Chaos, Solitons and Fractals 41 (2009) 2193–2199

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Structural stability of finite dispersion-relation preserving schemes

Claire David *, Pierre SagautUniversité Pierre et Marie Curie-Paris 6, Institut Jean Le Rond d’Alembert, UMR CNRS 7190, Boîte courrier no. 162, 4 place Jussieu, 75252 Paris, Cedex 05, France

a r t i c l e i n f o

Article history:Accepted 19 August 2008

0960-0779/$ - see front matter � 2009 Published bdoi:10.1016/j.chaos.2008.08.028

* Corresponding author. Fax: +33 144 27 52 59.E-mail address: david@lmm.jussieu.fr (C. David).

a b s t r a c t

The goal of this work is to determine classes of travelling solitary wave solutions for a dif-ferential approximation of a finite difference scheme by means of an hyperbolic ansatz. It isshown that spurious solitary waves can occur in finite-difference solutions of nonlinearwave equation. The occurrance of such a spurious solitary wave, which exhibits a very longlife time, results in a non-vanishing numerical error for arbitrary time in unboundednumerical domains. Such a behavior is referred here to has a structural instability of thescheme, since the space of solutions spanned by the numerical scheme encompasses typesof solutions (solitary waves in the present case) that are not solution of the original contin-uous equations. This paper extends our previous work about classical schemes to disper-sion-relation preserving schemes [1].

� 2009 Published by Elsevier Ltd.

1. Introduction: the DRP scheme

The Burgers equation

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 [2] forstudying turbulence in 1939.

i, n denoting natural integers, a linear finite difference scheme for this equation can be written under the form

Xalmum

l ¼ 0; ð2Þ

where

uml ¼ uðlh;msÞ ð3Þ

l 2 fi� 1; i; iþ 1g; m 2 fn� 1;n;nþ 1g; j ¼ 0; . . . ;nx; n ¼ 0; . . . ;nt . The alm are real coefficients, which depend on the meshsize h, and the time step s.

The Courant–Friedrichs–Lewy number (cfl) is defined as r ¼ cs=h.A numerical scheme is specified by selecting appropriate values of the coefficients alm. Then, depending on them, one can

obtain optimum schemes, for which the error will be minimal.m being a strictly positive integer, the first derivative ou=ox is approximated at the lth node of the spatial mesh by

ouox

� �l’Xm

k¼�m

ckuniþk: ð4Þ

y Elsevier Ltd.

2194 C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 2193–2199

Following the method exposed by Tam and Webb in [5], the coefficients ck are determined requiring the Fourier transform ofthe finite difference scheme (4) to be a close approximation of the partial derivative ou=oxð Þl. Eq. (4) is a special case of

ouox

� �l

’Xm

k¼�m

ckuðxþ khÞ; ð5Þ

where x is a continuous variable and can be recovered setting x ¼ lh.Denote by x the phase. Applying the Fourier transform, referred to byb, to both sides of (5), yields

jxu ’Xm

k¼�m

ckejkxhhatu; ð6Þ

j denoting the complex square root of �1.Comparing the two sides of (6) enables us to identify the wavenumber k of the finite difference scheme (4) and the quan-

tity ð1=jÞPm

k¼�mckejkxh, i.e., the wavenumber of the finite difference scheme (4) is thus

k ¼ �jXm

k¼�m

ckejkxh: ð7Þ

To ensure that the Fourier transform of the finite difference scheme is a good approximation of the partial derivative ðou=oxÞlover the range of waves with wavelength longer than 4h, the a priori unknown coefficients ck must be choosen so as to min-imize the integrated error

E ¼Z p

2

�p2

jkh� khj2dðkhÞ

¼Z p

2

�p2

jkhþ jXm

k¼�m

ckejkxhhj2dðkhÞ

¼Z p

2

�p2

jfþ jXm

k¼�m

ck cosðkfÞ þ j sinðkfÞf gj2df

¼Z p

2

�p2

f�Xm

k¼�m

ck sinðkfÞ" #2

þXm

k¼�m

ck cosðkfÞ" #2

8<:9=;df

¼ 2Z p

2

0f�

Xm

k¼�m

ck sinðkfÞ" #2

þXm

k¼�m

ck cosðkfÞ" #2

8<:9=;df: ð8Þ

The conditions that E is a minimum are

oE

oci¼ 0; i ¼ �m; . . . ;m; ð9Þ

i.e.,

Z p2

0�f sinðifÞ þ

Xm

k¼�m

ck cos ðk� iÞfð Þ( )

df ¼ 0: ð10Þ

Changing i into �i, and k into �k in the summation yields

Z p2

0f sinðifÞ þ

Xm

k¼�m

c�k cos ð�kþ iÞfð Þ( )

df ¼ 0; ð11Þ

i.e.,

Z p2

0f sinðifÞ þ

Xm

k¼�m

c�k cos ðk� iÞfð Þ( )

df ¼ 0: ð12Þ

Thus

Z p2

0

Xm

k¼�m

c�k þ ckf g cos ðk� iÞfð Þdf ¼ 0; ð13Þ

which yields

p2

c�i þ cif g þXm

k–i;k¼�m

c�k þ ck

k� i

n osin ðk� iÞp

2

� �¼ 0; ð14Þ

C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 2193–2199 2195

which can be considered as a linear system of 2mþ 1 equations, the unknowns of which are the c�i þ ci; i ¼ �m; . . . ;m. Thedeterminant of this system is not equal to zero, while it is the case of its second member: the Cramer formulae give then, fori ¼ �m; . . . ;m:

c�i þ ci ¼ 0 ð15Þ

or

c�i ¼ �ci: ð16Þ

For i ¼ 0, one of course obtains

c0 ¼ 0: ð17Þ

All this ensures

Xm

k¼�m

ck ¼ 0: ð18Þ

The values of the ck coefficients are obtained by substituting relation (16) into (10)

Xm

k¼�m

ck ¼ 0: ð19Þ

m being a strictly positive integer, a 2mþ 1-points DRP scheme ([5]) is thus given by

�unþ1i þ un

i þsh

Xm

k¼�m

ckuniþk ¼ 0; ð20Þ

where the ck; k 2 f�m;mg, are the coefficients of the considered scheme and satisfy the relation (16).Considering again the um

l terms as functions of the mesh size h and time step s, expanding them at a given order by meansof their Taylor series expansion, and neglecting the oðspÞ and oðhqÞ terms, for given values of the integers p, q, leads to thefollowing differential approximation (see [6]):

�unþ1i þ un

i þsh

Xm

k¼�m

ckFniþk u;

oruoxr

;osuots ; h; s

� �¼ 0; ð21Þ

where Fniþk denotes the function of u, oru=oxr ; osu=ots, h, s obtained by means of the above Taylor expansion, r, s being integers.

For sake of simplicity, a non-dimensional form of Eq. (21) will be used:

�~unþ1i þ ~un

i þXm

k¼�m

ckFniþk ~u;

or ~uoxr

;osuo~ts

� �¼ 0: ð22Þ

Depending on this differential approximation (22), solutions, as solitary waves, may arise.The paper is organized as follows. DRP schemes are analyzed in Section 2. The general method is exposed in Section 3.

Classical DRP schemes are studied in Section 4, where it is shown that out of the two studied schemes, only one leads tosolitary waves. A related class of travelling wave solutions of Eq. (21) is thus presented, by using a hyperbolic ansatz.

2. Analysis of DRP schemes

Consider uni 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: ð23Þ

It ensures

unþ1i � un

i

s¼ utðih;nsÞ þ

s2

uttðih;nsÞ þ oðsÞ: ð24Þ

In the same way, for k 2 f�m;mg, consider uniþk as a function of the mesh size h and expand it at the fourth order by means of

its Taylor series expansion

uniþk ¼ uððiþ kÞh; nsÞ ¼ uðh;nsÞ þ khuxðih;nsÞ þ

k2h2

2uxxðih;nsÞ þ

k4h4

4!uxxxxðih;nsÞ þ oðh4Þ: ð25Þ

Eq. (21) can thus be written as ( )

�utðih; nsÞ �

s2

uttðih;nsÞ þ oðsÞ þ sh

Xm

k¼�m

ck uðih;nsÞ þ khuxðih;nsÞ þk2h2

2uxxðih;nsÞ þ

k4h4

4!uxxxxðih;nsÞ þ oðh4Þ ¼ 0;

ð26Þ

2196 C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 2193–2199

i.e., at x ¼ ih and t ¼ ns:

�ut �s2

utt þ oðsÞ þ sh

Xm

k¼�m

ck uþ khux þk2h2

2þ uxx

k4h4

4!uxxxx þ oðh4Þ

( )¼ 0: ð27Þ

Eq. (19) ensures then

�ut �s2

utt þ oðsÞ þ sh

Xm

k¼�m

kck hux þ oðh4Þn o

¼ 0: ð28Þ

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

�ut �s2

utt þ sXm

k¼�m

kckux ¼ 0: ð29Þ

For the sake of simplicity, this latter equation can be adimensionalized in the following way: set

u ¼ U0~u;

t ¼ s0~t

x ¼ h0~x;

ð30Þ

where

U0 ¼h0

s0: ð31Þ

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

Reh ¼U0hl

: ð32Þ

For j 2 IN, the change of variables (30) leads to

ut ¼U0

s0~u~t ;

utj ¼U0

sj0

~u~tj ;

uxj ¼U0

hj0

~u~xj :

ð33Þ

Eq. (29) becomes

�U0

s0~u~t �

s2

U0

s20

~u~t~t þ 2sXm

k¼1

kckU0

h0~u~x ¼ 0: ð34Þ

Multiplying (34) by s0=U0 yields

�~u~t �s

2s0~u~t~t þ 2

ss0

h0

Xm

k¼1

kckh~u~x ¼ 0: ð35Þ

For h ¼ h0, due to r ¼ U0sh , Eq. (35) becomes

�~u~t �s

2s0~u~t~t þ 2

sh0

lReh

Xm

k¼1

kck~u~x ¼ 0; ð36Þ

which simplifies in

�~u~t �r2

~u~t~t þ2r

lReh

Xm

k¼1

kck~u~x ¼ 0: ð37Þ

3. Solitary waves

Approximated solutions of the Burgers equation (1) by means of the difference scheme (20) strongly depend on the valuesof the time and space steps. For specific values of s and h, Eq. (37) can, for instance, exhibit travelling wave solutions whichcan represent great disturbances of the searched solution.

C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 2193–2199 2197

We presently aim at determining the conditions, depending on the values of the parameters s and h, which give birth totravelling wave solutions of (37).

Following Feng [3] and our previous work [4], in which travelling wave solutions of the CBKDV equation were exhibited ascombinations of bell-profile waves and kink-profile waves, we aim at determining travelling wave solutions of (37) (see [7–15]).

Following [3], we assume that Eq. (37) has travelling wave solutions of the form

~uð~x;~tÞ ¼ ~uðnÞ; n ¼ ~x� v~t; ð38Þ

where v is the wave velocity. Substituting (38) into Eq. (22) leads to

fFð~u; ~uðrÞ; ð�vÞs~uðsÞÞ ¼ 0: ð39Þ

Performing an integration of (39) with respect to n leads to an equation of the form

fFPn ð~u; ~uðrÞ; ð�vÞs~uðsÞÞ ¼ C; ð40Þ

where C is an arbitrary integration constant, which will be the starting point for the determination of solitary wavessolutions.

It is important to note that, contrary to other works, the integration constant is not taken equal to zero, which would leadto a loss of solutions.

4. Travelling solitary waves

4.1. Hyperbolic ansatz

The discussion in the preceding section provides us useful information when we construct travelling solitary wave solu-tions for Eq. (39). Based on these results, in this section, a class of travelling wave solutions of the equivalent equation (29) issearched as a combination of bell-profile waves and kink-profile waves of the form

~uð~x;~tÞ ¼Xn

i¼1

Ui tanhi½Cið~x� v~tÞ� þ Vi sechi½Cið~x� v~t þ x0Þ�� �

þ V0; ð41Þ

where the Ui0s; Vi0s; Ci0s; ði ¼ 1; . . . ;nÞ; V0 and v are constants to be determined.In the following, c is taken equal to 1.

4.2. Theoretical analysis

Substitution of (52) into Eq. (40) leads to an equation of the form

Xi;j;k

Ai tanhiðCinÞsechjðCinÞsinhkðCinÞ ¼ C; ð42Þ

Ai being the real constant.The difficulty for solving Eq. (42) lies in finding the values of the constants Ui; Vi; Ci; V0 and v by solving the over-deter-

mined algebraic equations. Following [3], after balancing the higher-order derivative term and the leading nonlinear term,we deduce n ¼ 1.

Then, following [4] we replace sechðC1nÞ by 2=ðeC1n þ e�C1nÞ; sinhðC1nÞ by ðeC1n � e�C1nÞ=2; tanhðC1nÞ byðeC1n � e�C1nÞ=ðeC1n þ e�C1nÞ, and multiply both sides by ð1þ e2nC1 Þ2, so that Eq. (42) can be rewritten in the following form:

X4

k¼0

PkðU1;V1;C1; v;V0ÞekC1n ¼ 0; ð43Þ

where the Pk ðk ¼ 0; . . . ;4Þ are polynomials of U1; V1; C1; V0 and v.Depending whether (42) admits or no consistent solutions, spurious solitary waves solutions may, or not, appear.

4.3. Numerical scheme analysis

Eq. (39) is presently given by

�v~u0ðnÞ � v2r2

~u00ðnÞ þ 2rlReh

Xm

k¼1

kck~u0ðnÞ ¼ 0: ð44Þ

Performing an integration of (44) with respect to n yields

�v~uðnÞ � v2r2

~u0ðnÞ þ 2rlReh

Xm

k¼1

kck~uðnÞ ¼ C; ð45Þ

2198 C. David, P. Sagaut / Chaos, Solitons and Fractals 41 (2009) 2193–2199

i.e.,

2rlReh

Xm

k¼1

kck � v

( )~uðnÞ � v2r

2~u0ðnÞ ¼ C; ð46Þ

where C is an arbitrary integration constant.Substitution of (52) for n ¼ 1 into Eq. (46) leads to

2rlReh

Xm

k¼1

kck � v

( )U1 tanh½C1n� þ V1 sech½C1n� þ V0f g � v2r

2U1 sech2½C1n� � V1

sinh½C1n�cosh2½C1n�

( )¼ C; ð47Þ

i.e.,

2rlReh

Xm

k¼1

kck � v

( )U1

eC1n � e�C1n

eC1n þ e�C1nþ 2V1

eC1n þ e�C1nþ V0

� �� v2r

2U1

2eC1n þ e�C1n

� �2

� 2V1eC1n � e�C1n

eC1n þ e�C1nð Þ2

( )¼ C: ð48Þ

Multiplying both sides by ð1þ e2C1nÞ2 yields

2rlReh

Xm

k¼1

kck � v

( )U1 e4C1n � 1�

þ 2V1 e3C1n þ eC1n�

þ V0 1þ e2C1n� 2

n o� v2C1rC1

24U1 � 2V1 e3C1n � 1

� �¼ C; ð49Þ

which is a fourth-order equation in eC1n. This equation being satisfied for any real value of n, one therefore deduces that thecoefficients of ekC1n; k ¼ 0; . . . ;4 must be equal to zero, i.e.,

22r

lReh

Xm

k¼1

kck � v

( )�U1 þ V0f g � v2C1r

24U1 þ 2V1f g ¼ C;

2rlReh

Xm

k¼1

kck � v

( )2V1 ¼ 0;

22r

lReh

Xm

k¼1

kck � v

( )V0 ¼ 0;

22r

lReh

Xm

k¼1

kck � v

( )V1 þ v2C1rV1 ¼ 0;

2rlReh

Xm

k¼1

kck � v

( )U1 þ V0f g ¼ 0:

ð50Þ

v ¼ ð2r=lRehÞPm

k¼1kck; V1–0 leads to the trivial null solution. Therefore, V1 is necessarily equal to zero, which implies

v ¼ 2rlReh

Xm

k¼1

kck;

U1 ¼ �C

2C1v2r;

V0 2 R;C1 2 R:

ð51Þ

All DRP schemes admit thus kink-profile travelling solitary waves solutions, given by

~uð~x;~tÞ ¼ � C

2C12r

lReh

Pmk¼1kck

� �2r

tanh C1 ~x� 2rlReh

Xm

k¼1

kck

!~t

!" #þ V0: ð52Þ

5. Conclusions

The analysis of the nonlinear equivalent differential equation for finite-differenced DRP schemes for the Burgers equationhas been carried out. We show that all DRP schemes admit spurious travelling solitary waves solutions, which make them, asregards this point, structurally instable.

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