Adv. Studies Theor. Phys., Vol. 7, 2013, no. 9, 407 - 418
HIKARI Ltd, www.m-hikari.com
Numerical Simulation of KdV equation
M. Akdi and M. B. Sedra1
Université Ibn Tofail, Faculté des Sciences
Département de Physique, LHESIR
Kénitra, Morocco
c 2013 M. Akdi and M. B. Sedra. This is an open access article distributed
Copyright under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
It is by now well known that the 2d- non-linear KdV differential
equation u + u + uu = 0 is one of the most popular and famous
equations of mathematical physics. The major content of the recent
work deals with the establishing of a numerical schemes, based on the
finite difference, to descretize this famous equation. The latter, in it
self, is not the principal focus of our work. It’s exact solvability (the
existence of an explicit analytic solution) is behind our motivation to
develop a numerical strong approach to be tested first on this famous
equation before used it for other models. The principal result of the
this work is the presentation of a proof for the unconditional stability
of our schemes. Other important properties are presented.
1
Introduction
KdV integrable hierarchy systems, describing non linear phenomena, are associated to non linear differential equations that can be solved exactly [1, 2, 3, 4].
The subject of nonlinear phenomena play an important role in many areas of
sciences more notably in applied mathematics and physics. Solving explicitly
these kind of non linear differential equations is of fundamental importance
since it is not an easy job to do it in general. The property of solvability is
traced to the fact that there exist an appropriately large number of conserved
quantities in involution making the system soluble. Recall that the KdV equation, a prototype of integrable models in two dimensions, can be obtained as a
Dif f S 1 flow if we identify the Hill’s operator L = ∂ 2 +u(x, t) with the space of
1
[email protected]
408
M. Akdi and M. B. Sedra
quadratic differentials [5]. We know also that the second hamiltonian structure
of the KdV system is nothing but the classical form of the Virasoro algebra
or conformal symmetry [1, 5, 6]. We focus in this work to contribute much
more to this important research topic by performing numerical treatment of
the KdV equation.
1.1
The Leapfrog method
This is one of the well known method widely used to solve numerically initial boundary value problems for partial differential equations (PDEs). The
importance of this method comes from its simplicity and essentially from his
very good stability when computing PDEs. We start by recalling the essential properties of the Leapfrog difference scheme (LFS). Consider a first order
system of equations such that ∂u
= F [u], we assume that we have to solve
∂t
this first order equation writhing the midpoint method on an interval [t0 , tk ]
with initial value u0 = u(t0 ) with the iterative definition tn = t0 + nΔt and
un = u(tn ). The mid point method implies
un+1 − un−1
∂u
=
∂t
2Δt
1.2
(1)
The Crank Nicolson method
φn
+φn
This method is expressed by the following formula φnm = m+1 2 m−1 or φnm =
n
φn+1
m +φm
depending on whether the temporal or spatial variation is in order.
2
2
2.1
Numerical scheme for the KdV equation
Treatment of the linear part
The Leapfrog scheme is expressed in the case of time derivative as (1)
n+ 21
un+1 −un−1
while
2Δt
n
φn+1
m +φm
2
∂u
∂t
=
the Crank Nicolson scheme gives φm =
As we are interested here to only the linear part of the KdV equation
∂u
∂3u
+
.
∂t
∂x3
of the KdV potential u can be
It’s easily shown that the time derivative ∂u
∂t
written as
1
∂u
=
∂t
2
n−1
un+1
un+1 − un−1
m+1 − um+1
m
+ m
2Δt
2Δt
(1)
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Numerical Simulation of KdV equation
The combined Leapfrog-Crank Nicolson schemes lead us to write
n+1
n+1
n
n+1
n
n
um+2 + unm+2 − 3 un+1
+
3
(u
+
u
+
u
)
−
u
+
u
∂3u
m+1
m
m
m−1
m+1
m−1
=
∂x3
2Δx3
(2)
and finally the linear part of the KdV equation
∂u ∂ 3 u
+ 3 =
∂t
∂x
n−1
n+1
n−1
un+1
m+1 − um+1 + (um − um )
4Δt
+
2.2
un+1
m+2
+
unm+2
(3)
n+1
n+1
n
n+1
n
n
− 3 um+1 + um+1 + 3 (um + um ) − um−1 + um−1
2Δx3
Treatment of the non linear part
The non linear part of the KdV equation can be set as follows:
u.
∂u
1 ∂ (u2 )
= .
.
∂x
2 ∂x
(4)
We propose the following discretization scheme for the spatial derivative, we
obtain
1 n 2
∂u
n 2
=
. um+1 − (um )
u.
∂x
2Δx
2.3
(5)
Global scheme
On the basis of previous considerations we can establish a global scheme for
3
+ u ∂u
+ ∂∂xu3 = 0, we obtain the
the discretization of the KdV equation ∂u
∂t
∂x
following result:
1
2
1
.
Δx3
n−1
n−1
un+1
un+1 − um
m+1 − um+1
+ m
2Δt
2Δt
1 n 2
n 2
. um+1 − (um ) +
+
2Δx
n−1
n−1
n−1
un+1 + um+1
un+1 + un−1
un+1
un+1 + um
m+2 + um+2
m−1
− 3. m+1
+ 3. m
− m−1
2
2
2
2
=0
(6)
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M. Akdi and M. B. Sedra
3
Stability study
Let W = ν = (ν 0 , ..., ν M ) ∈ RM +1 , ν 0 = ν M +1 = 0 and let’s define the following finite difference operators for a function φ ∈ W as
∇− φm =
φm − φm−1 +
φ
φ
− φm −
− φm−1
, ∇ φm = m+1
, ∇ φm = m+1
Δx
Δx
2Δx
and
∇− φm =
3.1
n
n
φm−1 − 2φm + φm−1 ∂φ
φn+1
φn+1
n+ 12
m − φm
m − φm
=
,
φ
,
=
m
Δx2
∂t
Δx
2
Stability for the linear part
For the study of the stability for this part of the KdV equation, we will use the
previous derived results and apply an algebraic analysis based on the following
steps. We start by setting
∂ û
(ξ) = iξ û (ξ)
∂x
(1)
and by means of the Leap Forg-Crank Nicolson schemes we can write
−
û+
∂ û
ξ − ûξ
(ξ) =
∂t
2Δt
(2)
û+ + û−
∂ 3 û
ξ
3
3 ξ
(ξ) = −iξ û (ξ) = −iξ
3
∂x
2
(3)
and
This leads us to write the linear part of the KdV equation as follows
−
−
û+
û+
ξ − ûξ
ξ + ûξ
− iξ 3
=0
2Δt
2
Let’s consider û+
ξ =
Finally, since
1+iΔtξ 3 −
û and
1−iΔtξ 3 ξ
(4)
−
by use of the norme, we obtain û+
ξ = ûξ .
1 + iΔtξ 3 =1
|g (Δt, ξ)| = 1 − iΔtξ 3 (5)
we can conclude, for this linear part of the KdV equation, tat the kind of
scheme applied is unconditionally stable.
411
Numerical Simulation of KdV equation
Linearized KdV equation and the g(θ)-amplification
factor
3.2
To investigate the stability of nonlinear KdV equation, we propose to adopt an
approach having an indicative character. This means that we will discuss the
stability criterion applied to this famous equation, through its linearization.
The ensuing results will be of great importance for us in the sense that if the
discretized scheme adopted is presented as unstable, it will be discarded for
the remainder of the study. We guess that this approach is of great relevance
for the study of the stability criterion. Reconsider the LF-CN scheme of the
linearized KdV equation namely
1
.
2
+
n−1
un+1
un+1 − un−1
m+1 − um+1
m
+ m
2Δt
2Δt
1
.
Δx3
+ μ.
n−1
un+1
un+1 + un−1
m+1 + um+1
m−1
− m−1
2Δx
2Δx
(6)
un+1
m+2
+
2
n−1
um+2
− 3.
un+1
m+1
+
2
n−1
um+1
+ 3.
un+1
m
+
2
un−1
m
−
un+1
m−1
+
2
n−1 um−1
=0
and then proceed to study stability by using the Fourier transform method.
We introduce in the above scheme the following form of the discretization
unm = g n eimθ in order to determine the amplification factor g, we obtain
g n+1 − g n−1 ei(m+1)θ + eimθ +
Δt n+1
g
+ g n−1 ei(m+1)θ − ei(m−1)θ +
2μ
Δx
i(m+2)θ
Δt n+1
n−1
e
2 3 g
+g
− 3.ei(m+1)θ + 3.eimθ − ei(m−1)θ = 0 (7)
Δx
Δt
Δt
Performing the following notations A = 2μ Δx
and B = 2 Δx
3 , we can write
g2 − 1
ei(m+1)θ + eimθ + A g 2 + 1 ei(m+1)θ − ei(m−1)θ +
B g 2 + 1 ei(m+2)θ − 3ei(m+1)θ + 3eimθ − ei(m−1)θ = 0
(8)
Thus
g 2 − 1 eiθ eiθ + 1 + g 2 + 1 A ei2θ − 1 + B ei3θ − 3ei2θ + 3eiθ − 1 = 0
(9)
The extraction of the g 2 (θ) amplification factor is an easy exercise. We need
only to specify the following situations:
412
M. Akdi and M. B. Sedra
First situation:
The solution to the previous equation is given by
eiθ eiθ + 1 − A ei2θ − 1 − B ei3θ − 3ei2θ + 3eiθ − 1
g (θ) = iθ iθ
e (e + 1) + A (ei2θ − 1) + B (ei3θ − 3ei2θ + 3eiθ − 1)
(10)
A e2iθ − 1 + B 3eiθ − 3e2iθ + e3iθ − 1 + eiθ eiθ + 1 = 0
(11)
2
if
Second situation:
The set of solutions is set to be C if
eiθ eiθ + 1 − A ei2θ − 1 − B ei3θ − 3ei2θ + 3eiθ − 1 = 0
(12)
eiθ eiθ + 1 + A ei2θ − 1 + B ei3θ − 3ei2θ + 3eiθ − 1 = 0
(13)
and
Third situation:
There is no possible solution if
eiθ eiθ + 1 − A ei2θ − 1 − B ei3θ − 3ei2θ + 3eiθ − 1 = 0
(14)
eiθ eiθ + 1 + A ei2θ − 1 + B ei3θ − 3ei2θ + 3eiθ − 1 = 0
(15)
and
Next, we will focus on the first case of solutions since it’s the only case
giving us an explicit expression of the g 2 , θ−dependent function namely:
eiθ eiθ + 1 − A ei2θ − 1 − B ei3θ − 3ei2θ + 3eiθ − 1
g (θ) = iθ iθ
e (e + 1) + A (ei2θ − 1) + B (ei3θ − 3ei2θ + 3eiθ − 1)
2
To explore much more the obtained formula for g (θ)2 , we set
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Numerical Simulation of KdV equation
X = eiθ eiθ + 1
Y = A ei2θ − 1 + B ei3θ − 3ei2θ + 3eiθ − 1
(16)
giving rise to
g (θ)2 =
X −Y
X +Y
(17)
Identifying the real and imaginary part of the complex numbers X and Y
g 2(θ) =
Re(X − Y ) + iIm(X − Y )
Re(X + Y ) + iIm(X + Y )
(18)
the module of the g 2 function reads
[Re(X − Y ) + iIm(X − Y )] [Re(X + Y ) − iIm(X + Y )]
|g (θ)| =
(19)
[Re(X + Y ) + iIm(X + Y )] [Re(X + Y ) − iIm(X + Y )]
(Re(X)2 − Re(Y )2 ) + i (Im(X)2 − Im(Y )2 ) + 2i (ImXReY − ImY ReX)
=
(Re (X) + Re (Y ))2 + (Im (X) + Im (Y ))2
The stability condition is given by |g (θ)| ≤ 1, implying
Re (X) Re (Y ) + Im (X) .Im (Y ) ≥ 0
(20)
or
Re (X) = cos (2θ) + cos (θ)
Im (X) = sin (2θ) + sin (θ)
(21)
and
Re (Y ) = (A − 3B) cos (2θ) + B (cos (3θ) + cos (θ)) − A − B
Im (Y ) = (A − 3B) sin (2θ) + B (sin (3θ) + sin (θ)) − A − B
Refereing to several research works on the stability using the Schur polynome
we can deduce some necessary constraints on the stability scheme similar to
the condition |g (θ)| ≤ 1. We can summarize as follows
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M. Akdi and M. B. Sedra
⎧
⎪
|μ| ≤ 2π
,
∀ Δt
⎪
L
⎪
⎪
⎪
|μ|
⎪
√ ≤ 2π ≤
⎪
|μ|,
then Δt < 2π 2 12 2π 4 ;
⎨
L
3
( L ) |μ| −( L )
√
if :
|μ|
2π
2πN
3 3 1
2
⎪
√ ≤
≤
,
then
Δt
<
;
⎪
L
L
2 |μ|3
⎪
3
⎪
⎪
⎪
|μ|
⎪
≤ √
,
then Δt 2 < 2πN 2 12 2πN 4 .
⎩ 2πN
L
3
( L ) |μ| −( L )
where N is the order of the Schur polynomial. This shows the conditional
character of the stability of our scheme.
As a first summary, we can say that the stability study for the KdV equation for the different cases discussed, has allowed us to illustrate the specific
criterion of stability for each case. These criteria, in absence of the nonlinear
part, are often restrictive and conditionals. In the following, we will develop
a numerical scheme in the case of global KdV equation and present the corresponding stability study.
3.3
Stability for the improved scheme
Consider {v, w} ∈ W 2 , the scalar product as well as the norme are defined as
follows
v |w
Δx = Δx
M
vm wm ,
m=1
vΔx
M
2 ,
= Δx.
vm
|v|1,Δx
M
+ 2
=
Δx.
∇ vm
(22)
m=1
m=1
Let ψ : W × W −→ W , the bilinear function defined as follows
(ψ (v, w))m =
1
(vm−1 + vm + vm+1 ) (wm+1 − wm−1 ) .
6
By means of the schemes discussed previously, we can write
(23)
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Numerical Simulation of KdV equation
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
+
∇ v |w
Δx = − v ∇− w Δx
Δv |w
Δx
∇v |v
Δx = 0
M = −Δx
∇+ vm ∇+ wm .
(24)
m=1
− Δv |v
Δx = |v|21,Δx
ψ (v, v) |v
Δx = 0
In fact, given ∀v ∈ W, v0 = vM +1 = 0, we have
M
+ +
∇ v m .wm = − v ∇− w Δx .
∇ v |w
Δx = Δx.
m=1
M
∇v |v
Δx = Δx.
Δv |w
Δx = Δx.
M
m=1
m=1
(∇v)m .vm = 0.
M
+ + ∇ vm . ∇ wm .
(Δv)m .wm = −Δx.
m=1
ψ (v, v) |v
Δx = Δx.
M
m=1
(ψ (v, v))m .vm = 0 .
Consider also the following descretization
∂
û
∂t
+ D 3 û + 13 (û.∇û + ∇û2 )
û = ϕ (x) ,
for t = 0
(25)
with D3 = ∇+ ∇∇− and
û.∇û =
1
1
(û.∇û + 2û.∇û) =
û.∇û + ∇û2
3
3
(26)
we obtain the following result
∂
û |û
Δx + D 3 û |û
Δx +
∂t
1
û.∇û + ∇û2 |û
Δx = 0 .
3
(27)
416
M. Akdi and M. B. Sedra
We show also that
1
1
um+1 − um−1 u2m+1 − u2m−1
2
û.∇û + ∇û =
+
um ×
3
3
2.Δx
2.Δx
1
(um . (um+1 − um−1 ) + (um+1 + um−1 ) (um+1 − um−1 ))
=
6.Δx
1
(um+1 + um + um−1 ) (um+1 − um−1 )
=
6.Δx
= ψ (u, u)
By means of the following property
ψ (u, u) |u
Δx = 0 .
we have
3
D û |û
Δx = ∇+ ∇∇− û |û
Δx
= ∇+ ∇∇− û |û
Δx
= − ∇∇− û ∇− û Δx
= − ∇ ∇− û ∇− û
=
|ũ|21,Δx
Δx
∂
by setting ũ = ∇− û , it’s easily shown that ∂t
û |û
Δx = − |ũ|21,Δx giving
∂
û2Δx ≤ 0. This result confirm that the evolution of ûΔx decreases in
∂t
time. Thus ∀n ∈ [0, N] , we have
n+1 u − un ≤ 0
Δx
Δx
(28)
This is a proof of the unconditional stability of the established scheme. As a
conclusion, the descretization of the global KdV equation, on the basis of the
Crank-Nicolson scheme, is given by
n
∂Um
∂t
+
n+ 1
n+ 1
n+ 1
n+ 1
n+ 1
Um+12 − Um−12
Um+12 + Um 2 + Um−12
6.Δx
+
n+ 1
n+ 1
n+ 1
n+ 1
Um+22 − 2.Um+12 + 2.Um−12 − Um−22
and the complete descretization in time and space is given by
2.Δx3
(29)
=0
Numerical Simulation of KdV equation
417
n
un+1
m − um
+
Δt
n+1
1 n+1
n
n+1
n
n
n+1
n
um+1 + unm+1 + un+1
m + um + um−1 + um−1 × um+1 + um+1 − um−1 − um−1 +
24.Δx
n+1
n+1
1 n+1
n
n
n
n+1
n
u
+
2.
u
−
u
=0
+
u
−
2.
u
+
u
+
u
−
u
m+2
m+2
m+1
m+1
m−1
m−1
m−2
m−2
4.Δx3
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Received: February 9, 2013