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Chaos, Solitons and Fractals 41 (2009) 655–660

www.elsevier.com/locate/chaos

Spurious solitons and structural stability of finite-differenceschemes for non-linear wave equations

Claire David *, Pierre Sagaut

Universite Pierre et Marie Curie-Paris 6, Institut Jean Le Rond d’Alembert, UMR CNRS 7190,

Boıte courrier no 162, 4 place Jussieu, 75252 Paris Cedex 05, France

Accepted 26 February 2008

Abstract

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of afinite-difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of non-linear wave equation. The occurrence of such a spurious solitary wave, which exhibits a verylong life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such abehavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numer-ical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original con-tinuous equations.� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The Burgers equation:

0960-0doi:10.

* CoE-m

ut þ cuux � luxx ¼ 0; ð1Þ

c; l being real constants, plays a crucial role in the history of wave equations. It was named after its use by Burgers [1]for studying turbulence in 1939.

A finite-difference scheme for the Burgers equation can be written under the following general form:

F ðuml ; h; sÞ ¼ 0; ð2Þ

where the discrete solution is denoted

uml ¼ uðldx;mdtÞ ð3Þ

l 2 fi� 1; i; iþ 1g;m 2 fn� 1; n; nþ 1g; j ¼ 1; . . . ; nx; n ¼ 1; . . . ; nt; h, s denoting, respectively, the mesh size and timestep, and r the Courant–Friedrichs–Lewy number ðcflÞ coefficient, defined as r ¼ cs=h.

A numerical scheme is specified by selecting appropriate expression of the function F in Eq. (2).

779/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.1016/j.chaos.2008.02.038

rresponding author. Fax: +33 1 44 27 52 59.ail address: david@lmm.jussieu.fr (C. David).

656 C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 655–660

Considering the uml terms as functions of the mesh size h and time step s, expanding them at a given order by means

of their Taylor series expansion, and neglecting the oðspÞ and oðhqÞ terms, for given values of the integers p; q, leads to adifferential approximation of the Burgers equation (see [4]):

F u;oruoxr

;osuots

; h; s� �

¼ 0; ð4Þ

r; s being integers.For sake of simplicity, a non-dimensional form of Eq. (4) will be used:

fF ~u;or~uo~xr

;os~uo~ts

� �¼ 0; ð5Þ

Depending on this differential approximation (4), solutions, as solitary waves, may arise.The paper is organized as follows. Two specific schemes are exhibited in Section 2. The general method is exposed in

Section 3. In Section 4, it is shown that out of the two studied schemes, only one leads to solitary waves. A related classof traveling wave solutions of Eq. (4) is thus presented, by using a hyperbolic ansatz. The stability of this class of solu-tions is discussed in the same section.

2. Analysis of some usual finite-difference schemes

2.1. Finite-difference second-order centered scheme in space, Euler-time scheme

For the finite second-order accurate centered scheme in space and Euler-time scheme, the function F in Eq. (2) takesthe form:

F ðuml ; h; sÞ ¼

unþ1i � un

i

sþ cun

i

uniþ1 � un

i�1

2h� l

uniþ1 � 2un

i þ uni�1

h2¼ 0: ð6Þ

Consider unþ1i as a function of the time step s, and expand it at the second-order by means of its Taylor series:

unþ1i ¼ uðih; ðnþ 1ÞsÞ ¼ uðih; nsÞ þ sutðih; nsÞ þ s2

2uttðih; nsÞ þ oðs2Þ: ð7Þ

It ensures

unþ1i � un

i

s¼ utðih; nsÞ þ s

2uttðih; nsÞ þ oðsÞ: ð8Þ

In the same way, consider uniþ1 and un

i�1 as functions of the mesh size h, and expand them at the fourth-order by means oftheir Taylor series expansion:

uniþ1 ¼ uððiþ 1Þh; nsÞ ¼ uðih; nsÞ þ huxðih; nsÞ þ h2

2uxxðih; nsÞ þ h3

3!uxxxðih; nsÞ þ h4

4!uxxxxðih; nsÞ þ oðh4Þ; ð9Þ

uni�1 ¼ uðði� 1Þh; nsÞ ¼ uðih; nsÞ � huxðih; nsÞ þ h2

2uxxðih; nsÞ � h3

3!uxxxðih; nsÞ þ h4

4!uxxxxðih; nsÞ þ oðh4Þ: ð10Þ

It ensures

uniþ1 � 2un

i þ uni�1

h2¼ uxxðih; nsÞ þ 2h2

4!uxxxxðih; nsÞ þ oðh2Þ ð11Þ

and

uniþ1 � un

i�1

2h¼ uxðih; nsÞ þ h2

3!uxxxðih; nsÞ þ oðh3Þ ð12Þ

Eq. (4) can thus be written as

utðih; nsÞ þ s2

uttðih; nsÞ þ oðsÞ þ cuðih; nsÞ uxðih; nsÞ þ h2

3!uxxxðih; nsÞ þ oðh3Þ

� �

� l uxxðih; nsÞ þ 2h2

4uxxxxðih; nsÞ þ oðh2Þ

� �¼ 0 ð13Þ

C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 655–660 657

i.e., at x ¼ ih and t ¼ ns

ut þs2

utt þ oðsÞ þ cu ux þh2

3!uxxx þ oðh3Þ

� �� l uxx þ

2h2

4!uxxxx þ oðh2Þ

� �� �ðx;tÞ¼ 0: ð14Þ

The first differential approximation of the Burgers equation (1) is thus obtained neglecting the oðsÞ and oðh2Þ terms,yielding

ut þs2

utt þ cu ux þh2

3!uxxx

� �� l uxx þ

h2

12uxxxx

� �� �ðx;tÞ¼ 0 ð15Þ

that we will keep as

ut þ cuux � luxx þs2

utt þh2

6uuxxx � l

h2

12uxxxx ¼ 0: ð16Þ

For sake of simplicity, this latter equation can be adimensionalized through in the following way.Set

u ¼ U 0~u;

t ¼ s0~t;

x ¼ h0~x;

8><>: ð17Þ

where

U 0 ¼h0

s0

: ð18Þ

In the following, Reh will denotes the mesh Reynolds number, defined as

Reh ¼U 0hl: ð19Þ

The change of variables (17) leads to

ut ¼ U0

s0~u~t;

uxk ¼ U0

hk0

~u~xk :

8<: ð20Þ

Multiplying (16) by s0

U0yields

~u~t þ cU 0s0

h0

~u~u~x � ls0

h20

~u~x~x þs

2s0

~u~t~t þh2U 0s0

6h30

~u~u~x~x~x � lh2s0

12h40

~u~x~x~x~x ¼ 0: ð21Þ

Relations (18) and (19) ensure

~u~t þ c~u~u~x �h

h0Reh~u~x~x þ

s2s0

~u~t~t þh2

6h20

~u~u~x~x~x �h3

12Rehh30

~u~x~x~x~x ¼ 0: ð22Þ

For h ¼ h0, due to r ¼ U0sh , Eq. (23) becomes

~u~t þ c~u~u~x �1

Reh~u~x~x þ r~u~t~t þ

1

6~u~u~x~x~x �

1

12Reh~u~x~x~x~x ¼ 0: ð23Þ

2.2. The Lax-Wendroff scheme

For the Lax-Wendroff scheme, the function F of (2) takes the form

F ðuml ; h; sÞ ¼

unþ1i � un

i

sþ cun

i

uniþ1 � un

i�1

2h

� �� lþ c2s

2

� �un

iþ1 � 2uni þ un

i�1

h2

� �¼ 0: ð24Þ

unþ1i �un

is is expressed by means of (8), and

uniþ1�2un

i þuni�1

h2 by means of (11), leading to

uniþ1 � 2un

i þ uni�1

h2¼ uxxðih; nsÞ þ 2h2

4!uxxxxðih; nsÞ þ oðh2Þ: ð25Þ

658 C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 655–660

Eq. (11) also yields

uniþ1 � un

i�1

2h¼ uxðih; nsÞ þ h2

3!uxxxðih; nsÞ þ oðh3Þ: ð26Þ

Eq. (24) can thus be written as

utðih; nsÞ þ s2

uttðih; nsÞ þ oðsÞ þ auðih; nsÞ uxðih; nsÞ þ h2

3!uxxxðih; nsÞ þ oðh3Þ

� �

� lþ c2s2

� �uxxðih; nsÞ þ 2h2

4!uxxxxðih; nsÞ þ oðh2Þ

� �¼ 0 ð27Þ

i.e., at x ¼ ih and t ¼ ns

ut þs2

utt þ oðsÞ þ cu ux þh2

3!uxxx þ oðh3Þ

� �� lþ c2s

2

� �uxx þ

2h2

4!uxxxx þ oðh2Þ

� �� �ðx;tÞ¼ 0: ð28Þ

The first differential approximation of the Burgers equation (1) is thus obtained neglecting the oðsÞ and oðh2Þ terms:

ut þs2

utt þ cu ux þh2

3!uxxx

� �� lþ c2s

2

� �uxx þ

h2

12uxxxx

� �� �ðx;tÞ¼ 0 ð29Þ

that we will keep as

ut þ cuux � lþ c2s

2h2

� �uxx þ

s2

utt þh2

6uuxxx � lþ c2s

2

� �h2

12uxxxx ¼ 0: ð30Þ

Eq. (30) is adimensionalized as in Section 2.1, leading to

~u~t þ c~u~ux �1

Rehþ c2r

2

� �~u~x~x þ

s2

~u~t~t þ1

6~u~u~x~x~x �

1

Rehþ c2r

2

� �1

12~u~x~x~x~x ¼ 0: ð31Þ

3. Solitary waves

Approximated solutions of the Burgers equation (1) by means of the difference scheme (2) strongly depend on thevalues of the time and space steps. For specific values of s and h, Eq. (5) can, for instance, exhibit traveling wave solu-tions which can represent great disturbances of the searched solution.

We presently aim at determining the conditions, depending on the values of the parameters s and h, which give birthto traveling wave solutions of (16).

Following Feng [2] and our previous work [3], in which traveling wave solutions of the cBKDV equation were exhib-ited as combinations of bell-profile waves and kink-profile waves, we aim at determining traveling wave solutions of (5)(see [5–13]).

Following [2], we assume that Eq. (5) has traveling wave solutions of the form

~uð~x;~tÞ ¼ ~uðnÞ; n ¼ ~x� v~t; ð32Þ

where v is the wave velocity. Substituting (32) into Eq. (5) leads to

fFð~u; ~uðrÞ; ð�vÞs~uðsÞÞ ¼ 0: ð33Þ

Performing an integration of (33) with respect to n and setting the integration constant to zero leads to an equation ofthe form:

fFP

ð~u; ~uðrÞ; ð�vÞs~uðsÞÞ ¼ 0; ð34Þ

which will be the starting point for the determination of solitary waves solutions.

4. Traveling solitary waves

4.1. Hyperbolic ansatz

The discussion in the preceding section provides us useful information when we construct traveling solitary wavesolutions for Eq. (33). Based on these results, in this section, a class of traveling wave solutions of the equivalentEq. (16) is searched as a combination of bell-profile waves and kink-profile waves of the form

TableThe re

Sets 1,

Set 3

Set 4

C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 655–660 659

~uð~x;~tÞ ¼Xn

i¼1

ðU itanhi½Cið~x� v~tÞ� þ V isechi½Cið~x� v~t þ x0Þ�Þ þ V 0; ð35Þ

where the U 0is; V 0is; C0is ði ¼ 1; . . . ; nÞ; V 0 and v are constants to be determined.In the following, c is equal to 1.

4.2. Theoretical analysis

Substitution of (35) into Eq. (33) leads to an equation of the form

Xi;j;k

AitanhiðCinÞsechjðCinÞsinhkðCinÞ ¼ 0 ð36Þ

the Ai being real constants.The difficulty for solving Eq. (36) lies in finding the values of the constants U i; V i; Ci; V 0 and v by solving the over-

determined algebraic equations. Following [2], after balancing the higher-order derivative term and the leading non-lin-ear term, we deduce n ¼ 1. Then, following [3] we replace sechðC1nÞ by 2

eC1nþe�C1n ; sinhðC1nÞ by eC1n�e�C1n

2; tanhðC1nÞ by

eC1n�e�C1n

eC1 ;nþe�C1n, and multiply both sides by ðenC1 þ e�nC1Þ5e5nC1 , so that Eq. (36) can be rewritten in the following form:

X10

k¼0

P kðU 1; V 1;C1; v; V 0ÞekC1n ¼ 0; ð37Þ

where the P kðk ¼ 0; . . . ; 10Þ, are polynomials of U 1; V 1;C1; V 0 and v.

4.3. Numerical scheme analysis

4.3.1. Finite-difference second-order centered scheme in space, explicit Euler-time integration

Eq. (33) is presently given by

�v~u0ðnÞ þ c~uðnÞ~u0ðnÞ � 1

Reh~u00ðnÞ þ v2 s

2~u00ðnÞ þ 1

6~uðnÞ~uð3ÞðnÞ � 1

Reh

1

12~uð4ÞðnÞ ¼ 0: ð38Þ

Performing an integration of (38) with respect to n and setting the integration constant to zero yields

�v~uðnÞ þ c2

~u2ðnÞ þ v2 r2� 1

Reh

� �~u0ðnÞ þ 1

6~uðnÞ~uð2ÞðnÞ � 1

2~u02ðnÞ

� �� 1

12Reh~uð3ÞðnÞ ¼ 0: ð39Þ

The related system (37) has consistent solutions, which are given in Table 1.For sake of simplicity, we use e to denote 1 or �1.In the following, we shall denote:

r1;2 ¼ 484Reh729

r3 ¼5Rehð17C2

1�12Þ

6C21ð4C2

1�9Þ2

r4 ¼ �Rehð64C6

1�384C4

1þ551C2

1�156Þ

6C21ð4C2

1�9Þ2 ¼ � RehðC2

1�4Þð8C2

1�13Þð8C2

1�3Þ

6C21ð4C2

1�9Þ2

8>>>><>>>>:

ð40Þ

1lated system (37)

r v U1 V 1 C1 V 0

2 484Reh729 e 108

11ffiffiffiffi11p

Rehe 108

5ffiffiffiffi11p

Reh0 �e 6ffiffiffiffi

11p �e 108

5ffiffiffiffi11p

Reh

5Rehð17C21�12Þ

6C21ð4C2

1�9Þ2 � 2ð4C31�9C1Þ

5Reh� 18C1

5Reh0 2 R 18C1

5Reh

� Rehð64C61�384C4

1þ551C21�156Þ

6C21ð4C2

1�9Þ2� 5C1

13�8C21

�C1

Reh� 2ð8C3

1�9C1ÞRehð8C2

1�13Þ 0 2 R 18C1

Rehð8C21�13Þ

660 C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 655–660

4.3.2. The Lax-Wendroff scheme

Eq. (33) is then given by

�v~u0ðnÞ þ c~uðnÞu0ðnÞ � 1

Rehþ c2r

2

� �~u00ðnÞ þ v2 r

2~u00ðnÞ þ 1

6~uðnÞuð3ÞðnÞ � 1

Rehþ c2r

2

� �1

12uð4ÞðnÞ ¼ 0: ð41Þ

Performing an integration with respect to n and setting the integration constant to zero yields

�v~uðnÞ þ c2

~u2ðnÞ þ v2 r2� 1

Rehþ c2r

2

� �� �u0ðnÞ þ 1

6~uðnÞ~uð2ÞðnÞ � 1

2~u02ðnÞ

� �� 1

Rehþ c2r

2

� �1

12~uð3ÞðnÞ ¼ 0: ð42Þ

The related system (37) does not admit consistent solutions.

5. Conclusions

The analysis of the non-linear equivalent differential equation for finite-differenced Burgers equation has been car-ried out. It is shown that some finite-difference schemes can lead to the occurrence of spurious traveling solitary waves,which are not solutions of the exact continuous original equation. It is proposed to refer these schemes as structurallyinstable schemes. Such spurious solitary waves have constant energy, and therefore the numerical error norm does notvanish at arbitrary long integration times on unbounded numerical domains.

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