An Explicit Finite-difference

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An explicit finite-difference scheme for the solutionof the kadomtsev-petviashvili equationA. G. Bratsos a & E. H. Twizell ba aDepartment of Mathematics , Technological Educational Institution (T.E.I.) of Athens ,Egaleo, Athens, 122 10, Greeceb bDepartment of Mathematics and Statistics , Brunei University , Uxbridge, Middlesex,UB8 3PH, EnglandPublished online: 19 Mar 2007.

To cite this article: A. G. Bratsos & E. H. Twizell (1998) An explicit finite-difference scheme for the solutionof the kadomtsev-petviashvili equation, International Journal of Computer Mathematics, 68:1-2, 175-187, DOI:10.1080/00207169808804685

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Intern. J. Computer Math., Vol. 68, pp. 175- 187 Reprints available directly from the publisher Photocopying permitted by license only

0 1998 OPA (Overseas Publishers Association) Amsterdam B.V. Published under license

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Printed in India.

AN EXPLICIT FINITE-DIFFERENCE SCHEME FOR THE SOLUTION OF THE

KADOMTSEV-PETVIASHVILI EQUATION

A. G. BRATSOSa and E. H. TWIZELL~

"Department of Mathematics, Technological Educational Institution (T.E.I.) of Athens, 122 10 Egaleo, Athens, Greece; b~epartment of Mathematics

and Statistics, Brunel University, Uxbridge, Middlesex, England, UB8 3PH

(Received 24 August 1996)

A finite-difference method is used to transform the initial/boundary-value problem associated with the nonlinear Kadomtsev-Petviashvili equation, into an explicit scheme.

The numerical method is developed by replacing the time and space derivatives by central- difference approximants. The resulting finite-difference method is analysed for local truncation error, stability and convergence. The results of a number of numerical experiments are given.

Keywords: Kadomtsev-Petviashvili equation; finite-difference method; explicit scheme

C. R. Categories: G.1.8

1. INTRODUCTION

The equation which initiated the search for soliton solutions is that obtained by Korteweg and de Vries [5] and generalized by Kadomtsev and Petviashvili [4]. The latter will be referred to from now on as the Kadomtsev-Petviashvili nonlinear equation, or the KP equation, which may be written as

(u, + 6 uu, + uXx,),+3 u,, = 0

that is

where u = u (x, y, t ) is a sufficiently-often differentiable function.

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176 A. G. BRATSOS AND E. H. TWIZELL

Initial conditions associated with the partial differential equation (PDE) given in (1) will be assumed to have the form

with initial velocity

Boundary conditions will be assumed to be of the form

d u ( L o , y , t ) - - a u ( x , L o , t ) = 0; t > to, d x a~

and

It was shown by Hirota [3] that equation ( 1 ) has soliton solution which can be expressed in the form

where the function f = f ( x , y, t ) is defined for the single soliton by

exp (81 + * I ) f = l + - 11 + n ~

2 in which 01 = 11 x - 1 y - 41: t , Q l = nl x + n! y - 4ni t with 11, nl, al real constants.

Let kl = - ( I : - ny) y - 4(1: + n:)t and X = al / (11 + nI) . Then f can be written as

f = 1 + X exp [ ( I 1 + nl ) x + k l ] (8)

so that,

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Relation (6) requires f > 0, which implies

a1 1 +- exp (01 + QI) > 0 with 11 + nl # 0.

11 + nl

Since the function exp ( e l + Q1) is by definition always positive, inequality (10) is obviously satisfied when

If a l = 0, then fl = 1 is a constant function, so ul = 0, which is an obvious solution df (1). For a l # 0, (1 1) gives a l > 0 and Il + nl > 0 or a l < 0 and ll + nl < 0. The case of al(ll + nl) < 0 will be examined.

In Figure 1 the surfaces show the theoretical solution u at time t = 0 and t = 12 withal = 6, l1 = 1 a n d n l = 1.

2. THE NUMERICAL METHOD AND ITS ANALYSIS

Following Bratsos [I], to obtain a numerical solution the time and space partial derivatives are replaced by central - difference replacements. To this effect the region R = R x [t > to] where R is a square defined by the lines x, y = Li; i = 0, 1 with boundary dR and initial axis t = to, is covered with a rectangular mesh, G, of points with coordinates (x, y, t) = (xk, ym, tn) = ( L o + k h , L o + m h , t o + n l ) w i t h k , m = O , l , . . . , N + l a n d n = 0 , 1 , . . . , defined as follows: both intervals L o < x < L1 are divided into N + 1

KP for c=O XP t or c-12

FIGURE 1 The surfaces show the theoretical solution u at time t = 0 and t = 12 respectively when al = 6, lI = 1 and nl = 1.

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subintervals each of width h, where N is an even number, so that (N + l)h = L1 - Lo and the time variable t into equally spaced steps of length I. The square R together with its boundary d R have thus been superimposed by a square grid of N~ points within R and N + 2 equally spaced points along each side of d R . The solution u(x, y, t) of the KP equation at the typical mesh point (xk, ym, t,) is u(xk, y,, t,) which may be denoted, when convenient, by u i, ,. The solution of an approximating difference scheme at the same point will be denoted by U;,,: for the purpose of analysing stability, the numerical value of U;,, actually obtained (subject, for instance, to computer round-off errors) will be denoted by a;,,. Let

T denoting transpose. Clearly, there are N' values of the solution to be determined at each time level.

Replacing in (1) the space and time derivatives by central-differecne approximants gives the following finite-difference scheme for the general mesh point (xk, ym, t,) of the grid of G,

Let r = l/h and q = l/h3. Then (13) gives the following three-level, eleven- point, explicit finite-difference scheme,

fo rk , m = 1,2 ,..., N. Relations (4) give to first order D

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KADOMTSEV-PETVIASHVILI EQUATION 179

for m = 1,2,. . . , N and ( 5 ) gives

Then ( 1 5 ) - ( 1 8 ) , when applied to the mesh point (xk , ym, t n ) give the following explicit scheme for evaluating the solution vector U ( t , + 1 ) = ~ n + l

2

for k = 1

for m = 1,

- 12r ( U T , , - , + U T , m + l ) + u;;!, - B

f o r m = 2 , 3 ,..., N-1,

~ n + l - A

2 . N - - 6r ( B - u ; . N ) ~ - 4 q ( U ; , N + U ; , N )

+ 8(2q - 3 r u;,,)(B + U';, ,) + 2 4 ( r - q + 2r

- 1 2 r ( U ; , N - , + B ) + u';;; - B

for m = N ;

f o r k = 3 3 , . . . , N - 3

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for rn = 1,

for rn = N ;

f o r k = N - 1

form = 1,

for rn= 2,3,. . ., N - 1,

for rn = N,

for k = N

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for m = 1,

f o r m = 2,3 , . . . , N - 1,

f o r m = N ;

f o r k = N - 2 , N - 4 , . . . , 4

for m = l ,

f o r m = 2,3 ,..., N - 1 ,

for m = N; and,

for k = 2

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A. G. BRATSOS AND E. H. TWIZELL

for m = 1.

for m = N.

The local truncation error of the method is

which tends to zero as h, I+ 0 simultaneously, so the difference scheme (13) is consistent with (1).

For the stability analysis of the mthod let Z i , , = U ; , , - u;,,. Then (1 3) gives

Let U K = max {u:,,); k, m = 1, 2, . . . , N. Then the linearization gives

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So (21) can be written as

If Z;l-,m = e anle ;gkhe iymh, i = fl, where a is a complex number and /?, y are real, then substituting into (22), cancelling both sides by ean'ei4khe'7mh and using Euler's formula em = cos x + i sin x, gives the following stability equation of the method

The von Neumann necessary criterion for stability requires

It1 5 1 (24)

Assuming that sin p h # 0, equation (23) is of the form

Let J1, J2 be the roots of (25). Then

so, using a well known property of the modulus of complex number and condition (24), relation (26) leads to

that is

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Condition (27) for (23) becomes

IBI 5 1,

which leads to the following restriction

3. NUMERICAL EXPERIMENTS

The IBVP (1 - 5) was solved numerically with initial time to = 0, boundary line L o = -80, L l = 90 when h = 10, L l = 95 when h = 5, L l = 92 when h = 4 and L = 91 when h = 3. For the initial condition (2) the case U (x, y, 0) = u (x, y, 0), x, y E (LO, L was examined, that is the numerical solution is equal to the theoretical solution for to = 0, while for the numerical solution at the first time step t = I the numerical solution was taken to be the theoretical solution, that is U ( x , y, 1) = u (x, y, I ) , x , y E (Lo, L1). It was deduced from the experiments that the most accurate results were obtained for boundary value B= 0.

Let the error, e = en, be the value of u;,, - U;,, with the maximum modulus (k, m = 1,2,. . . , N) at time level t = nl for n = 0 , l . . .; let the corresponding percentage relative error be e = 8, = en x 100/u;,,, the mean value of the relative errors be 5 = i$, = (C ;= &,)In and let (x, y), to denote the (x, y) coordinates of the point at which e = en occurs.

The KP equation was solved numerically using the values al = 6, nl = 1 and l1 = 1. From Table I, in which the most significant numerical results for

TABLE I Results of the method when a, = 6, 1, = 1 and n , = 1

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different values of h and 1 are given, the following may be deduced

- the best results were obtained for h = 3 and 1 =

- the refining of I did not give any improvement in accuracy, so the value 1 = lop5 should be used for the numerical results,

- even values of h did not give accurate results, while odd values of h with respect to E give accurate results,

- using h = 3 and 1 = lo-' the method gives nonlinear blow-up at time t = 0.18

In Figure 2 the curve shows the relative errors E from time t = 0 to t = 0.72E- 03 when h = 3,1= with al = 6, l1 = 1 and nl = 1, while in Figure 3 the surface shows the numerical solution U at time t = 0.72 E - 03

The results for different values of the parameters al, l1 and nl are given in Tables 11-IV. With respect to E, the following may be deduced

i) from Table 11, for different values of a , when 1, = nl = 1, results with a l > 1012 or a1 E (6, lo3) were not accurate,

ii) from Table 111, for different values of nl when a1 = 6 and 11, results with n l ~ ( 1 , 1.2), nl E[O.I, 0.91 and n1 1 lop4 were not accurate, while the method diverges when nl > 1.5,

iii) finally, from Table IV, for different values of 11 = 1 when al = 6 and nl = I , accurate results were obtained only for 1, E [I, 1.21, while the method diverges when n, 2 1.5.

In Tables I1 - IV, the large relative errors when appeared are due to the surface generated by the numerical method is not coincident with that

relative error ( % )

FIGURE 2 The curve shows the relative error E from time t = 0 to t = 0.72E - 03, with a, = 6 , 1, = 1 a n d q = 1.

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A. G. BRATSOS AND E. H. TWIZELL

FIGURE 3 The surface shows the numerical solution U at time-t = 0.72E - 03, with a , = 6, 1, = 1 andn, = 1.

TABLE I1 Results of the method when h= 3, I = 0.10E-04, nl = 1 and 1, = 1 at time t=0.72E-03

TABLE I11 Results of the method when h= 3, 1= 0.IOE-04, a, = 6 and I ,= 1 at time t ~0 .72E-03

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TABLE IV Results of the method when h = 3, 1= 0.10E-04, a, = -6 and nl = 1 at time t=0.72E-03

relating to the theoretical solution. When the theoretical solution is very small (remote from the soliton peak), this leads to large relative errors near the peak of the soliton generated by the numerical method.

Acknowledgements

The authors would like to thank Professor T. Mitsui, Nagoya University, Japan, for his useful comments to the theoretical solution of KP.

References

[I] Bratsos, A. G. (1993). Numerical Solutions of Nonlinear Partial Differential Equations, Ph. D. Thesis, Brunel University.

[2] Freeman, N. C. (1984). Soliton solutions of Non-linear Evolution Equations, IMA .lournu1 of Applied Mathematics, 32, 125- 145.

[3] Hirota, R. (1980). Direct methods in soliton theory. In Solitons (Bullough, R. K. and Caudrey, P. J., eds). Berlin: Springer-Verlang.

[4] Kadomtsev, B. B. and Petviashvili, V. I. (1970). On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15, 539 - 541.

[5] Korteweg, D. J. and de Vries, G. (1985). On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave, Phil. Mag. Ser., 5(39), 422 - 443.

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An Explicit Finite-difference