second equation. After subtracting (4.3.2) from (4.3.3) we get:
70
dE(1) = 4 M(xx)E(1) + P T M(yy)P E(1) +
dt
3h
1 M(x) P T M(y)P + P T M(y)P M(x)i E(2) + T(1)
1
2
1
6 2
dE(2) = M(xx)E(2) + 4 P T M(yy)E(2)P +
dt
3
1 hM(x) P T M(y)P + P T M(y)P M(x)i E(1) + T(2)
1
2
1
6 2
(4.3.4)
Taking the scalar product of E(1) with the rst equation and of E(2) with the second
one one gets after adding the two equations:
1 d ;k E(1) k2 + k E(2) k2 = 4 ;E(1) M(xx)E(1) + ;E(1) P T M(yy) P E(1) +
2 dt
3
1 E(1) hM(x) P T M(y)P + P T M(y)P M(x)i E(2) +
2
1
2
1
;6E(2) M(xx)E(2) + 4 ;E(2) P T M(yy) P E(2) +
3
1 E(2) hM(x) P T M(y)P + P T M(y)P M(x)i E(1) +
2
1
2
1
6;
;
+ E(1) T(1) + E(2) T(2)
(4.3.5)
Then after doing the same manipulations leading to equation (4.1.14) we get:
1 d ;k E(1) k2 + k E(2) k2 E(1) 4 M(xx) + 1 M(x)T M(x) + M(x)M(x)T E(1) +
1
1
2
2
2 dt
3
12
(y)T (y) (y) (y)T (1)
1
(
1)
T
(
yy
)
PE +
E P M + 12 M1 M1 + M2 M2
M 121 M(1x)T M(1x) + M(2x)M(2x)T E(2) +
4
(y)T (y) (y) (y)T (2)
1
(
2)
T
(
yy
)
E P 3 M + 12 M1 M1 + M2 M2
PE +
E(2)
(xx) +
k E( ) k k T( ) k + k E( ) k k T( ) k :
1
1
2
2
(4.3.6)
71
We can now dene the error vector E as a 2N long vector whose rst N entries are
the entries of E(1) and the other N entries are the ones of E(2). In a similar way we can
dene the truncation-error vector, T. Using these denitions the norms of E and T are:
q
k E k= p12 k E( ) k2 + k E( ) k2
1
2
(4.3.7)
and
q
1
(4.3.8)
k T k= p2 k T(1) k2 + k T(2) k2:
If we use the second order dierentiation matrices presented in the previous section,
see equations (4.2.1) to (4.2.9), then using the arguments presented in the last paragraph
of that section and using, e.g., 43 = 16 (1+ 21), etc, we may write the following inequality:
1 d k E k2 ;0:02 11 k E k2 + k E k k T k
2 dt
6
Now by using the denition:
c0 = 0:02 116 we get that
k E k k Tc0kM (1 ; e;c t)
\constant" k T kM = max0 t k T( ) k. This
0
(4.3.9)
(4.3.10)
where the
\constant" is function of the
exact solution (u(1) u(2))T and its derivatives.
We can also apply the same procedure to the diusion part of the Navier-Stokes
equations in three-dimensions.
72
Consider the problem
@u(1) = 4 u(1) + u(1) + u(1) + 1 u(2) + 1 u(3) (x y z) 2 t 0
@t 3 xx yy zz 3 xy 3 xz
@u(2) = u(2) + 4 u(2) + u(2) + 1 u(1) + 1 u(3) (x y z) 2 t 0
xx 3 yy
zz
@t
3 xy 3 yz
@u(3) = u(3) + u(3) + 4 u(3) + 1 u(1) + 1 u(2) (x y z) 2 t 0
xx
yy 3 zz
@t
3 xz 3 yz
u(1)(x y z 0) = u(1)
(x y z) 2
0 (x y z )
u(2)(x y z 0) = u(2)
0 (x y z )
u(3)(x y z 0) = u(3)
0 (x y z )
(x y z) 2
(x y z) 2
u(1)(x y z t)j@ = u(1)
B (t)
u(2)(x y z t)j@ = u(2)
B (t)
u(3)(x y z t)j@ = u(3)
B (t)
(4.3.11)
We write the scheme, analogous to equation (4.3.3).
dV(1) = 4 M(xx)V(1) + P T M(yy) P V(1) + P T M(zz) P V(1) +
2
3
2
3
dt
3h
i
1 M(x) P T M(y)P + P T M(y)P M(x) V(2) +
6h 2 2 1 2 2 2 2 1 i
1 M(x) P T M(z)P + P T M(z)P M(x) V(3) +
6 2 3 1 3 3 2 3 1
G(11x) + P2T G(11y) + P3T G(11z) +
G(12x) + P2T G(12y) + P3T G(12z) +
G(13x) + P2T G(13y) + P3T G(13z)
73
dV(2) =
dt
M(xx)V( ) + 34 P2T M(yy)P2V( ) + P3T M(zz) P3V( ) +
1 hM(x) P T M(y)P + P T M(y)P M(x)i V( ) +
6h 2 2 1 2 2 2 2 1
1 P T M(y)P P T M(z)P + P T M(z)P P T M(y)P i V( ) +
6 2 2 2 3 1 3 3 2 3 2 1 2
2
2
2
1
3
G(21x) + P2T G(21y) + P3T G(21z) +
G(22x) + P2T G(22y) + P3T G(22z) +
G23(x) + P2T G(23y) + P3T G(23z)
dV(3) =
dt
M(xx)V( ) + P2T M(yy)P2V( ) + 34 P3T M(zz) P3V( ) +
1 hM(x) P T M(z) P + P T M(z)P M(x)i V( ) +
6h 2 3 1 3 3 2 3 1
1 P T M(y)P P T M(z)P + P T M(z)P P T M(y)P i V( ) +
6 2 2 2 3 1 3 3 2 3 2 1 2
3
3
3
1
2
G(31x) + P2T G(31y) + P3T G(31z) +
G(32x) + P2T G(32y) + P3T G(32z) +
G33(x) + P2T G(33y) + P3T G(33z)
(4.3.12)
Then after doing the same manipulations leading to equation (4.3.9) and the analogous denitions for E and T,
q
1
k E k= p3 k E(1) k2 + k E(2) k2 + k E(3) k2
and
we get, as before:
q
1
k T k= p3 k T(1) k2 + k T(2) k2 + k T(3) k2
74
1 d k E k2 ;0:02 11 k E k2 + k E k k T k
2 dt
6
Now by using the denition:
c0 = 0:02 116 we get that
k E k k Tc0kM (1 ; e;c t)
\constant" k T kM = max0 t k T( ) k.
0
where the
75
(4.3.13)
(4.3.14)
Chapter 5
Bounded error schemes for the wave
equation
5.1 Description of the method
We consider the following problem
@ 2u = @ 2u + f (x t) ; x ; t 0
L
R
@t2 @x2
u(x 0) = u0(x)
@ u(x 0) = u (x)
t0
@t
u(;L t) = gL (t)
(5:1:1d)
u(;R t) = gR (t)
(5:1:1e)
(5:1:1a)
(5:1:1b)
(5:1:1c)
and f (x t) 2 C 2.
Let us discretize (5.1.1) spatially on the same grid and use the same notation as in
in the previous chapters.
We would like to construct a semi-discrete nite dierence scheme to solve (5.1.1)
without rst writing it as a system of rst order P.D.E's. The reason for constructing a
direct solver for the wave-equation is demonstrated for the two-dimensional case:
76
Consider,
@ 2u
@t2
u(x y t)
u(x y 0)
@ u(x y 0)
@t
=
=
=
=
@ 2u + @ 2u
@x2 @y2
uB (x y t) (x y) 2 @
u0(x y)
ut0 (x y)
Converting the above to a system of rst order P.D.E's, using the standard substitution: vI = ut vII = ux vIII = uy we get:
0 vI 1 2 0
@ @ vII A = 4 1
3 0
1 2
3 0
1
1 0 @
vI
0 0 1 @
vI
II
0 0 5 @ v A + 4 0 0 0 5 @ vII A (5.1.2)
@t vIII
0 0 0 @x vIII
1 0 0 @y vIII
Unlike the one-dimensional case, where the right hand side of the system can be
diagonalized, this is not the case here, as the two matrices cannot be diagonalized
simultaneously. This means that boundary values must be assigned to vII and vIII .
These values cannot be derived directly from the original wave equation, nor from a
procedure analogous to the 1-D characteristic decomposition. Additionally it should
also be noted that if we solve the wave equation in d-dimensional space on a grid
with N grid points by converting it to the d-dimensional version of (5.1.2), then after
discretization we have an O.D.E system with (1+ d)N 'variables'. When (5.1.1) is solved
by the method to be proposed shortly, see (5.1.4), one gets an O.D.E system with only
N 'variables', or 2N 'variables' if one solves (5.1.11).
Since, unlike the previous cases, equation (5.1.1) has a second time derivative, attempts to apply naively the methods presented in chapter 1 will fail. The reason is that
if we write a discrete approximation to equation (5.1.1),
and the numerical scheme,
d2u = Du + f (t) + T
e
dt2
d2v = Dv ; (A v ; g ) ; (A v ; g )] + f (t)
L L
L
R R
R
dt2
as in chapter 1, the equation for the error-vector becomes,
77
(5.1.3)
(5.1.4)
d2 = D ; A
L L
dt2
= M +T
where
; RAR
]+T
(5.1.5)
M = D ; L AL ; RAR
is a negative-denite matrix and
T = Te ; L TL ; RTR = (T1 : : : Tm : : : TN )T :
If the matrix M can be diagonalized, as can be shown for the M 's used in chapters
2 and 3, then
M = Q;1$Q
with the diagonal matrix, $, having the eigenvalues of M . Using the denition
equation (5.1.5) becomes,
d2 = $ + QT
dt2
= $ + T^ :
is:
=Q
,
(5.1.6)
This is an un-coupled system of O.D.E's. The general solution for the mth equation,
p
p
Z
p
^
t
1
m (t) = cm1 exp m t +cm 2 exp ; m t + p
sinh m (t ; s) Tm(s) ds :
m 0
Recalling that at t = 0, = t = 0, we have, at t = 0, = t = 0 and the
solution for (5.1.6) is:
78
Z t p
1
m (t) = p
sinh m (t ; s) T^m(s) ds :
(5.1.7)
m 0
that unless all the eigenvalues of M are real and non-positive some of the
pmNote
's will have a positive real part. In that case at least one of the m may grow
exponentially in time. In order to prevent this, we have to demand that M , in addition
to being negative-denite, also possess only real eigenvalues. Furthermore in order to
use the one-dimensional scheme as a building block for multi-dimensional schemes in
the way presented in the second section of chapter 1, M should be built in a way that
veries that these properties will be carried over to the multi-dimensional dierentiating
matrix. One way to achieve this goal is to construct M as a negative-denite symmetric
matrix. Then, an estimate on the error bound can be derived directly from the solution
(5.1.7),
jm(t)j pj1mj T^mM t
where T^mM = max0st jT^m(s)j. Then, for a normalized Q
k k = k k c10 k T^ M kt
(5.1.8)
p
where c0 = minm=1:::N jmj. Therefore k k grows at most linearly with t. Alterna-
tively one may use an energy method in a way similar to that presented in chapter 1.
Taking the scalar product of t with (5.1.5) one gets:
( t
tt) = ( t M
) + ( t T):
Since M is a symmetric matrix this equation can be written as:
1 d k k2 ;( M ) = ( T) :
t
2 dt t
After integrating from 0 to t one gets:
79
1 k
2
t k ;( M )
2
=
=
Using Schwarz's inequality, ;( M )
= t = 0 one gets:
1c k
20
Zt
Zt
0
0
( s T)ds
( T)s ; ( Ts)] ds:
c0 k k2 and the fact that at t = 0,
Zt
k ( T) ; 0 ( Ts)ds
Zt
k k k T k + 0 k k k Ts k ds:
(5.1.9)
Denoting k T kM = max0 t k T( ) k, k Tt kM = max0 t k T ( ) k,
k kM = max0 t k ( ) k and by tM the time where k k gets that maximum
value, (0 tM t), we can evaluate equation (5.1.9) at t = tM by:
2
k kM 2 c20 k kM k T kM +k kM k Tt kM
After a division by k kM one gets:
k k k kM
c20 k T kM + k Tt kM
c20 k T kM + k Tt kM
tM ] :
tM ]
t]
(5.1.10)
i.e., a linear growth in time. It is not surprising that the bound (5.1.10) is not as 'sharp'
as (5.1.8), since the latter was derived directly from the exact solution (5.1.7).
As mentioned before, the multi-dimensional case
@ 2u = r2u + f (x : : : x t)
1
d
@t2
on complex shapes is completely analogous to the method indicated in chapter 1 and
the proofs go over in the same manner.
80
Note also that instead of solving (5.1.4) directly as a 2nd order ODE system in time
one can solve
dw = Dv ; (A v ; g ) ; (A v ; g )] + f
L L
L
R R
R
dt
dv = w :
dt
(5.1.11)
The number of 'variables' has increased from N to 2N but one gains in the simplicity
of the time integration.
5.2 Construction of the scheme
This section is devoted to the task of constructing a symmetric negative denite M for
the case of m = 2, i.e., a second order accurate nite dierence algorithm.
Let
82
3
>
1
;
2
1
0
>
66 1 ;2 1 0
77
>
>
66 0 1 ;2 1
77
>
>
6
77
>
<66 0 0 1 ;. 2 . 1 .
1
77
.. .. ..
D = h2 >66
7
>
6
7
1
;
2
1
0
0
>
7
>666
1 ;2 1 0 77
>
1 ;2 1 5
>
:4
1 ;2 1
20
66 c2
66
c3
6
+ 66
66
4
...
cN ;2
cN ;1
0
32 0 0 0 0
3
77 66 1 ;3 3 ;1
77
77 66 ;1 4 ;6 4 ;1
77
77 66
77
... ... ... ... ...
77 66
77
;
1
4
;
6
4
;
1
75 64
;1 3 ;3 1 75
0
81
0
0
0
2 0
66 0
66 ;1
;c~ 666
66
4
where
and
0 0
1 ;2 1
2 0 ;2 1
... ... ... ... ...
;1 2 0 ;2 1
;1 2 ;1 0
0 0 0 0
39
>
77>
>
77>
>
77=
77>
75>
>
>
>
(5.2.1)
ck = c2 + cNN;1;;3c2 (k ; 2) (5.2.2)
c~ = cNN;1;;3c2 :
(5.2.3)
Note, that as in chapter 3, we had to resort to using connectivity terms in (5.2.1).
21
66 2 (2 + L.)(1 + L) ;L (2. + L)
..
..
AL = 64
1
1 ( + 2 )
2 L. L
..
1
2
(2
+
L )(1 + L ) ;L (2 + L ) (L + L )
2
2
2
66 0. : : :
AR = 64 ..
0
...
0 ::: 0
1 ( + 2 )
2 R. R
..
1 ( + 2 )
2 R R
0 :::
...
0 :::
3
07
... 77 5
0
(5.2.4)
1 (2 + )(1 + ) 3
R
R 7
2
77 :
...
...
5
;R(2 + R) 21 (2 + R)(1 + R)
(5.2.5)
;R(2 + R)
L = h12 diag L1 L2 L3 0 : : : 0 0] (5.2.6)
R = h12 diag 0 0 : : : 0 RN ;2 RN ;1 RN (5.2.7)
In order to make the matrix M = D ; LAL ; R AR symmetric we choose:
82
c2 = (1 ; 2L) L
cN ;1 = (1 ; 2R) R
L2 = 3 ; 1L +; 2 L L1
L
;
2
+
L + L L1
L3 =
2 + L
3
;
R ; 2 R RN
RN ;1 =
1 + R
RN ;2 = ;2 + 2R++ R RN
(5.2.8)
R
L1 and RN will be determine later.
We now decompose M as follows:
M = h12 M1 + (1 ; )M2 + M3 + M4 + M5]
where:
2 ;2 1
3
66 1 ;2 1
77
66
77
1 ;2 1
77 ... ... ...
M1 = 666
77
66
1 ;2 1
7
4
1 ;2 1 5
1 ;2
20 0 0
3
66 0 0 0
77
66 0 0 ;1 1
77
66
77
1 ;2 1
6
77 ... ... ...
M2 = 66
77
66
1 ;2 1
7
66
0 1 ;1 0 0 77
4
0 0 05
0 0 0
83
(5.2.9)
(5.2.10)
(5.2.11)
8
20 0 0
>
>
66 0 ;1 1
>
>
>
<666 0 1 ;. 2
..
M3 = ; >66
>
66
>
>
>
:4
20
66 c2
66
c3
66
66
64
...
cN ;2
cN ;1
0
1
...
1
3
77
77
77
...
;2 1 0 777
1 ;1 0 5
32 0 0 0
777 666 0 ;1 1
77 66 0 1 ;. 2
..
77 66
77 66
54
2 11
66 mm4142
66 m143
M4 = 66
64
0
0 0
1
...
1
39
>
77>
>
77>
>
77= ...
;2 1 0 777>>>
1 ;1 0 5>
3
77
77
77
7
05
0
0 0
m142 m413
m242 m243 0
m243 m343
0
where:
m141 = 1 + 2 ; (1 + L) (22 + L) L1
m142 = ;2 ; + L (2 + L) L1
m143 = 1 ; L (1 +2L) L1
2
2
m242 = 2 + 7L ; 4 (1 + L)2;(1+L (2)+ L) (1 + 4L1 )
L
2
m243 = 1 ; ; 32L + 2L + L2L1
2
3
2
m343 = 2 + ;4 + L ; 2L(2 ;+ L ) (1 + L) L1
L
and
84
(5.2.12)
(5.2.13)
2
3
66 0
77
0
66
77
6
N ;2N ;2 mN ;1N ;2 mNN ;2 7
77
5
5
M5 = 666 m5
66 0 mN ;1N ;2 mN ;1N ;1 mNN ;1 777
5
5
5
64
75
where:
;2
mNN
5
;1
mNN
5
(5.2.14)
mNN
5
mNN
= 1 + 2 ; (1 + R) (22 + R) RN
5
;1 = ;2 ; + R (2 + R ) R
mNN
5
N
R (1 + R ) RN
NN
;
2
m5
= 1;
2
2
2
7
;
mN5 ;1N ;1 = 2 + R 4 (1 + R)2;(1R+ (2 )+ R) (1 + 4RN )
R
2
3
mN5 ;1N ;2 = 1 ; ; 2R + 2R + R2RN
2
3
2
mN5 ;2N ;2 = 2 + ;4 + R ; 2R(2;+ R )(1 + R) RN :
R
The matrix M1 is negative-denite and bounded away from 0 by h22 by the argument leading to eq. (2.4.31), see appendix to chapter 2. M2 is non-positive denite,
see eq. (2.4.34) and (2.4.35) in that appendix. From (5.2.2), (5.2.3) and (5.2.8) follows that ck 0, k = 1 : : : N , therefore, the matrix M3 is non-positive. For a given
value of 0 1, L1 and RN can be found such that the matrices M4 and M5 will
be non-positive, for all L and R . For example: for = 1=10 L1 = RN = 4, for
= 1=2 L1 = RN = 9 and for = 8=10 L1 = RN = 24. This completes the proof
that M is indeed a negative-denite matrix, bounded away from 0 by 2. Therefore
the norm of the error vector k k can grow at most linearly in time, see equations (5.1.8)
and (5.1.10).
85
Summary
In this work a methodology for constructing nite-dierence semi-discrete schemes,
for initial boundary value problems (IBVP), on complex, multi-dimensional shapes was
presented. Its starting point is the development of one-dimensional schemes on a uniform
mesh with boundary points that do not necessarily coincide with the extremal nodes of
the grid. The 1-D construction was done in such a way that the coecient matrix of the
corresponding ODE system ( which represents the error evolution in time ) is negative
denite and bounded away from 0 by a constant independent of the size of the matrix,
or is at least non-positive denite. This construction was carried out by imposing the
boundary conditions using simultaneous approximation terms (SAT). Using the fact
that the coecient matrix is negative, or non-positive, denite it was proved that the
one dimensional scheme is error-bounded and that this scheme can be used as a building
block for multi-dimensional, error-bounded algorithms. The general theory was given
in chapter 1.
In chapters 2 and 3 the methodology presented in chapter 1 was used to develop
second and fourth order accurate approximations for @ 2=@x2 and a second order accurate approximation for @=@x. Using these approximations error-bounded schemes were
constructed for the one and multi-dimensional diusion and linear advection-diusion
equations. Numerical examples show that the method is eective even where standard
schemes, stable by traditional denitions, fail.
In chapter 4 the methodology presented in chapter 1 was adopted to construct errorbounded schemes for parabolic equations and systems containing mixed-derivatives. In
chapter 5 the method was modied to solve the wave equation.
86
The ecacy of the method was demonstrated via the various numerical examples.
The operators developed herein can be used as 'o the shelf operators' for other differential equations of mathematical physics. However the severe restrictions on the
approximations, i.e. the negative or non-positive deniteness in standard L2 norm,
make the construction far from being trivial. The development of fourth order accurate
approximations for @=@x as well as schemes for hyperbolic systems is left for future
research. The imposition of other boundary conditions such as Neumann or absorbing
boundary conditions is also left for future study.
87
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