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A decoupled and conservative difference scheme with fourth-order accuracy for the Symmetric Regularized Long Wave equations
Applied Mathematics and Computation 219 (2013) 9461–9468
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A decoupled and conservative difference scheme with fourth-order accuracy for the Symmetric Regularized Long Wave equations q Tao Nie Section of Basic Courses, Nanjing College of Chemical Technology, Nanjing 210048, China
a r t i c l e
i n f o
a b s t r a c t
Keywords: Symmetric Regularized Long Wave equation Conservation Decoupled scheme High-order convergence
In this paper, a decoupled finite difference scheme with fourth-order accuracy is proposed to solve the Symmetric Regularized Long Wave equation. The scheme is proved to conserve the total energy in the discrete level. Without any restrictions on the grid ratios, the convergence of the difference scheme is proved by utilizing the energy method to be of forth-order in space and second-order in time. A numerical example is given to support the theoretical analysis. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction A weakly nonlinear analysis of the cold-electron plasma equations appropriate for a strongly magnetized non-relativistic electron beam yields the (1 + 1)-dimensional Symmetric Regularized-Long-Wave (SRLW) equation, expressed as a first-order system
ut uxxt þ qx þ uux ¼ 0; qt þ ux ¼ 0; x 2 R:
x 2 R;
ð1:1Þ ð1:2Þ
This system has been shown to describe weakly nonlinear (1 + 1)-dimensional ion-acoustic and space-charge waves, where q and u are the dimensionless electron charge density and the fluid velocity, respectively, (see [1] and references therein). The SRLW equation is explicitly symmetry in the x and t derivatives and is very similar to the Regularized-Long-Wave equation (RLW) [2] which describes shallow water waves and plasma drift waves, the RLW equation is
ut uxxt þ ux þ uux ¼ 0;
x 2 R:
Many finite difference schemes have been presented for the RLW equations [3–12]. For the numerical study of the SRLW equation, there also are many results. In [13], Guo studied the numerical solutions of SRLW equations by the spectral method. In [14], Zheng et al. presented a Fourier pseudo-spectral method with a restraint operator for the SRLW equations, and proved the stability and optimum error estimates. In [15], Ren considered Chebyshev pseudo-spectral method for SRLW equations, he constructed the semi-discrete and fully discrete chebyshev pseudo-spectral schemes and established the corresponding errors estimates. In [16], Shang and Guo analyzed a Chebyshev pseudo-spectral scheme for multi-dimensional generalized SRLW equations. In [17], an Euler mid-point scheme in time and a Fourier pseudo-spectral method in space were
q
This work is supported by the National Natural Science Foundation of China, NO. 11201239. E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.076
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
used to the multi-symplectic formulation of the SRLW equation. It made a multi-symplectic Fourier pseudo-spectral scheme, and several discrete conservation laws of this scheme were proved. In [18,19], some conservative difference schemes with second-order accuracy were proposed and analyzed. In [20], mixed finite element method was used and analyzed to solve the dissipative SRLW equations with damping term. In this paper we consider the SRLW equations (1.1) and (1.2) with the following initial-boundary conditions
qðx; 0Þ ¼ q0 ðxÞ; x 2 R; qðx; tÞ ! 0; as jxj ! 1; t > 0;
uðx; 0Þ ¼ u0 ðxÞ;
ð1:3Þ
uðx; tÞ ! 0;
ð1:4Þ
where u0 ðxÞ and q0 ðxÞ are two given smooth functions which tend to zero rapidly as jxj ! 1. The initial-boundary value problem (1.1)–(1.4) possesses the following conservative quantity[1]:
EðtÞ ¼
1 1 ðjjujj2L2 þ jjux jj2L2 þ jjqjj2L2 Þ ¼ ðjju0 jj2L2 þ jjðu0 Þx jj2L2 þ jjq0 jj2L2 Þ ¼ Eð0Þ; 2 2
ð1:5Þ
where EðtÞ is the called energy. This paper aims to construct a new difference scheme which has the following three advantages: 1. The new scheme is linearized and decoupled in implementation. It is obvious that, the SRLW equation is a coupled system, and almost all of the proposed schemes in the literature are coupled and nonlinear in implementation, and then too much CPU time should be used. Our scheme is decoupled and linearized in practical computation, so it is expected to be more efficient. 2. The new scheme preserve the total energy in the discrete level. In [21], Li and Vu-Quoc said, ‘‘in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation.’’ Zhang et al. pointed out in [22] that the nonconservative schemes may easily show nonlinear blow-up. Thus, a scheme preserving the conservative laws of the initial problem is very important. 3. The new scheme is, without any restrictions on the grid ratios, convergent at forth-order in space and second-order in time, respectively. The remainder of this paper is arranged as follows. In Second 2, a decoupled difference scheme is proposed and analyzed. The scheme is proved to preserve the discrete energy and be convergent at forth-order in space and second-order in time. In Section 3, numerical results are provided to verify the theoretical analysis. 2. A decoupled conservative scheme In this section, we propose a decoupled, linearized, conservative finite difference scheme for the initial-boundary value problem (1.1)–(1.4) and give numerical analysis. 2.1. Finite difference scheme and its conservative law We take a finite interval ½xl ; xr ½0; T as the computing domain, where xl and xr are large enough such that the following initial-boundary value problem
ut uxxt þ qx þ uux ¼ 0; ðx; tÞ 2 ðxl ; xr Þ ð0; T; qt þ ux ¼ 0; ðx; tÞ 2 ðxl ; xr Þ ð0; T; uðx; 0Þ ¼ u0 ðxÞ;
qðx; 0Þ ¼ q0 ðxÞ; x 2 ½xl ; xr ; qðxl ; tÞ ¼ qðxr ; tÞ ¼ 0; t 2 ð0; T
uðxl ; tÞ ¼ uðxr ; tÞ ¼ 0;
ð2:1Þ ð2:2Þ ð2:3Þ ð2:4Þ
is consistent with the problem (1.1)–(1.4). Before giving the finite difference scheme, some notations are firstly introduced. For a positive integer N, let time-step s ¼ ft n ¼ nsjn ¼ 0; 1; . . . ; Ng. Given a tems ¼ T=N; tn ¼ ns; n ¼ 0; 1; 2; . . . ; N, denote Xs ¼ ftn ¼ nsjn ¼ 1; 2; . . . ; N 1g and X poral discrete function fun jt n 2 Xs g, we denote ðun Þ^t ¼ ðunþ1 un1 Þ=2s. For a positive integer J, let space-step h ¼ ðxr xl Þ=J, xj ¼ jh; j ¼ 1; 0; 1; 2; ; J; J þ 1. Denote the grid Xh ¼ fxj ¼ xl þ jhjj ¼ 1; 2; ; J 1g and the extended discrete grid XEh ¼ fxj ¼ jhjj ¼ 1; 0; 1; 2; 3; . . . ; J; J þ 1g. Given a grid function u ¼ fuj jxj 2 Xh g, denote
ðuj Þx ¼ ðujþ1 uj Þ=h;
ðuj Þx ¼ ðuj uj1 Þ=h;
ðuj Þ^x ¼ ðujþ1 uj1 Þ=2h; Let V h ¼ fuju ¼ fuj jxj 2 X product of them as
E hg
ðuj Þ€x ¼ ðujþ2 uj2 Þ=4h: and u1 ¼ u0 ¼ uJ ¼ uJþ1 ¼ 0g. For any grid functions u; v 2 V h , denote the discrete inner
T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
hu; v i ¼ h
J1 X uj v j :
9463
ð2:5Þ
j¼1
The discrete L2 norm jjv jj, the discrete semi-H1 norms jjv x jj; jjv ^x jj; jjv €x jj and the discrete L1 norm (or discrete maximum norm) jjv jj1 of v 2 V h are defined, respectively, as follows
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X jjv x jj ¼ th jðv j Þx j2 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffi jjv jj2 ¼ hv ; v i;
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X jjv €x jj ¼ th jðv j Þ€x j2 ;
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X jjv ^x jj ¼ th jðv j Þ^x j2 ;
j¼0
j¼1
jjv jj1 ¼ max jv j j: 16j6J1
j¼1
ð2:6Þ
For simplicity, we denote unj uðxj ; tn Þ; U nj uðxj ; tn Þ; qnj qðxj ; t n Þ; /nj qðxj ; t n Þ, respectively. Let C denote a positive constant independent of discretization parameters, but may have different values at different occurrences. Based on the introduced notations, we give the following decoupled finite difference scheme.
2 1 n nþ1 4 ðU nj Þ^t þ ½U nj ðU nþ1 þ U n1 Þ^x þ ððU nþ1 þ U n1 ÞU nj Þ^x ½U ðU þ U n1 Þ€x þ ððU nþ1 þ U n1 ÞU nj Þ€x ðU nj Þxx^t j j j j j j j 9 18 j j 3 1 n 4 n 1 n þ ðU j Þ^x^x^t þ ð/j Þ^x ð/j Þ€x ¼ 0; ðxj ; t n Þ 2 Xh Xs ; 3 3 3 4 n 1 n n ð/j Þ^t þ ðU j Þ^x ðU j Þ€x ¼ 0; ðxj ; t n Þ 2 Xh Xs ; 3 3 4 1 1 1 @ 2 u0 ðU j Þxx ðU j Þ^x^x U 1j ¼ ðxj Þ u0 ðxj Þ þ swðxj Þ; xj 2 Xh ; 3 3 @x2 1 /j ¼ q0 ðxj Þ swðxj Þ; xj 2 Xh ; /0j
¼ q0 ðxj Þ;
/n 2 V h ;
U 0j
¼ u0 ðxj Þ;
Un 2 V h ;
E h;
xj 2 X
t n 2 Xs ;
ð2:7Þ ð2:8Þ ð2:9Þ ð2:10Þ ð2:11Þ ð2:12Þ
where
wðxÞ ¼
@ q0 @u0 ðxÞ þ u0 ðxÞ; @x @x
wðxÞ ¼
@u0 ðxÞ: @x
ð2:13Þ
Obviously, this scheme is decoupled in practical computation. Firstly, ðU 0 ; /0 Þ is obtained directly from (2.11), and ðU 1 ; /1 Þ is computed by using (2.9) and (2.10), then ðU 2 ; /2 Þ is computed by using (2.7) and (2.8). If ðU n1 ; /n1 Þ and ðU n ; /n Þ are known, then U nþ1 and /nþ1 can be computed by (2.7) and (2.8) respectively at the same time. Corresponding to the invariant (1.5) possessed by the continuous problem (1.1)–(1.4), the difference scheme (2.7)–(2.12) satisfy the following discrete conservative law: Lemma 2.1. suppose that u0 2 H1 ðRÞ; q0 2 L2 ðRÞ, then the scheme (2.7)–(2.12) is conservative in the sense
En E0 ;
n ¼ 0; 1; 2; . . . ; N 1;
ð2:14Þ
where
En ¼
1 4 1 4 1 2s jj/nþ1 jj2 þ jjU nþ1 jj2 þ jjU nþ1 jj2 jjU ^nþ1 jj2 þ jj/n jj2 þ jjU n jj2 þ jjU nx jj2 jjU ^nx jj2 þ ðh/n^x ; U nþ1 i x x 4 3 3 3 3 3
s
þ h/nþ1 ; U ^nx iÞ ðh/€nx ; U nþ1 i þ h/nþ1 ; U n€x iÞ: 6
ð2:15Þ
Proof. Computing the inner product of (2.7) with 12 ðU nþ1 þ U n1 Þ yields
1 n 2 2 n 2 1 n 2 2 n nþ1 1 1 jjU jj^t þ jjU x jj^t jjU ^x jj^t þ h/^x ; U þ U n1 i h/n€x ; U nþ1 þ U n1 i þ hnðU n1 ; U n ; U nþ1 Þ; U nþ1 þ U n1 i ¼ 0: 2 3 6 3 6 2 ð2:16Þ Direct computation gives
hnðU n1 ; U n ; U nþ1 Þ; U nþ1 þ U n1 i ¼ 0; where
nðU n1 ; U nj ; U nþ1 Þ¼ j j
2 n nþ1 1 n nþ1 ½U ðU þ U n1 Þ^x þ ððU nþ1 þ U n1 ÞU nj Þ^x ½U ðU þ U n1 Þ€x þ ððU nþ1 þ U n1 ÞU nj Þ€x : j j j j j j 9 j j 18 j j
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This together with (2.16) gives
1 n 2 2 n 2 1 n 2 2 n nþ1 1 jjU jj^t þ jjU x jj^t jjU ^x jj^t þ h/^x ; U þ U n1 i h/n€x ; U nþ1 þ U n1 i ¼ 0: 2 3 6 3 6
ð2:17Þ
Computing the inner product of (2.8) with 12 ð/nþ1 þ /n1 Þ, we obtain
1 n 2 2 nþ1 1 jj/ jj^t þ h/ þ /n1 ; U ^nx i h/nþ1 þ /n1 ; U n€x i ¼ 0: 2 3 6
ð2:18Þ
Adding (2.18) to (2.17) and noticing
h/n^x ; U nþ1 þ U n1 i þ h/nþ1 þ /n1 ; U ^nx i ¼ h/n^x ; U nþ1 i þ h/n^x ; U n1 i þ h/nþ1 ; U n^x i þ h/n1 ; U n^x i ¼ h/^nx ; U nþ1 i i þ h/nþ1 ; U n^x i h/^n1 ; U n i; h/n ; U n1 ^x x
ð2:19Þ
h/n€x ; U nþ1 þ U n1 i þ h/nþ1 þ /n1 ; U €nx i ¼ h/n€x ; U nþ1 i þ h/n€x ; U n1 i þ h/nþ1 ; U n€x i þ h/n1 ; U n€x i ¼ h/€nx ; U nþ1 i h/n ; U n1 i þ h/nþ1 ; U n€x i h/€n1 ; U n i; €x x
ð2:20Þ
yield
1 4 1 2s s ðjj/nþ1 jj2 þ jjU nþ1 jj2 þ jjU nx jj2 jjU n^x jj2 Þ þ ðh/^nx ; U nþ1 i þ h/nþ1 ; U ^nx iÞ ðh/€nx ; U nþ1 i þ h/nþ1 ; U €nx iÞ 4 3 3 3 6 1 4 1 2s s ; U n i þ h/n ; U ^n1 iÞ ðh/n1 ; U n i þ h/n ; U €n1 iÞ: ¼ ðjj/n1 jj2 þ jjU n1 jj2 þ jjU nx jj2 jjU ^nx jj2 Þ þ ðh/n1 ^x €x x x 4 3 3 3 6 This gives (2.14).
ð2:21Þ
h
2.2. a priori estimate and convergence of the difference solution On the solution of the proposed scheme, there is a priori estimate as follows: Lemma 2.2. Suppose that u0 2 H1 ; q0 2 L2 , then the following inequalities
jj/n jj 6 C;
jjU n jj 6 C;
jjU nx jj 6 C;
jjU n jj1 6 C
hold. Proof. Cauchy–Schwartz inequality and tedious calculation give
jj/n€x jj2 6 jj/n^x jj2 6 jj/nx jj2
ð2:22Þ
1 ¼ jh/ 6 ðjj/n jj2 þ jjU nþ1 jj2 Þ; x 2 1 ij 6 ðjj/n jj2 þ jjU nþ1 jj2 Þ; jh/n€x ; U nþ1 ij ¼ jh/n ; U €nþ1 x x 2 1 jh/nþ1 ; U n^x ij 6 ðjj/nþ1 jj2 þ jjU nx jj2 Þ; 2 1 n nþ1 jh/ ; U €x ij 6 ðjj/nþ1 jj2 þ jjU nx jj2 Þ; 2 jh/n^x ; U nþ1 ij
n
; U ^nþ1 ij x
ð2:23Þ ð2:24Þ ð2:25Þ ð2:26Þ
this together with (2.14) gives
1 s 1 ðjj/nþ1 jj2 þ jjU nþ1 jj2 þ jj/n jj2 þ jjU nx jj2 Þ þ ðjjU nþ1 jj2 þ jjU n jj2 Þ 6 C: x 4 3 4 If
s < 34, then from (2.27) we obtain jj/n jj 6 C;
jjU n jj 6 C;
jjU nx jj 6 C:
This together with the discrete Sobolev equality gives
jjU n jj1 6 C:
ð2:27Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
Remark 2.1. Lemma 2.2 implies that the proposed scheme is unconditionally stable. The truncation error of the proposed scheme is defined as
2 1 n nþ1 4 1 ðunj Þ^t þ ½unj ðunþ1 þ un1 Þ^x þ ððunþ1 þ un1 Þunj Þ^x ½u ðu þ un1 Þ€x þ ððunþ1 þ un1 Þunj Þ€x ðunj Þxx^t þ ðunj Þ^x^x^t j j j j j j j 9 18 j j 3 3 4 n 1 n n þ ð/j Þ^x ð/j Þ€x ¼ rj ; ðxj ; t n Þ 2 Xh Xs ; 3 3 4 n 1 n ðqj Þ^t þ ðuj Þ^x ðunj Þ€x ¼ rnj ; ðxj ; tn Þ 2 Xh Xs ; 3 3 4 1 1 1 @ 2 u0 ðuj Þxx ðuj Þ^x^x u1j ¼ ðxj Þ u0 ðxj Þ þ swðxj Þ þ r0j ; xj 2 Xh ; 3 3 @x2 q1j ¼ q0 ðxj Þ swðxj Þ þ r0j ; xj 2 Xh ; 0 j
E h;
u0j
q ¼ q0 ðxj Þ; ¼ u0 ðxj Þ; xj 2 X qn 2 V h ; un 2 V h ; tn 2 Xs :
ð2:28Þ ð2:29Þ ð2:30Þ ð2:31Þ ð2:32Þ ð2:33Þ
Using Taylor’s expansion yields. Lemma 2.3. Suppose that u0 2 H1 ; q0 2 L2 , uðx; tÞ 2 C 6;3 ; qðx; tÞ 2 C 6;3 , then the truncation error of the proposed scheme satisfies 4
jrnj j þ jrnj j ¼ Oðs2 þ h Þ; as
s ! 0; h ! 0.
Theorem 2.1. Suppose that uðx; tÞ 2 C 6;3 ; qðx; tÞ 2 C 6;3 , then the solution of the scheme (2.7)–(2.12) converges to the solution of 4 the problem (2.1)–(2.4) at the order Oðs2 þ h Þ in the discrete L1 norm for U n , and in the discrete L2 norm for /n . Proof. Denote
enj ¼ unj U nj ;
gnj ¼ /nj qnj :
ð2:34Þ
Subtracting (2.7)–(2.12) from (2.28)–(2.33) yields the following error function:
4 1 4 1 ðenj Þ^t þ nðun1 ; unj ; unþ1 Þ nðU n1 ; U nj ; U nþ1 Þ ðenj Þxx^t þ ðenj Þ^x^x^t þ ðgnj Þ^x ðgnj Þ€x ¼ r nj ; j j j j 3 3 3 3 4 n 1 n n n ðgj Þ^t þ ðej Þ^x ðej Þ€x ¼ rj ; ðxj ; t n Þ 2 Xh Xs ; 3 3 4 1 1 1 ðe Þ ðe Þ e1j ¼ r 0j ; xj 2 Xh ; 3 j xx 3 j ^x^x g1j ¼ r0j ; xj 2 Xh ; 0
g ¼ 0; n
e 2 Vh;
e0j
¼ 0;
ðxj ; t n Þ 2 Xh Xs ;
E h;
xj 2 X
n
ð2:35Þ ð2:36Þ ð2:37Þ ð2:38Þ ð2:39Þ
n
g 2 V h ; t 2 Xs :
ð2:40Þ 1
Computing the discrete product of (2.37) with e gives
4 1 2 1 1 2 jje jj jje^x jj þ jje1 jj2 ¼ hr0 ; e1 i: 3 x 3
ð2:41Þ
This together with Lemma 2.3 and Cauchy-Schwartz inequality gives 4
jje1x jj2 þ jje1 jj2 6 Oðh þ s2 Þ:
ð2:42Þ 1
Computing the discrete product of (2.38) with g gives
jjg1 jj2 ¼ hr0 ; g1 i:
ð2:43Þ
This together with Lemma 2.3 and Cauchy–Schwartz inequality gives 4
jjg1 jj2 6 Oðh þ s2 Þ: Computing the inner product of (2.35) with e
ð2:44Þ nþ1
n1
þe
yields
4 1 4 1 jjen jj2 þ jjenx jj2 jjen^x jj2 þ hgn1 ; enþ1 þ en1 i hgn1 ; enþ1 þ en1 i þ hnðun1 ; un ; unþ1 Þ 3 3 3 ^x 3 €x ^t nðU n1 ; U n ; U nþ1 Þ; enþ1 þ en1 i ¼ ðr n ; enþ1 þ en1 Þ:
ð2:45Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
This together with Lemma 2.2 and Cauchy–Schwartz inequality gives
4 1 4 1 jjen jj2 þ jjenx jj2 jjen^x jj2 þ hgn1 ; enþ1 þ en1 i hgn1 ; enþ1 þ en1 i ^x 3 3 3 3 €x ^t 6 jjr n jj2 þ C jjen1 jj2 þ jjen jj2 þ jjenþ1 jj2 þ jjen1 jj2 þ jjenx jj2 þ jjenþ1 jj2 : x x
ð2:46Þ
Computing the inner product of (2.36) with gnþ1 þ gn1 yields
4 1 jjgn jj^2t hg^nx ; gnþ1 þ gn1 i þ hg€nx ; gnþ1 þ gn1 i ¼ hrn ; gnþ1 þ gn1 i: 3 3
ð2:47Þ
This together with Cauchy–Schwartz inequality gives
4 1 jjgn jj^2t hg^nx ; gnþ1 þ gn1 i þ hg€nx ; gnþ1 þ gn1 i 6 jjrn jj2 þ jjgnþ1 jj2 þ jjgn1 jj2 : 3 3
ð2:48Þ
Adding 2.48 to 2.46 yields
ðjjen jj2 þ jjenx jj2 þ jjgn jj2 Þ^t 6 jjr n jj2 þ jjrn jj2 þ C jjen1 jj2 þ jjen jj2 þ jjenþ1 jj2 þ jjen1 jj2 þ jjenx jj2 þ jjenþ1 jj2 : x x
ð2:49Þ
This together with Lemma 2.3 and the discrete Gronwall’s inequality gives 4
jjen jj 6 Oðh þ s2 Þ;
4
jjgn jj 6 Oðh þ s2 Þ;
4
jjenx jj 6 Oðh þ s2 Þ;
ð2:50Þ
when s is small enough. 4 It follows from the discrete Sobolev inequality that jjen jj1 6 Oðh þ s2 Þ. h Remark 2.2. All results above in this paper are correct for periodic boundary-initial value problem of the SRLW equation, and the initial functions should be periodic ones correspondingly.
3. Numerical experiments Denote
k n 4 2k ðu þ unj2 Þ; Bnj ¼ 2 ðunj þ unj1 Þ; 36 j 9 3h 5 4 2k 1 k n ðuj þ unjþ2 Þ; C nj ¼ 1 þ 2 ; Dnj ¼ 2 þ ðunj þ unjþ1 Þ; Enj ¼ 2 9 36 2h 3h 12h e n ¼ 1 k ðun þ un Þ; B e n ¼ 4 þ 2k ðun þ un Þ; A j j2 j j1 2 2 36 j 9 j 12h 3h en ¼ 1 þ 5 ; D e n ¼ 4 2k ðun þ un Þ; E e n ¼ 1 þ k ðun þ un Þ; C j j j1 j j2 2 2 2 9 j 36 j 2h 3h 12h 4k k e n un1 þ B e n un1 þ D e n un1 þ C e n un1 þ e E nj un1 ð/n /nj1 Þ þ ð/njþ2 /nj2 Þ; F nj ¼ A j j2 j j1 j j j jþ1 jþ2 3 jþ1 6
Anj ¼
1
12h
2
þ
ð3:1Þ
where k ¼ hs, then the proposed scheme can be written as the following form
Hunþ1 ¼ F; /nþ1 j
ðxj ; tn Þ 2 Xh Xs ; 4k k ¼ /n1 ðunjþ1 unj1 Þ þ ðunjþ2 unj2 Þ; j 3 6
ð3:2Þ ðxj ; tn Þ 2 Xh Xs ;
where
0
C1
D1
E1
B B2 B B B A3 B B 0 B H¼B B B B B 0 B B @ 0
C2
D2
B3
C3
A4
B4 .. .
C4 .. .
...
0
...
0
...
0
0
0
0
E2
0
0
...
D3
E3
0
...
D4 .. .
E4 .. .
... .. .
AJ3
BJ3
C J3
DJ3
0
AJ2
BJ2
C J2
0 C C C 0 C C 0 C C C C C C EJ3 C C C DJ2 A
0
0
AJ1
BJ1
C J1
...
0
1
0
ð3:3Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468 Table 1 Errors of the numerical solution computed by the proposed scheme at t ¼ 2. h
s
e
order1
g
order2
0.4 0.2 0.1
0.4 0.1 0.025
1.5076e2 1.0768e3 6.7112e5
– 3.81 4.00
2.5328e2 1.6322e3 1.0187e4
– 3.95 4.00
Table 2 Errors of the numerical solution computed by the proposed scheme with extrapolation technique at t ¼ 8 h
s
Fe
order3
Fg
order4
0.8 0.4 0.2 0.1
0.8 0.4 0.2 0.1
4.0104e2 2.7777e3 1.8476e4 1.3051e5
– 3.86 3.90 3.82
3.6881e2 2.3139e3 1.5127e4 1.0189e5
– 3.99 3.94 3.89
Table 3 Discrete energy En computed by the proposed scheme with h ¼ 0:2; s ¼ 0:1 at various tn . tn
En
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
13.55949629317673 13.55949629317681 13.55949629317669 13.55949629317668 13.55949629317670 13.55949629317672 13.55949629317669 13.55949629317669 13.55949629317669 13.55949629317673
is a penta-diagonal matrix. The system of linear algebra Eqs. (3.2) can be solved by using Thomas algorithm, and (3.3) can be solved explicitly. In order to achieve higher order numerical accuracy, the Richardson extrapolation is also a practical method. It obtains high-order accurate resolutions by using certain combinations of difference solutions with various grid parameters. The main advantage of the global extrapolations is that they preserve the numerical stability of lower-order methods used initially. Using the extrapolation technique on the difference schemes, one can obtain higher order accuracy. Let unj ðh; sÞ denote h s 2n the numerical solution unj with mesh-sizes h and s at the point ðxj ; tn Þ, and u2n 2j ð2 ; 2Þ denote the numerical solution u2j with mesh-sizes 2h and 2s at the point ðxj ; tn Þ. Denote
4 2n h s 1 unj ðh; sÞ u2j ; 3 2 2 3
ðuFE Þnj ¼
ð3:4Þ
and
ðeFE Þnj ¼ U nj ðuFE Þnj :
ð3:5Þ
Simple but tedious computation gives that 4
ðeFE Þnj ¼ Oðh þ s4 Þ:
ð3:6Þ
Choosing
u0 ðxÞ ¼
pffiffiffi pffiffiffi ! 2 3 2 2 x ; sech 2 4
3 2
q0 ðxÞ ¼ sech2
pffiffiffi ! 2 x ; 4
ð3:7Þ
then the problem (1.1)–(1.4) has the following exact solution
uðx; tÞ ¼
! pffiffiffi pffiffiffi 3 2 1 2 2 x t ; sech 2 2 4
3 2
qðx; tÞ ¼ sech2
! pffiffiffi 1 2 x t : 2 4
In implementation, we chose the computational domain ½xl ; xr as ½40; 40 and the iterative tolerance as 108 .
ð3:8Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
Denote
eðh; sÞ ¼ jjU N ðh; sÞ uN ðh; sÞjj1 ; gðh; sÞ ¼ jj/N ðh; sÞ qN ðh; sÞjj1 ; h s Feðh; sÞ ¼ jj4=3U N 1=3U N ðh; sÞ uN ðh; sÞjj1 ; ; 2 2 h s 1=3/N ðh; sÞ qN ðh; sÞjj1 ; ; F gðh; sÞ ¼ jj4=3/N 2 2 h s h s = logð2Þ; order2 ¼ log gðh; sÞ=g ; = logð2Þ; order1 ¼ log eðh; sÞ=e ; 2 4 2 4 h s h s = logð2Þ; order4 ¼ log F gðh; sÞ=F g ; = logð2Þ; order3 ¼ log Feðh; sÞ=Fe ; 2 2 2 2 then we list the numerical results in the Table 1, Table 2 and Table 3. Table 1 shows that the convergence of the proposed scheme is forth-order in space and second-order in time, this verifies Theorem 2.1. Table 2 shows the high accuracy of the proposed extrapolation. Table 1 and Table 2 also verify the good stability of the proposed scheme. Table 3 shows that the proposed scheme conserves the discrete energy very well, this verifies Lemma 2.1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
C.E. Seyler, D.L. Fenstermacher, A symmetric regularized- long-wave equation, Phys. Fluids 27 (1984) 4–7. D.H. Peregrine, Calculations of the development of an unduiar bore, J. Fluid Mech. 25 (1966) 321–330. M.E. Alexender, J.Ll. Morris, Galerkin method applied to some model equation for nonlinear dispersive waves, J. Comput. Phys. 30 (1979) 428–451. P.J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Philos. Soc. 85 (1979) 143–159. J.M. Sarna, I. Christie, Petro–Galerkin methods for nonlinear dispersive waves, J. Comput. Phys. 39 (1981) 94–102. L.R.T. Gardner, G.A. Gardner, Solitary waves of regularized long wave equation, J. Comput. Phys. 91 (1990) 441–459. Q. Chang, G. Wang, B. Guo, Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion, J. Comput. Phys. 93 (1991) 360–375. T. Achouri, N. Khiari, K. Omrani, On the convegence of difference schemes for the Benjanmin–Bona–Mahony(BBM) Equation, Appl. Math. Comput. 182 (2006) 999–1005. Luming Zhang, A finite difference scheme for generalized regularized long-wave equation, Appl. Math. Comput. 168 (2005) 962–972. J.I. Ramos, Explicit finite difference methods for the EW and RLW equations, Appl. Math. Comput. 179 (2006) 622–638. A. Araujoa, A. Duranb, Error propagation in the numerical integration of solitary waves. The regularized long wave equation, Appl. Numer. Math. 36 (2001) 197–217. D. Bhardwaj, R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl. 40 (2000) 1397–1404. Bo-ling Guo, The spectral method for symmetric regulaized wave equations, J. Comput. Math. 5 (1987) 297–306. Jiadong Zheng, Rufen Zhang, Benyu Guo, The fourier pseudo spectral method for the SRLW equation, Appl. Math. Mech. 10 (1989) 801–810. Zongxiu Ren, Chebyshev Pseudo-spectral method for SRLW equations, Chin. J. Eng. Math. 12 (1995) 34–40. Ya-dong Shang, Bo-ling Guo, Analysis of chebyshev pseudospectral method for multi-dimentional generalized SRLW equations, Appl. Math. Mech. 24 (2003) 1035–1048. Linghua Kong, Wenping Zeng, Ruxun Liu, Lingjian Kong, A muitisymplectic fourier pseudo-spectral scheme for the SRLW equation and conservation laws, Chin. J. Comput. Phys. 23 (2006) 25–31. Tingchun Wang, Luming Zhang, Fangqi Chen, Conservative schemes for the symmetric regularized long wave equations, Appl. Math. Comput. 190 (2007) 1063–1080. Yan Bai, Lu-ming Zhang, A conservative finite difference scheme for symmetric regularized long wave equations, J. Appl. Math. Mech. 30 (2007) 248– 255. Youcai Xu, Bing Hu, Xiaoping Xie, Jinsong Hu, Mixed finite element analysis for dissipative SRLW equations with damping term, Appl. Math. Comput. 218 (2012) 4788–4797. S. Li, L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein–Gordon equation, SIAM J. Numer. Anal. 32 (1995) 1839–1875. Fei Zhang, Vı´ctor M. Pérez-Garcı´a, Luis Vázquez, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. Math. Comput. 71 (1995) 165–177.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A decoupled and conservative difference scheme with fourth-order accuracy for the Symmetric Regularized Long Wave equations q Tao Nie Section of Basic Courses, Nanjing College of Chemical Technology, Nanjing 210048, China
a r t i c l e
i n f o
a b s t r a c t
Keywords: Symmetric Regularized Long Wave equation Conservation Decoupled scheme High-order convergence
In this paper, a decoupled finite difference scheme with fourth-order accuracy is proposed to solve the Symmetric Regularized Long Wave equation. The scheme is proved to conserve the total energy in the discrete level. Without any restrictions on the grid ratios, the convergence of the difference scheme is proved by utilizing the energy method to be of forth-order in space and second-order in time. A numerical example is given to support the theoretical analysis. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction A weakly nonlinear analysis of the cold-electron plasma equations appropriate for a strongly magnetized non-relativistic electron beam yields the (1 + 1)-dimensional Symmetric Regularized-Long-Wave (SRLW) equation, expressed as a first-order system
ut uxxt þ qx þ uux ¼ 0; qt þ ux ¼ 0; x 2 R:
x 2 R;
ð1:1Þ ð1:2Þ
This system has been shown to describe weakly nonlinear (1 + 1)-dimensional ion-acoustic and space-charge waves, where q and u are the dimensionless electron charge density and the fluid velocity, respectively, (see [1] and references therein). The SRLW equation is explicitly symmetry in the x and t derivatives and is very similar to the Regularized-Long-Wave equation (RLW) [2] which describes shallow water waves and plasma drift waves, the RLW equation is
ut uxxt þ ux þ uux ¼ 0;
x 2 R:
Many finite difference schemes have been presented for the RLW equations [3–12]. For the numerical study of the SRLW equation, there also are many results. In [13], Guo studied the numerical solutions of SRLW equations by the spectral method. In [14], Zheng et al. presented a Fourier pseudo-spectral method with a restraint operator for the SRLW equations, and proved the stability and optimum error estimates. In [15], Ren considered Chebyshev pseudo-spectral method for SRLW equations, he constructed the semi-discrete and fully discrete chebyshev pseudo-spectral schemes and established the corresponding errors estimates. In [16], Shang and Guo analyzed a Chebyshev pseudo-spectral scheme for multi-dimensional generalized SRLW equations. In [17], an Euler mid-point scheme in time and a Fourier pseudo-spectral method in space were
q
This work is supported by the National Natural Science Foundation of China, NO. 11201239. E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.076
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
used to the multi-symplectic formulation of the SRLW equation. It made a multi-symplectic Fourier pseudo-spectral scheme, and several discrete conservation laws of this scheme were proved. In [18,19], some conservative difference schemes with second-order accuracy were proposed and analyzed. In [20], mixed finite element method was used and analyzed to solve the dissipative SRLW equations with damping term. In this paper we consider the SRLW equations (1.1) and (1.2) with the following initial-boundary conditions
qðx; 0Þ ¼ q0 ðxÞ; x 2 R; qðx; tÞ ! 0; as jxj ! 1; t > 0;
uðx; 0Þ ¼ u0 ðxÞ;
ð1:3Þ
uðx; tÞ ! 0;
ð1:4Þ
where u0 ðxÞ and q0 ðxÞ are two given smooth functions which tend to zero rapidly as jxj ! 1. The initial-boundary value problem (1.1)–(1.4) possesses the following conservative quantity[1]:
EðtÞ ¼
1 1 ðjjujj2L2 þ jjux jj2L2 þ jjqjj2L2 Þ ¼ ðjju0 jj2L2 þ jjðu0 Þx jj2L2 þ jjq0 jj2L2 Þ ¼ Eð0Þ; 2 2
ð1:5Þ
where EðtÞ is the called energy. This paper aims to construct a new difference scheme which has the following three advantages: 1. The new scheme is linearized and decoupled in implementation. It is obvious that, the SRLW equation is a coupled system, and almost all of the proposed schemes in the literature are coupled and nonlinear in implementation, and then too much CPU time should be used. Our scheme is decoupled and linearized in practical computation, so it is expected to be more efficient. 2. The new scheme preserve the total energy in the discrete level. In [21], Li and Vu-Quoc said, ‘‘in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation.’’ Zhang et al. pointed out in [22] that the nonconservative schemes may easily show nonlinear blow-up. Thus, a scheme preserving the conservative laws of the initial problem is very important. 3. The new scheme is, without any restrictions on the grid ratios, convergent at forth-order in space and second-order in time, respectively. The remainder of this paper is arranged as follows. In Second 2, a decoupled difference scheme is proposed and analyzed. The scheme is proved to preserve the discrete energy and be convergent at forth-order in space and second-order in time. In Section 3, numerical results are provided to verify the theoretical analysis. 2. A decoupled conservative scheme In this section, we propose a decoupled, linearized, conservative finite difference scheme for the initial-boundary value problem (1.1)–(1.4) and give numerical analysis. 2.1. Finite difference scheme and its conservative law We take a finite interval ½xl ; xr ½0; T as the computing domain, where xl and xr are large enough such that the following initial-boundary value problem
ut uxxt þ qx þ uux ¼ 0; ðx; tÞ 2 ðxl ; xr Þ ð0; T; qt þ ux ¼ 0; ðx; tÞ 2 ðxl ; xr Þ ð0; T; uðx; 0Þ ¼ u0 ðxÞ;
qðx; 0Þ ¼ q0 ðxÞ; x 2 ½xl ; xr ; qðxl ; tÞ ¼ qðxr ; tÞ ¼ 0; t 2 ð0; T
uðxl ; tÞ ¼ uðxr ; tÞ ¼ 0;
ð2:1Þ ð2:2Þ ð2:3Þ ð2:4Þ
is consistent with the problem (1.1)–(1.4). Before giving the finite difference scheme, some notations are firstly introduced. For a positive integer N, let time-step s ¼ ft n ¼ nsjn ¼ 0; 1; . . . ; Ng. Given a tems ¼ T=N; tn ¼ ns; n ¼ 0; 1; 2; . . . ; N, denote Xs ¼ ftn ¼ nsjn ¼ 1; 2; . . . ; N 1g and X poral discrete function fun jt n 2 Xs g, we denote ðun Þ^t ¼ ðunþ1 un1 Þ=2s. For a positive integer J, let space-step h ¼ ðxr xl Þ=J, xj ¼ jh; j ¼ 1; 0; 1; 2; ; J; J þ 1. Denote the grid Xh ¼ fxj ¼ xl þ jhjj ¼ 1; 2; ; J 1g and the extended discrete grid XEh ¼ fxj ¼ jhjj ¼ 1; 0; 1; 2; 3; . . . ; J; J þ 1g. Given a grid function u ¼ fuj jxj 2 Xh g, denote
ðuj Þx ¼ ðujþ1 uj Þ=h;
ðuj Þx ¼ ðuj uj1 Þ=h;
ðuj Þ^x ¼ ðujþ1 uj1 Þ=2h; Let V h ¼ fuju ¼ fuj jxj 2 X product of them as
E hg
ðuj Þ€x ¼ ðujþ2 uj2 Þ=4h: and u1 ¼ u0 ¼ uJ ¼ uJþ1 ¼ 0g. For any grid functions u; v 2 V h , denote the discrete inner
T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
hu; v i ¼ h
J1 X uj v j :
9463
ð2:5Þ
j¼1
The discrete L2 norm jjv jj, the discrete semi-H1 norms jjv x jj; jjv ^x jj; jjv €x jj and the discrete L1 norm (or discrete maximum norm) jjv jj1 of v 2 V h are defined, respectively, as follows
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X jjv x jj ¼ th jðv j Þx j2 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffi jjv jj2 ¼ hv ; v i;
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X jjv €x jj ¼ th jðv j Þ€x j2 ;
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X jjv ^x jj ¼ th jðv j Þ^x j2 ;
j¼0
j¼1
jjv jj1 ¼ max jv j j: 16j6J1
j¼1
ð2:6Þ
For simplicity, we denote unj uðxj ; tn Þ; U nj uðxj ; tn Þ; qnj qðxj ; t n Þ; /nj qðxj ; t n Þ, respectively. Let C denote a positive constant independent of discretization parameters, but may have different values at different occurrences. Based on the introduced notations, we give the following decoupled finite difference scheme.
2 1 n nþ1 4 ðU nj Þ^t þ ½U nj ðU nþ1 þ U n1 Þ^x þ ððU nþ1 þ U n1 ÞU nj Þ^x ½U ðU þ U n1 Þ€x þ ððU nþ1 þ U n1 ÞU nj Þ€x ðU nj Þxx^t j j j j j j j 9 18 j j 3 1 n 4 n 1 n þ ðU j Þ^x^x^t þ ð/j Þ^x ð/j Þ€x ¼ 0; ðxj ; t n Þ 2 Xh Xs ; 3 3 3 4 n 1 n n ð/j Þ^t þ ðU j Þ^x ðU j Þ€x ¼ 0; ðxj ; t n Þ 2 Xh Xs ; 3 3 4 1 1 1 @ 2 u0 ðU j Þxx ðU j Þ^x^x U 1j ¼ ðxj Þ u0 ðxj Þ þ swðxj Þ; xj 2 Xh ; 3 3 @x2 1 /j ¼ q0 ðxj Þ swðxj Þ; xj 2 Xh ; /0j
¼ q0 ðxj Þ;
/n 2 V h ;
U 0j
¼ u0 ðxj Þ;
Un 2 V h ;
E h;
xj 2 X
t n 2 Xs ;
ð2:7Þ ð2:8Þ ð2:9Þ ð2:10Þ ð2:11Þ ð2:12Þ
where
wðxÞ ¼
@ q0 @u0 ðxÞ þ u0 ðxÞ; @x @x
wðxÞ ¼
@u0 ðxÞ: @x
ð2:13Þ
Obviously, this scheme is decoupled in practical computation. Firstly, ðU 0 ; /0 Þ is obtained directly from (2.11), and ðU 1 ; /1 Þ is computed by using (2.9) and (2.10), then ðU 2 ; /2 Þ is computed by using (2.7) and (2.8). If ðU n1 ; /n1 Þ and ðU n ; /n Þ are known, then U nþ1 and /nþ1 can be computed by (2.7) and (2.8) respectively at the same time. Corresponding to the invariant (1.5) possessed by the continuous problem (1.1)–(1.4), the difference scheme (2.7)–(2.12) satisfy the following discrete conservative law: Lemma 2.1. suppose that u0 2 H1 ðRÞ; q0 2 L2 ðRÞ, then the scheme (2.7)–(2.12) is conservative in the sense
En E0 ;
n ¼ 0; 1; 2; . . . ; N 1;
ð2:14Þ
where
En ¼
1 4 1 4 1 2s jj/nþ1 jj2 þ jjU nþ1 jj2 þ jjU nþ1 jj2 jjU ^nþ1 jj2 þ jj/n jj2 þ jjU n jj2 þ jjU nx jj2 jjU ^nx jj2 þ ðh/n^x ; U nþ1 i x x 4 3 3 3 3 3
s
þ h/nþ1 ; U ^nx iÞ ðh/€nx ; U nþ1 i þ h/nþ1 ; U n€x iÞ: 6
ð2:15Þ
Proof. Computing the inner product of (2.7) with 12 ðU nþ1 þ U n1 Þ yields
1 n 2 2 n 2 1 n 2 2 n nþ1 1 1 jjU jj^t þ jjU x jj^t jjU ^x jj^t þ h/^x ; U þ U n1 i h/n€x ; U nþ1 þ U n1 i þ hnðU n1 ; U n ; U nþ1 Þ; U nþ1 þ U n1 i ¼ 0: 2 3 6 3 6 2 ð2:16Þ Direct computation gives
hnðU n1 ; U n ; U nþ1 Þ; U nþ1 þ U n1 i ¼ 0; where
nðU n1 ; U nj ; U nþ1 Þ¼ j j
2 n nþ1 1 n nþ1 ½U ðU þ U n1 Þ^x þ ððU nþ1 þ U n1 ÞU nj Þ^x ½U ðU þ U n1 Þ€x þ ððU nþ1 þ U n1 ÞU nj Þ€x : j j j j j j 9 j j 18 j j
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This together with (2.16) gives
1 n 2 2 n 2 1 n 2 2 n nþ1 1 jjU jj^t þ jjU x jj^t jjU ^x jj^t þ h/^x ; U þ U n1 i h/n€x ; U nþ1 þ U n1 i ¼ 0: 2 3 6 3 6
ð2:17Þ
Computing the inner product of (2.8) with 12 ð/nþ1 þ /n1 Þ, we obtain
1 n 2 2 nþ1 1 jj/ jj^t þ h/ þ /n1 ; U ^nx i h/nþ1 þ /n1 ; U n€x i ¼ 0: 2 3 6
ð2:18Þ
Adding (2.18) to (2.17) and noticing
h/n^x ; U nþ1 þ U n1 i þ h/nþ1 þ /n1 ; U ^nx i ¼ h/n^x ; U nþ1 i þ h/n^x ; U n1 i þ h/nþ1 ; U n^x i þ h/n1 ; U n^x i ¼ h/^nx ; U nþ1 i i þ h/nþ1 ; U n^x i h/^n1 ; U n i; h/n ; U n1 ^x x
ð2:19Þ
h/n€x ; U nþ1 þ U n1 i þ h/nþ1 þ /n1 ; U €nx i ¼ h/n€x ; U nþ1 i þ h/n€x ; U n1 i þ h/nþ1 ; U n€x i þ h/n1 ; U n€x i ¼ h/€nx ; U nþ1 i h/n ; U n1 i þ h/nþ1 ; U n€x i h/€n1 ; U n i; €x x
ð2:20Þ
yield
1 4 1 2s s ðjj/nþ1 jj2 þ jjU nþ1 jj2 þ jjU nx jj2 jjU n^x jj2 Þ þ ðh/^nx ; U nþ1 i þ h/nþ1 ; U ^nx iÞ ðh/€nx ; U nþ1 i þ h/nþ1 ; U €nx iÞ 4 3 3 3 6 1 4 1 2s s ; U n i þ h/n ; U ^n1 iÞ ðh/n1 ; U n i þ h/n ; U €n1 iÞ: ¼ ðjj/n1 jj2 þ jjU n1 jj2 þ jjU nx jj2 jjU ^nx jj2 Þ þ ðh/n1 ^x €x x x 4 3 3 3 6 This gives (2.14).
ð2:21Þ
h
2.2. a priori estimate and convergence of the difference solution On the solution of the proposed scheme, there is a priori estimate as follows: Lemma 2.2. Suppose that u0 2 H1 ; q0 2 L2 , then the following inequalities
jj/n jj 6 C;
jjU n jj 6 C;
jjU nx jj 6 C;
jjU n jj1 6 C
hold. Proof. Cauchy–Schwartz inequality and tedious calculation give
jj/n€x jj2 6 jj/n^x jj2 6 jj/nx jj2
ð2:22Þ
1 ¼ jh/ 6 ðjj/n jj2 þ jjU nþ1 jj2 Þ; x 2 1 ij 6 ðjj/n jj2 þ jjU nþ1 jj2 Þ; jh/n€x ; U nþ1 ij ¼ jh/n ; U €nþ1 x x 2 1 jh/nþ1 ; U n^x ij 6 ðjj/nþ1 jj2 þ jjU nx jj2 Þ; 2 1 n nþ1 jh/ ; U €x ij 6 ðjj/nþ1 jj2 þ jjU nx jj2 Þ; 2 jh/n^x ; U nþ1 ij
n
; U ^nþ1 ij x
ð2:23Þ ð2:24Þ ð2:25Þ ð2:26Þ
this together with (2.14) gives
1 s 1 ðjj/nþ1 jj2 þ jjU nþ1 jj2 þ jj/n jj2 þ jjU nx jj2 Þ þ ðjjU nþ1 jj2 þ jjU n jj2 Þ 6 C: x 4 3 4 If
s < 34, then from (2.27) we obtain jj/n jj 6 C;
jjU n jj 6 C;
jjU nx jj 6 C:
This together with the discrete Sobolev equality gives
jjU n jj1 6 C:
ð2:27Þ
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Remark 2.1. Lemma 2.2 implies that the proposed scheme is unconditionally stable. The truncation error of the proposed scheme is defined as
2 1 n nþ1 4 1 ðunj Þ^t þ ½unj ðunþ1 þ un1 Þ^x þ ððunþ1 þ un1 Þunj Þ^x ½u ðu þ un1 Þ€x þ ððunþ1 þ un1 Þunj Þ€x ðunj Þxx^t þ ðunj Þ^x^x^t j j j j j j j 9 18 j j 3 3 4 n 1 n n þ ð/j Þ^x ð/j Þ€x ¼ rj ; ðxj ; t n Þ 2 Xh Xs ; 3 3 4 n 1 n ðqj Þ^t þ ðuj Þ^x ðunj Þ€x ¼ rnj ; ðxj ; tn Þ 2 Xh Xs ; 3 3 4 1 1 1 @ 2 u0 ðuj Þxx ðuj Þ^x^x u1j ¼ ðxj Þ u0 ðxj Þ þ swðxj Þ þ r0j ; xj 2 Xh ; 3 3 @x2 q1j ¼ q0 ðxj Þ swðxj Þ þ r0j ; xj 2 Xh ; 0 j
E h;
u0j
q ¼ q0 ðxj Þ; ¼ u0 ðxj Þ; xj 2 X qn 2 V h ; un 2 V h ; tn 2 Xs :
ð2:28Þ ð2:29Þ ð2:30Þ ð2:31Þ ð2:32Þ ð2:33Þ
Using Taylor’s expansion yields. Lemma 2.3. Suppose that u0 2 H1 ; q0 2 L2 , uðx; tÞ 2 C 6;3 ; qðx; tÞ 2 C 6;3 , then the truncation error of the proposed scheme satisfies 4
jrnj j þ jrnj j ¼ Oðs2 þ h Þ; as
s ! 0; h ! 0.
Theorem 2.1. Suppose that uðx; tÞ 2 C 6;3 ; qðx; tÞ 2 C 6;3 , then the solution of the scheme (2.7)–(2.12) converges to the solution of 4 the problem (2.1)–(2.4) at the order Oðs2 þ h Þ in the discrete L1 norm for U n , and in the discrete L2 norm for /n . Proof. Denote
enj ¼ unj U nj ;
gnj ¼ /nj qnj :
ð2:34Þ
Subtracting (2.7)–(2.12) from (2.28)–(2.33) yields the following error function:
4 1 4 1 ðenj Þ^t þ nðun1 ; unj ; unþ1 Þ nðU n1 ; U nj ; U nþ1 Þ ðenj Þxx^t þ ðenj Þ^x^x^t þ ðgnj Þ^x ðgnj Þ€x ¼ r nj ; j j j j 3 3 3 3 4 n 1 n n n ðgj Þ^t þ ðej Þ^x ðej Þ€x ¼ rj ; ðxj ; t n Þ 2 Xh Xs ; 3 3 4 1 1 1 ðe Þ ðe Þ e1j ¼ r 0j ; xj 2 Xh ; 3 j xx 3 j ^x^x g1j ¼ r0j ; xj 2 Xh ; 0
g ¼ 0; n
e 2 Vh;
e0j
¼ 0;
ðxj ; t n Þ 2 Xh Xs ;
E h;
xj 2 X
n
ð2:35Þ ð2:36Þ ð2:37Þ ð2:38Þ ð2:39Þ
n
g 2 V h ; t 2 Xs :
ð2:40Þ 1
Computing the discrete product of (2.37) with e gives
4 1 2 1 1 2 jje jj jje^x jj þ jje1 jj2 ¼ hr0 ; e1 i: 3 x 3
ð2:41Þ
This together with Lemma 2.3 and Cauchy-Schwartz inequality gives 4
jje1x jj2 þ jje1 jj2 6 Oðh þ s2 Þ:
ð2:42Þ 1
Computing the discrete product of (2.38) with g gives
jjg1 jj2 ¼ hr0 ; g1 i:
ð2:43Þ
This together with Lemma 2.3 and Cauchy–Schwartz inequality gives 4
jjg1 jj2 6 Oðh þ s2 Þ: Computing the inner product of (2.35) with e
ð2:44Þ nþ1
n1
þe
yields
4 1 4 1 jjen jj2 þ jjenx jj2 jjen^x jj2 þ hgn1 ; enþ1 þ en1 i hgn1 ; enþ1 þ en1 i þ hnðun1 ; un ; unþ1 Þ 3 3 3 ^x 3 €x ^t nðU n1 ; U n ; U nþ1 Þ; enþ1 þ en1 i ¼ ðr n ; enþ1 þ en1 Þ:
ð2:45Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
This together with Lemma 2.2 and Cauchy–Schwartz inequality gives
4 1 4 1 jjen jj2 þ jjenx jj2 jjen^x jj2 þ hgn1 ; enþ1 þ en1 i hgn1 ; enþ1 þ en1 i ^x 3 3 3 3 €x ^t 6 jjr n jj2 þ C jjen1 jj2 þ jjen jj2 þ jjenþ1 jj2 þ jjen1 jj2 þ jjenx jj2 þ jjenþ1 jj2 : x x
ð2:46Þ
Computing the inner product of (2.36) with gnþ1 þ gn1 yields
4 1 jjgn jj^2t hg^nx ; gnþ1 þ gn1 i þ hg€nx ; gnþ1 þ gn1 i ¼ hrn ; gnþ1 þ gn1 i: 3 3
ð2:47Þ
This together with Cauchy–Schwartz inequality gives
4 1 jjgn jj^2t hg^nx ; gnþ1 þ gn1 i þ hg€nx ; gnþ1 þ gn1 i 6 jjrn jj2 þ jjgnþ1 jj2 þ jjgn1 jj2 : 3 3
ð2:48Þ
Adding 2.48 to 2.46 yields
ðjjen jj2 þ jjenx jj2 þ jjgn jj2 Þ^t 6 jjr n jj2 þ jjrn jj2 þ C jjen1 jj2 þ jjen jj2 þ jjenþ1 jj2 þ jjen1 jj2 þ jjenx jj2 þ jjenþ1 jj2 : x x
ð2:49Þ
This together with Lemma 2.3 and the discrete Gronwall’s inequality gives 4
jjen jj 6 Oðh þ s2 Þ;
4
jjgn jj 6 Oðh þ s2 Þ;
4
jjenx jj 6 Oðh þ s2 Þ;
ð2:50Þ
when s is small enough. 4 It follows from the discrete Sobolev inequality that jjen jj1 6 Oðh þ s2 Þ. h Remark 2.2. All results above in this paper are correct for periodic boundary-initial value problem of the SRLW equation, and the initial functions should be periodic ones correspondingly.
3. Numerical experiments Denote
k n 4 2k ðu þ unj2 Þ; Bnj ¼ 2 ðunj þ unj1 Þ; 36 j 9 3h 5 4 2k 1 k n ðuj þ unjþ2 Þ; C nj ¼ 1 þ 2 ; Dnj ¼ 2 þ ðunj þ unjþ1 Þ; Enj ¼ 2 9 36 2h 3h 12h e n ¼ 1 k ðun þ un Þ; B e n ¼ 4 þ 2k ðun þ un Þ; A j j2 j j1 2 2 36 j 9 j 12h 3h en ¼ 1 þ 5 ; D e n ¼ 4 2k ðun þ un Þ; E e n ¼ 1 þ k ðun þ un Þ; C j j j1 j j2 2 2 2 9 j 36 j 2h 3h 12h 4k k e n un1 þ B e n un1 þ D e n un1 þ C e n un1 þ e E nj un1 ð/n /nj1 Þ þ ð/njþ2 /nj2 Þ; F nj ¼ A j j2 j j1 j j j jþ1 jþ2 3 jþ1 6
Anj ¼
1
12h
2
þ
ð3:1Þ
where k ¼ hs, then the proposed scheme can be written as the following form
Hunþ1 ¼ F; /nþ1 j
ðxj ; tn Þ 2 Xh Xs ; 4k k ¼ /n1 ðunjþ1 unj1 Þ þ ðunjþ2 unj2 Þ; j 3 6
ð3:2Þ ðxj ; tn Þ 2 Xh Xs ;
where
0
C1
D1
E1
B B2 B B B A3 B B 0 B H¼B B B B B 0 B B @ 0
C2
D2
B3
C3
A4
B4 .. .
C4 .. .
...
0
...
0
...
0
0
0
0
E2
0
0
...
D3
E3
0
...
D4 .. .
E4 .. .
... .. .
AJ3
BJ3
C J3
DJ3
0
AJ2
BJ2
C J2
0 C C C 0 C C 0 C C C C C C EJ3 C C C DJ2 A
0
0
AJ1
BJ1
C J1
...
0
1
0
ð3:3Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468 Table 1 Errors of the numerical solution computed by the proposed scheme at t ¼ 2. h
s
e
order1
g
order2
0.4 0.2 0.1
0.4 0.1 0.025
1.5076e2 1.0768e3 6.7112e5
– 3.81 4.00
2.5328e2 1.6322e3 1.0187e4
– 3.95 4.00
Table 2 Errors of the numerical solution computed by the proposed scheme with extrapolation technique at t ¼ 8 h
s
Fe
order3
Fg
order4
0.8 0.4 0.2 0.1
0.8 0.4 0.2 0.1
4.0104e2 2.7777e3 1.8476e4 1.3051e5
– 3.86 3.90 3.82
3.6881e2 2.3139e3 1.5127e4 1.0189e5
– 3.99 3.94 3.89
Table 3 Discrete energy En computed by the proposed scheme with h ¼ 0:2; s ¼ 0:1 at various tn . tn
En
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
13.55949629317673 13.55949629317681 13.55949629317669 13.55949629317668 13.55949629317670 13.55949629317672 13.55949629317669 13.55949629317669 13.55949629317669 13.55949629317673
is a penta-diagonal matrix. The system of linear algebra Eqs. (3.2) can be solved by using Thomas algorithm, and (3.3) can be solved explicitly. In order to achieve higher order numerical accuracy, the Richardson extrapolation is also a practical method. It obtains high-order accurate resolutions by using certain combinations of difference solutions with various grid parameters. The main advantage of the global extrapolations is that they preserve the numerical stability of lower-order methods used initially. Using the extrapolation technique on the difference schemes, one can obtain higher order accuracy. Let unj ðh; sÞ denote h s 2n the numerical solution unj with mesh-sizes h and s at the point ðxj ; tn Þ, and u2n 2j ð2 ; 2Þ denote the numerical solution u2j with mesh-sizes 2h and 2s at the point ðxj ; tn Þ. Denote
4 2n h s 1 unj ðh; sÞ u2j ; 3 2 2 3
ðuFE Þnj ¼
ð3:4Þ
and
ðeFE Þnj ¼ U nj ðuFE Þnj :
ð3:5Þ
Simple but tedious computation gives that 4
ðeFE Þnj ¼ Oðh þ s4 Þ:
ð3:6Þ
Choosing
u0 ðxÞ ¼
pffiffiffi pffiffiffi ! 2 3 2 2 x ; sech 2 4
3 2
q0 ðxÞ ¼ sech2
pffiffiffi ! 2 x ; 4
ð3:7Þ
then the problem (1.1)–(1.4) has the following exact solution
uðx; tÞ ¼
! pffiffiffi pffiffiffi 3 2 1 2 2 x t ; sech 2 2 4
3 2
qðx; tÞ ¼ sech2
! pffiffiffi 1 2 x t : 2 4
In implementation, we chose the computational domain ½xl ; xr as ½40; 40 and the iterative tolerance as 108 .
ð3:8Þ
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T. Nie / Applied Mathematics and Computation 219 (2013) 9461–9468
Denote
eðh; sÞ ¼ jjU N ðh; sÞ uN ðh; sÞjj1 ; gðh; sÞ ¼ jj/N ðh; sÞ qN ðh; sÞjj1 ; h s Feðh; sÞ ¼ jj4=3U N 1=3U N ðh; sÞ uN ðh; sÞjj1 ; ; 2 2 h s 1=3/N ðh; sÞ qN ðh; sÞjj1 ; ; F gðh; sÞ ¼ jj4=3/N 2 2 h s h s = logð2Þ; order2 ¼ log gðh; sÞ=g ; = logð2Þ; order1 ¼ log eðh; sÞ=e ; 2 4 2 4 h s h s = logð2Þ; order4 ¼ log F gðh; sÞ=F g ; = logð2Þ; order3 ¼ log Feðh; sÞ=Fe ; 2 2 2 2 then we list the numerical results in the Table 1, Table 2 and Table 3. Table 1 shows that the convergence of the proposed scheme is forth-order in space and second-order in time, this verifies Theorem 2.1. Table 2 shows the high accuracy of the proposed extrapolation. Table 1 and Table 2 also verify the good stability of the proposed scheme. Table 3 shows that the proposed scheme conserves the discrete energy very well, this verifies Lemma 2.1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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