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Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation
Applied Mathematical Modelling xxx (2014) xxx–xxx
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation q Jinsong Hu a, Kelong Zheng b,⇑, Maobo Zheng c a
School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China School of Science, Southwest University of Science and Technology, Mianyang 621010, China c Chengdu Technological University, Chengdu 611730, China b
a r t i c l e
i n f o
Article history: Received 17 June 2013 Received in revised form 21 February 2014 Accepted 29 April 2014 Available online xxxx Keywords: SRLW equation Conservative difference scheme Richardson extrapolation Stability Convergence
a b s t r a c t Coupled with the Richardson extrapolation, a new conservative Crank–Nicolson finite dif4 ference scheme, which has the accuracy of Oðs2 þ h Þ without refined mesh for the symmetric regularized long wave equation is proposed. The corresponding conservative quantities are discussed, and the existence of numerical solution is proved by the Browder fixed point theorem. The convergence, unconditional stability and uniqueness of the scheme are also derived using the energy method. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Consider the following initial boundary value problem of the Symmetric Regularized Long Wave (SRLW) equation
ut þ qx þ uux uxxt ¼ 0;
ð1:1Þ
qt þ ux ¼ 0; x 2 ðxL ; xR Þ; t 2 ð0; T; uðx; 0Þ ¼ u0 ðxÞ; qðx; 0Þ ¼ q0 ðxÞ; x 2 ½xL ; xR ; uðxL ; tÞ ¼ uðxR ; tÞ ¼ 0; qðxL ; tÞ ¼ qðxR ; tÞ ¼ 0; t 2 ½0; T;
ð1:3Þ
ð1:2Þ ð1:4Þ
where u0 ðxÞ and q0 ðxÞ are two known smooth functions. SRLW equation has been shown to describe weakly nonlinear ð1 þ 1Þ-dimensional ion-acoustic and space-charge waves, where q and u are the dimensionless electron charge density and the fluid velocity, respectively [1]. Eliminating q from (1.1) and (1.2), we can get another form of SRLW equation,
1 utt uxx þ ðu2 Þxt uxxxt ¼ 0; 2
ð1:5Þ
which is explicitly symmetric about the x and t derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves [2–6]. The solitary wave solutions of Eqs. (1.1) and (1.2) are (v 2 > 1 is velocity), q This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department (No. 11ZB009), the Doctoral Program Research Fund of Southwest University of Science and Technology (No. 11zx7129) and the Applied Basic Research Project of Sichuan Province (No. 2013JY0096). ⇑ Corresponding author. Tel.: +86 13547116902. E-mail address: [email protected] (K. Zheng).
http://dx.doi.org/10.1016/j.apm.2014.04.062 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v2 1 ðx v tÞ; 2 v v2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðv 2 1Þ v2 1 21 qðx; tÞ ¼ sech ðx v tÞ; 2 2 v v2 uðx; tÞ ¼
3ðv 2 1Þ
2
sech
and physical boundary requires
uðx; tÞ ! 0;
qðx; tÞ ! 0; as jxj ! 1; t > 0:
ð1:6Þ
Hence, if xL 0 and xR 0, problem (1.1)–(1.4) is in accordance with the Cauchy problem of Eq. (1.1) and (1.2). It can be proved easily that SRLW equation has the following conserved laws,
Q 1 ðtÞ ¼
Z
xR
uðx; tÞdx ¼
xL xR
Q 2 ðtÞ ¼
Z
Z
xR
u0 ðxÞdx ¼ Q 1 ð0Þ;
ð1:7Þ
q0 ðxÞdx ¼ Q 2 ð0Þ;
ð1:8Þ
x
qðx; tÞdx ¼
xL
Z L xR xL
and
EðtÞ ¼ kuk2L2 þ kux k2L2 þ kqk2L2 ¼ Eð0Þ;
ð1:9Þ
where Q 1 ð0Þ; Q 2 ð0Þ and Eð0Þ are three positive constants which relate to the initial condition. SRLW equation (1.1) and (1.2) or (1.5) arises also in many other areas of mathematical physics [7]. Numerical investigation indicates that interactions of solitary waves are inelastic[8], thus the solitary wave of SRLW equation is not a solution. Research on the well-posedness for its solution and numerical methods has aroused more and more interest. In [9], Guo studied the existence, uniqueness and regularity of numerical solutions for the periodic initial value problem of generalized SRLWE by spectral method. In Ref. [10], Zheng et al. presented a Fourier pseudospectral method with a restraint operator for SRLW equation, and proved the stability and the optimum error estimates. In Ref. [11], Shang and Guo analyzed a chebyshev pseudospectral scheme for multi-dimensional generalized SRLW equations and other results[12–14]. In Ref. [15], an Euler midpoint scheme in time and a Fourier pseudospectral method in space are used to the multisymplectic formulation of SRLW equation, in which a multisymplectic Fourier pseudospectral scheme is proposed, and several discrete conservation laws of this scheme are proved. Wang and Zhang[16] studied SRLW equation by some conservative finite difference methods. For the dissipative SRLW equations with damping term, a Crank–Nicolson scheme is proposed by Hu et al. [17] and a mixed finite element method is presented by Xu et al. [18], respectively. In [19], Li and Vu-Quoc pointed out that ‘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’. Thus, the main purpose of this paper is to study the conservative schemes[21] for SRLW equation. Coupled with the Richardson extrapolation, a two-level nonlinear finite 4 difference scheme which has the accuracy of Oðs2 þ h Þ without refined mesh is proposed. The scheme well approximates three conserved quantities (1.7), (1.8) and (1.9). Moreover, the existence and the uniqueness of numerical solutions are discussed. Priori estimate, convergence and unconditional stability of the scheme are also proved. The remainder of this paper is arranged as follows. In Section 2, a nonlinear conservative difference scheme is proposed. In Section 3, we prove the existence of scheme by the Browder fixed point theorem. Priori estimate, convergence and stability are proved in Section 4, and numerical experiments to verify the theoretical analysis are reported in Section 5. 2. Nonlinear finite difference scheme L Let h ¼ xR x be the step size for the spatial grid such J that xj ¼ xL þ jh; ðj ¼ 1; 0; 1; . . . ; J; J þ 1Þ. Let s be the step size for the temporal direction, t n ¼ ns; ðn ¼ 0; 1; 2; . . . ; NÞ; N ¼ Ts . Denote unj uðxj ; tn Þ; qnj qðxj ; tn Þ and
Z 0h ¼ u ¼ ðuj Þju1 ¼ u0 ¼ uJ ¼ uJþ1 ¼ 0;
j ¼ 1; 0; 1; 2; . . . ; J; J þ 1 :
Define
ðunj Þx ¼ ðunj Þ€x ¼
unjþ1 unj
ðunj Þx ¼
;
h unjþ2 unj2
;
4h J1 X n n hu ; v i ¼ h unj v nj ;
unj unj1
ðunj Þt ¼
j¼1
h unþ1 j
unj
s
ðunj Þ^x ¼
; ;
kun k2 ¼ hun ; un i;
nþ12
uj
unjþ1 unj1
¼
2h þ unj
unþ1 j
2
; ;
kun k1 ¼ max junj j: 16j6J1
In this paper, C denotes a general positive constant which may have different values in different occurrences. Lemma 2.1. [20] For a mesh function u 2 Z 0h , by Cauchy–Schwarz inequality, we have
ku€x k2 6 ku^x k2 6 kux k2 : Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
The following difference scheme for problem (1.1)–(1.4) is considered,
n nþ1 2 4 n 1 n 4 nþ12 1 nþ12 4 nþ12 nþ12 1 nþ1 nþ1 uj unj uj þ uj þ q q þ uj þ ðuj 2 Þ uj 2 uj 2 j j ^x^xt ^x €x ^x €x t xxt 3 3 3 3 9 ^x 9 2 nþ12 þ uj ¼ 0; j ¼ 1; 2; . . . ; J 1; n ¼ 1; 2; . . . ; N 1;
4 nþ12 1 nþ12 uj u ¼ 0; ^x €x t 3 3 j u0j ¼ u0 ðxj Þ; q0j ¼ q0 ðxj Þ; 0 6 j 6 J;
qnj þ
n
u 2
ð2:1Þ
€x
Z 0h ;
n
q 2
Z 0h ;
ð2:2Þ ð2:3Þ
n ¼ 0; 1; 2; . . . ; N:
ð2:4Þ
From boundary condition (1.4) and physical boundary (1.6), discrete boundary condition (2.4) is reasonable. Based on scheme (2.1)–(2.4), the discrete versions of (1.7)–(1.9) are obtained as follows. Theorem 2.1. Scheme (2.1)–(2.4) is conservative in the senses
Q n1 ¼ h
J1 X unj ¼ Q n1 ¼ ¼ Q 01 ; 1
ð2:5Þ
j¼1
Q n2 ¼ h
J1 X
qnj ¼ Q n1 ¼ ¼ Q 02 ; 2
ð2:6Þ
j¼1
4 1 En ¼ kun k2 þ kunx k2 ku^nx k2 þ kqn k2 ¼ En1 ¼ ¼ E0 ; 3 3
ð2:7Þ
for n ¼ 1; 2; . . . ; N. Proof. Multiplying (2.1) and (2.2) with h respectively, then summing up for j from 1 to J 1, we have J1 X h ðunj Þt ¼ 0;
h
J1 X ðqnj Þt ¼ 0:
j¼1
j¼1
Q n1
By the definition of and Q n2 , Eqs. (2.5) and (2.6) are obtained from formula above. 1 Taking the inner product of (2.1) with 2unþ2 , according to boundary condition (2.4), we get
D E D E 4 1 8 D nþ12 nþ1 E 2 D nþ12 nþ1 E 1 1 1 1 kun k2t þ kunx k2t ku^nx k2t þ q^x ; u 2 q€x ; u 2 þ 2 uðunþ2 Þ; unþ2 2 jðunþ2 Þ; unþ2 ¼ 0; 3 3 3 3
where u
nþ1 uj 2
¼ 49
nþ1 uj 2
nþ1 nþ1 2 uj 2 þ uj 2 ^ x
and
^ x
j
nþ1 uj 2
¼ 19
nþ1 uj 2
nþ1 nþ1 2 uj 2 þ uj 2 € x
ð2:8Þ
.
€ x
Since
D
nþ1
1
E
D
1
nþ12
q^x 2 ; unþ2 ¼ qnþ2 ; u^x
E
;
D
nþ1
1
E
D
1
nþ12
q€x 2 ; unþ2 ¼ qnþ2 ; u€x
E
;
ð2:9Þ
J1 J1 J1 D 1 E 4 X nþ1 2 nþ1 4 X 2 nþ1 4 X nþ12 2 1 nþ1 nþ1 nþ1 nþ1 uj 2 ¼ h u unþ2 ; unþ2 ¼ h uj 2 uj 2 þ uj 2 uj 2 uj 2 þ h uj uj 2 ^x ^ x 9 j¼1 9 9 ^x ^x j¼1 j¼1 ¼
J1 J1 4 X nþ12 2 nþ12 4 X nþ12 2 nþ12 h uj uj h u uj ¼0 ^x ^x 9 j¼1 9 j¼1 j
ð2:10Þ
and
J1 J1 J1 D 1 E 1 X nþ1 2 nþ1 1 X 2 nþ1 1 X nþ12 2 1 nþ1 nþ1 nþ1 nþ1 uj 2 ¼ h j unþ2 ; unþ2 ¼ h uj 2 uj 2 þ uj 2 uj 2 uj 2 þ h uj uj 2 €x € x 9 j¼1 9 9 €x €x j¼1 j¼1 ¼
J1 J1 1 X nþ12 2 nþ12 1 X nþ12 2 nþ12 h uj uj h u uj ¼ 0; €x €x 9 j¼1 9 j¼1 j
ð2:11Þ
from (2.8)–(2.11), we have
4 1 8 D nþ1 nþ12 E 2 D nþ1 nþ12 E kun k2t þ kunx k2t ku^nx k2t q 2 ; u^x þ q 2 ; u€x ¼ 0: 3 3 3 3
ð2:12Þ
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 1
Then taking the inner product of (2.2) with 2qnþ2 , we have
kqn k2t þ
8 D nþ12 nþ1 E 2 D nþ12 nþ1 E u ;q 2 u ; q 2 ¼ 0: 3 ^x 3 €x
ð2:13Þ
Adding 2.12,2.13, we have
4 1 kunþ1 k2 kun k2 þ kunþ1 k2 kunx k2 kunþ1 k2 ku^nx k2 þ kqnþ1 k2 kqn k2 ¼ 0: ^x x 3 3 By the definition of En , (2.7) is obtained from formula above. h
3. Existence To prove the existence of solution for scheme (2.1)–(2.4), the following Browder fixed point theorem should be introduced. For the proof, see[22]. Lemma 3.1. Let H be a finite dimensional inner product space. Suppose that g : H ! H is continuous and there exists an a > 0 such that hgðxÞ; Then there exists x 2 H such that gðx Þ ¼ 0 and jjx jj 6 a. n xi > 0 for all x 2 H with jjxjj ¼ a. o 0 Let Z D ¼ v ¼ ðv 1 ; v 2 Þ ¼ ðv 1;j ; v 2;j Þjv 1 ; v 2 2 Z h and define
hv ; v 0 i ¼ hðv 1 ; v 2 Þ;
ðv 01 ; v 02 Þi ¼ hv 1 ; v 01 i þ hv 2 ; v 02 i;
kv k2 ¼ kv 1 k2 þ kv 2 k2 :
Theorem 3.1. There exists ðun ; qn Þ 2 Z 0h satisfying difference scheme (2.1)–(2.4). Proof. Consider the mathematical induction for the proof. Obviously, with the condition (2.3), the solution exists for n ¼ 0. Suppose that for n 6 N 1, ðu0 ; q0 Þ; ðu1 ; q1 Þ; . . . ; ðun ; qn Þ 2 Z D satisfy (2.1)–(2.4), then we prove that there exists ðunþ1 ; qnþ1 Þ satisfying (2.1)–(2.4). Define an operator g ¼ ðg 1 ; g 2 Þ as follows,
8 8 2 2 4 1 v 1xx þ unxx v 1^x^x þ un^x^x þ sv 2^x sv 2€x þ suðv 1 Þ sjðv 1 Þ; 3 3 3 3 3 3 4 1 n g 2 ðv Þ ¼ 2v 2 2q þ sv 1^x sv 1€x ; 8v ¼ ðv 1 ; v 2 Þ 2 Z D : 3 3
g 1 ðv Þ ¼ 2v 1 2un
ð3:1Þ ð3:2Þ
v ¼ ðv 1 ; v 2 Þ, similar to (2.9)–(2.11), we get hv 2€x ; v 1 i ¼ hv 2 ; v 1€x i; huðv 1 Þ; v 1 i ¼ 0; hjðv 1 Þ; v 1 i ¼ 0:
Taking the inner product of (3.1) and (3.2) with
hv 2^x ; v 1 i ¼ hv 2 ; v 1^x i;
From Lemma 2.1 and Cauchy–Schwarz inequality, we get
hgðv Þ; v i ¼ hg 1 ðv Þ; v 1 i þ hg 2 ðv Þ; v 2 i 8 8 2 2 ¼ 2kv 1 k2 2hun ; v 1 i þ kv 1x k2 hunx ; v 1x i kv 1^x k2 þ un^x ; v 1^x þ 2kv 2 k2 2hqn ; v 2 i 2kv 1 k2 3 3 3 3 8 2 4 n 2 1 n 2 kuk2 þ kv 1 k2 þ kv 1x k2 kux k þ kv 1x k2 kv 1^x k2 ku^x k þ kv 1^x k2 þ 2kv 2 k2 3 3 3 3 1 4 n 2 1 n 2 2 2 2 2 n 2 n 2 kq k þ kv 2 k P kv 1 k þ kv 2 k ku k þ kv 1x k kux k ku^x k kqn k2 P kv k2 3 3 3 4 n 2 1 n 2 n 2 n 2 ku k þ kux k þ ku^x k þ kq k : 3 3
Hence, for 8v 2 Z D ; hgðv Þ; v i P 0 when kv k2 ¼ kun k2 þ 43 kunx k2 þ 13 kun^x k2 þ kqn k2 þ 1. By Lemma 3.1, there exists v 2 Z D satisfying gðv Þ ¼ 0. Let unþ1 ¼ 2v 1 un ; qnþ1 ¼ 2v 2 qn , and it can be proved easily that ðunþ1 ; qnþ1 Þ is the solution of scheme (2.1)–(2.4). h
4. Convergence and unconditional stability Let v ðx; tÞ and /ðx; tÞ be the solutions of problem (1.1)–(1.4) and v nj ¼ uðxj ; tn Þ; /nj ¼ qðxj ; t n Þ, then the truncation errors of scheme (2.1)–(2.4) are obtained as follows:
1 nþ1 nþ1 nþ1 1 4 n 1 4 2 v j þ v nj ^^ þ /nþ /j 2 þ u v j 2 j v j 2 ; j ¼ 1;2;...;J 1; n ¼ 1;2;...;N 1; j ^x €x t xxt xxt 3 3 3 3 4 nþ1 1 nþ1 n n 2 2 vj ; j ¼ 1;2;...;J 1;n ¼ 1;2;...;N 1; sj ¼ /j þ v j ^x €x t 3 3
rnj ¼
v nj
ð4:1Þ ð4:2Þ
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
v 0j ¼ u0 ðxj Þ; /0j ¼ q0 ðxj Þ; 0 6 j 6 J; v n 2 Z0h ; /n 2 Z 0h ; n ¼ 0;1;2;...;N:
5
ð4:3Þ ð4:4Þ
4
Theorem 4.1. jrnj j þ jsnj j ¼ Oðs2 þ h Þ holds as s; h ! 0. Proof. Since v ðx; tÞ and /ðx; tÞ are the solutions of problem (1.1)–(1.4), we have
v t þ /x þ vv x v xxt ¼ 0; /t þ v x ¼ 0; x 2 ðxL ; xR Þ;
t 2 ð0; T:
v t , by Taylor expansion at the point ðxj ; tnþ Þ, we get 1 s2 v nþ1 ¼ vj þ v t jj þ v tt jnþ þ Oðs3 Þ; j j 2! 2 2 s nþ 1 s2 v nj ¼ v nþ þ v t jj þ v tt jnþ þ Oðs3 Þ: j j
Firstly, considering the term
s
nþ12
1 2
nþ12
1 2
1 2
1 2
1 2
2!
2
2
ð4:5Þ ð4:6Þ
It follows from (4.5) and (4.6) that 1 2
v t jnþ j
¼
v nþ1 v nj j
þ Oðs2 Þ ¼ ðv nj Þt þ Oðs2 Þ:
s
ð4:7Þ
Similarly, by Taylor expansion, we can obtain the following results, respectively, nþ12
/t jj
¼
/nþ1 /nj j
s nþ1 /jþ12
þ Oðs2 Þ ¼ ð/nj Þt þ Oðs2 Þ:
ð4:8Þ
nþ1 /j12
nþ1 1 2 1 2 nþ1 nþ1 4 4 h ð/xxx Þjj 2 þ Oðh Þ ¼ /j 2 h ð/xxx Þjj 2 þ Oðh Þ; ^x 6 6 2h nþ1 nþ1 nþ1 /jþ22 /j22 2 2 2 2 nþ1 nþ1 4 4 ¼ h ð/xxx Þjj 2 þ Oðh Þ ¼ /j 2 h ð/xxx Þjj 2 þ Oðh Þ; €x 3 3 4h
nþ12
ð/x Þjj
¼
nþ12
ð/x Þjj
ð4:9Þ ð4:10Þ
and nþ12
ðv xx Þjj
nþ12
ðv xx Þjj
1 2 nþ1 4 h ðv xxxx Þjj 2 þ Oðh Þ; 12 nþ1 1 2 nþ1 4 ¼ vj 2 h ðv xxxx Þjj 2 þ Oðh Þ: ^x^x 3 ¼
1 2
v nþ j
xx
ð4:11Þ ð4:12Þ
Thus, it follows from (4.9) and (4.10) that
4 nþ12 1 nþ12 nþ1 4 /j / ¼ ð/x Þjj 2 þ Oðh Þ: ^x €x 3 3 j
ð4:13Þ
By (4.11) and (4.12), we have
1 4 nþ12 1 nþ1 4 2 v j v nþ ¼ ðv xx Þjj 2 þ Oðh Þ: j ^x^x xx 3 3
ð4:14Þ
Moreover,
nþ12
u vj
nþ1 4 nþ1 nþ1 1 nþ1 nþ1 2 1 2 nþ12 2 2 2 v j 2 v j 2 ^ þ v nþ v v þ v j vj 2 ¼ j j j j €x x 9 9 ^x €x
2 2 1 nþ12 4 nþ12 1 nþ12 1 4 1 nþ12 nþ12 nþ1 4 þ ¼ ðuux Þjj 2 þ Oðh Þ: u u uj uj ¼ vj ^x €x 3 3 j 3 j 3 3 3 ^x €x
ð4:15Þ
4
Apparently, it follows from results above that jrnj j þ jsnj j ¼ Oðs2 þ h Þ holds. h Lemma 4.1. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of initial-boundary value problem (1.1)–(1.4) satisfy
kukL2 6 C;
kux kL2 6 C;
kqkL2 6 C;
kukL1 6 C:
Proof. It follows from (1.9) that
EðtÞ ¼ kuk2L2 þ kux k2L2 þ kqk2L2 ¼ Eð0Þ ¼ C;
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
which yields
kukL2 6 C;
kqkL2 6 C:
kux kL2 6 C;
By Sobolev inequality, kukL1 6 C holds. h Lemma 4.2. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of scheme (2.1)–(2.4) satisfy
kun k 6 C;
kunx k 6 C;
kqn k 6 C;
kun k1 6 C;
for n ¼ 1; . . . ; N. Proof. It follows from Theorem 2.1 and Lemma 2.1 that
kun k2 þ kunx k2 þ kqn k2 6 En ¼ E0 ¼ C; that is,
kun k 6 C;
kunx k 6 C;
kqn k 6 C:
By discrete Sobolev inequality[23], we have kun k1 6 C. h In fact, Lemma 4.2 indicates the stability of the solutions of scheme (2.1)–(2.4). Theorem 4.2. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of scheme (2.1)–(2.4) are stable by the L1 norm for un and by the L2 norm for qn . Theorem 4.3. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of difference scheme (2.1)–(2.4) converge to the 4 solutions of problem (1.1)–(1.4) with order Oðs2 þ h Þ by the L1 norm for un , and by the L2 norm for qn . Proof. Letting enj ¼ v nj unj ; gnj ¼ /nj qnj and subtracting (2.1)–(2.4) from (4.1)–(4.4), respectively, we have
nþ1 nþ1 nþ1 nþ1 4 1 4 nþ12 1 nþ1 rnj ¼ ðenj Þt ðenj Þxxt þ ðenj Þ^x^xt þ gj ^ gj 2 € þ u v j 2 u uj 2 j v j 2 þ j uj 2 ; x x 3 3 3 3 j ¼ 1; 2; . . . ; J 1; n ¼ 1; 2; . . . ; N 1;
snj ¼ gnj
e0j ¼ 0; n
e 2
4 nþ12 1 nþ12 þ ej e ; ^x €x t 3 3 j
ð4:16Þ
j ¼ 1; 2; . . . ; J 1; n ¼ 1; 2; . . . ; N 1;
gnj ¼ 0; 0 6 j 6 J;
Z 0h ;
n
g 2
Z 0h ;
ð4:17Þ ð4:18Þ
n ¼ 0; 1; 2; . . . ; N:
ð4:19Þ nþ12
Computing the inner product of (4.16) with 2e
, and using boundary condition (4.19), we get
D E D 1 E 4 1 8 D nþ12 nþ1 E 2 D nþ12 nþ1 E 1 1 1 r n ; 2enþ2 ¼ ken k2t þ kenx k2t ke^nx k2t þ g^x ; e 2 g€x ; e 2 þ 2 u v nþ2 u unþ2 ; enþ2 3 3 3 3 D 1 E nþ12 nþ2 nþ12 2 j v j u ;e :
ð4:20Þ
Similar to (2.9), we have
D
nþ1
1
E
D
1
nþ12
g^x 2 ; enþ2 ¼ gnþ2 ; e^x
E
;
D
nþ1
1
E
D
1
nþ12
g€x 2 ; enþ2 ¼ gnþ2 ; e€x
E
:
ð4:21Þ
According to Lemma 4.1, Lemma 4.2, Theorem 2.1 and Cauchy-Schwartz inequality, we get
D
1
1
1
J1 J1 nþ1 i nþ1 4 X 2 nþ1 2 nþ1 1 4 Xh nþ12 nþ12 nþ12 2 2 2 v j v j ^ unþ u þ h v uj 2 ej 2 e j j j j ^x x 9 j¼1 9 j¼1 ^x
E
u v nþ2 u unþ2 ; enþ2 ¼ h
J1 J1 h nþ1 i nþ1 4 X nþ1 i nþ1 1 4 Xh nþ12 nþ12 nþ1 nþ1 2 h v j ej ^ þ enþ u j 2 ej 2 h ej 2 uj 2 þ v j 2 ej 2 j ^x ^x x 9 j¼1 9 j¼1 6 C kenþ1 k2 þ ken k2 þ ke^xnþ1 k2 þ ken^x k2 6 C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 ; x
¼
ð4:22Þ
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
D
1
1
1
7
J1 J1 nþ1 i nþ1 1 X 2 nþ1 2 nþ1 1 1 Xh nþ12 nþ12 nþ12 2 v j v j € unþ u j 2 ej 2 þ h v uj 2 ej 2 j j €x x 9 j¼1 9 j¼1 €x
E
j v nþ2 j unþ2 ; enþ2 ¼ h
J1 J1 h nþ1 i nþ1 1 X nþ1 i nþ1 1 1 Xh nþ12 nþ12 nþ1 nþ1 2 h v j ej € þ enþ u j 2 ej 2 h ej 2 uj 2 þ v j 2 ej 2 j € €x x x 9 j¼1 9 j¼1 6 C kenþ1 k2 þ ken k2 þ ke€nþ1 k2 þ ken€x k2 6 C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 ; x x
¼
and
D
E 1 rn ; 2enþ2 ¼ r n ; enþ1 þ en 6 krn k2 þ kenþ1 k2 þ ken k2 :
ð4:23Þ
ð4:24Þ
Substituting (4.21)–(4.24) into (4.20), we get
4 1 8 D nþ1 nþ12 E 2 D nþ1 nþ12 E ken k2t þ kenx k2t ken^x k2t g 2 ; e^x þ g 2 ; e€x 6 krn k2 þ C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 : x 3 3 3 3
ð4:25Þ
1
Then computing the inner product of (4.17) with 2gnþ2 , we get
kgn k2t þ
8 D nþ12 nþ1 E 2 D nþ12 nþ1 E e ;g 2 e ; g 2 6 ksn k2 þ kgnþ1 k2 þ kgn k2 : 3 ^x 3 €x
ð4:26Þ
Adding (4.25) and (4.26), we have
4 1 ken k2t þ kenx k2t ken^x k2t þ kgn k2t 6 kr n k2 þ ksn k2 þ C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 þ kgnþ1 k2 þ kgn k2 : x 3 3
ð4:27Þ
Letting
4 1 Bn ¼ ken k2 þ kenx k2 ken^x k2 þ kgn k2 ; 3 3 and summing up (4.27) from 0 to n 1, we have
Bn 6 B0 þ C s
n1 n1 n X X X krl k2 þ C s ksl k2 þ C s kel k2 þ kelx k2 þ kgl k2 : l¼0
l¼0
l¼0
Noticing that
s
n1 X 4 2 krl k2 6 ns max kr l k2 6 T Oðs2 þ h Þ ; 06l6n1
l¼0
s
n1 X
4 2
ksl k2 6 ns max ksl k2 6 T Oðs2 þ h Þ ; 06l6n1
l¼0 0
4 2
and B ¼ Oðs2 þ h Þ , from Lemma 2.1, we get 4 2
ken k2 þ kenx k2 þ kgn k2 6 Bn 6 Oðs2 þ h Þ þ C s
n X kel k2 þ kelx k2 þ kgl k2 : l¼0
By discrete Gronwall inequality [23], we have 4
ken k 6 Oðs2 þ h Þ;
4
kenx k 6 Oðs2 þ h Þ;
4
kgn k 6 Oðs2 þ h Þ:
Finally, by discrete Sobolev inequality [23], we get 4
ken k1 6 Oðs2 þ h Þ: This completes the proof of Theorem 4.3. h Theorem 4.4. The solutions un ; qn of scheme (2.1)–(2.4) is unique. ~n ; q ~ n are the different solutions of scheme (2.1)–(2.4), and let Proof. Suppose that u
~en ¼ u ~ n un ;
g~ n ¼ q~ n qn :
Similar to the proof of Theorem 4.3, we obtain
k~en k ¼ 0;
~ n k ¼ 0; kg
~ n ¼ un ; q ~ n ¼ qn . h that is, u Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
8
J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
5. Numerical experiments Take
v ¼ 1:5, then the initial functions of problem (1.1)–(1.4) can be derived as follows,
u0 ðx; tÞ ¼
pffiffiffi 5 5 2 sech x; 2 6
5 3
q0 ðx; tÞ ¼ sech2
pffiffiffi 5 x: 6
ð5:1Þ
Scheme (2.1)–(2.4) is a nonlinear system of equations which can be solved by the Newton iteration. In numerical experiments, we take xL ¼ 50; xR ¼ 50 and T ¼ 10. The errors in the sense of L1 -norm of the numerical solutions are listed in Table 1 under different mesh steps h and s. Table 2 shows that the computational and the theoretical orders of the scheme are very close to each other. Furthermore, since we have shown in Theorem 2.1 that the numerical solution un satisfies invariants (2.5), (2.6) and (2.7), respectively, Table 3 is also presented to show the conservative laws Q n1 ; Q n2 and En . To better verify the unconditional stability, we also present error estimates for a lager s in Table 4.
Table 1 Error estimates of numerical solution with various h and s. ken k1
t=2 t=4 t=6 t=8 t = 10
kgn k1
s ¼ 0:2
s ¼ 0:05
h ¼ 0:1
h ¼ 0:05
s ¼ 0:0125 h ¼ 0:025
s ¼ 0:2 h ¼ 0:1
s ¼ 0:05 h ¼ 0:05
s ¼ 0:0125 h ¼ 0:025
1.0223604e2 1.8365924e2 2.4788018e2 3.0337987e2 3.5448953e2
6.4667574e4 1.1620124e3 1.5703416e3 1.9234880e3 2.2488626e3
4.0445600e5 7.2681438e5 9.8229106e5 1.2032503e4 1.4068627e4
7.8628740e3 1.4279499e2 1.8826206e2 2.2662662e2 2.6132566e2
4.97196559e4 9.0356974e3 1.1932482e3 1.4368754e3 1.6581380e3
3.1104326e5 5.6516632e5 7.4656508e5 8.9893676e5 1.0374639e4
Table 2 4 Numerical verification of theoretical accuracy Oðs2 þ h Þ. ken ðh; sÞk1 =ke4n
t=2 t=4 t=6 t=8 t = 10
h s 2 ; 4 k1
kgn ðh; sÞk1 =kg4n
h s 2 ; 4 k1
s ¼ 0:2
s ¼ 0:05
s ¼ 0:0125
h ¼ 0:1
h ¼ 0:05
h ¼ 0:025
s ¼ 0:2 h ¼ 0:1
s ¼ 0:05 h ¼ 0:05
s ¼ 0:0125 h ¼ 0:025
– – – – –
15.809475 15.805273 15.785112 15.772381 15.763058
15.988778 15.987746 15.986520 15.985767 15.984946
– – – – –
15.814416 15.803427 15.777275 15.772183 15.760187
15.984805 15.987678 15.983178 15.984165 15.982609
Table 3 Discrete mass and discrete energy with various h and s.
s ¼ 0:05; h ¼ 0:05 t=2 t=4 t=6 t=8 t = 10
s ¼ 0:0125; h ¼ 0:0125
Q n1
Q n2
En
Q n1
Q n2
En
13.4164052631 13.4164046882 13.4164046613 13.4164046443 13.4164037680
8.9442688298 8.9442687851 8.9442688177 8.9442688167 8.9442688106
34.7832795967 34.7832795967 34.7832795967 34.7832795967 34.7832795967
13.4164051942 13.4164046000 13.4164046122 13.4164047256 13.4164044001
8.9442687468 8.9442687048 8.9442687383 8.9442687335 8.9442687196
34.7832796466 34.7832796466 34.7832796466 34.7832796466 34.7832796466
Table 4 Error estimates of numerical solution with various h when
s ¼ 0:5.
n
kgn k1
ke k1
t=2 t=4 t=6 t=8 t = 10
s ¼ 0:5
s ¼ 0:5
s ¼ 0:5
h ¼ 0:2
h ¼ 0:1
h ¼ 0:05
s ¼ 0:5 h ¼ 0:2
s ¼ 0:5 h ¼ 0:1
s ¼ 0:5 h ¼ 0:5
5.9460011e2 1.0835575e1 1.5004266e1 1.8617672e1 2.1546069e1
5.9530277e2 1.0734165e1 1.5178814e1 1.8605148e1 2.1532498e1
5.9525561e2 1.0744797e1 1.5171686e1 1.8611669e1 2.1531646e1
5.0365296e2 8.5067424e2 1.1212972e1 1.3735943e1 1.6765103e1
5.0351574e2 8.4983794e2 1.1256341e1 1.3637629e1 1.6660025e1
5.0367228e2 8.4978534e2 1.1250638e1 1.3631451e1 1.6653422e1
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9
From these results, the stability and convergence of the scheme are verified and it shows that our proposed algorithm is effective and reliable. 6. Conclusions In this paper, with the help of the Richardson extrapolation, we propose a new conservative Crank–Nicolson finite difference scheme for the SRLW equation with second order temporal accuracy and four order partial accuracy. The discrete conservative quantities are discussed in detail and a local truncation error is derived by the Taylor expansion. Moreover, the existence of numerical solution is proved by the Browder fixed point theorem and the detailed analysis for the unconditional stability and convergence of the proposed scheme are also presented by the energy method. Some numerical experiments also verify this convergence in the maximum norm, second order accurate in time and four order accurate in space. In particular, such effective difference scheme applied to certain more complicated PDEs will be reported in our future works. Acknowledgments The authors are very grateful to both reviewers for carefully reading this paper and their comments, and thank Dr. Cheng Wang (University of Massachusetts, Dartmouth) for his valuable discussion. References [1] C.E. Seyler, D.C. Fenstermacler, A symmetric regularized long wave equation, Phys. Fluids 27 (1) (1984) 4–7. [2] C.J. Amick, J.L. Bona, M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differ. Equ. 81 (1) (1989) 1–49. [3] 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. [4] D. Bhardwaj, R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl. 40 (2000) 1397–1404. [5] V.G. Makhankov, Dynamics of classical solitons (in non-integrable systems), Phys. Rep. 35 (1) (1978) 1–128. A review section of physics letters (section C). [6] A.A. Soliman, Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method, Math. Comput. Simul. 70 (2) (2005) 119–124. [7] P.A. Clarkson, New similarity reductions and Painleve analysis for the symmetric regularized long wave and modified Benjamin–Bona–Mahoney equations, J. Phys. A: Math. Gen. 22 (18) (1989) 3821–3848. [8] J.L. Bogolubsky, Some examples of inelastic soliton interaction, Comput. Phys. Commun. 13 (1) (1977) 149–155. [9] B. Guo, The spectral method for symmetric regularized wave equations, J. Comput. Math. 5 (4) (1987) 297–306. [10] J. Zheng, R. Zhang, B. Guo, The Fourier pseudospectral method for the SRLW equation, Appl. Math. Mech. 10 (9) (1989) 801–810. [11] Y. Shang, B. Guo, Analysis of chebyshev pseudospectral method for multi-dimentional generalized SRLW equations, Appl. Math. Mech. 24 (10) (2003) 1035–1048. [12] Y. Shang, B. Guo, S. Fang, Long time behavior of the dissipative generalized symmetric regularized long wave equations, J. Partial Differ. Equ. 15 (2002) 35–45. [13] S. Fang, B. Guo, H. Qiu, The existence of global attractors for a system of multi-dimensional symmetric regularized long wave equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 61–68. [14] B. Guo, Y. Shang, Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term, Acta Math. Appl. Sin. 19 (2) (2003) 191–204. [15] L. Kong, W. Zeng, R. Liu, A multisymplectic Fourier pseudo-spectral scheme for the SRLW equation and conservation laws, Chin. J. Comput. Phys. 23 (1) (2006) 25–31. [16] T. Wang, L. Zhang, F. Chen, Conservative schemes for the symmetric regularized long wave equations, Appl. Math. Comput. 190 (2007) 1063–1080. [17] J. Hu, B. Hu, Y. Xu, C-N difference schemes for dissipative symmetric regularized long wave equations with damping term, Math. Prob. Eng. 2011 (2011) (Article ID 651642), 16 pp. [18] Y. Xu, B. Hu, X. Xie, J. Hu, Mixed finite element analysis for dissipative SRLW equations with damping term, Appl. Math. Comput. 218 (2012) 4788– 4797. [19] 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 (6) (1995) 1839–1875. [20] K. Zheng, J. Hu, High-order conservative Crank–Nicolson scheme for regularized long wave equation, Adv. Differ. Equ. 2013 (2013) 287. [21] F. Zhang, M.P. Victor, V. Luis, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. Math. Comput. 71 (2–3) (1995) 165–177. [22] F.E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc. Sympos. Appl. Math. 17 (1965) 24–49. [23] Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Method, Inter. Acad. Publishers, Beijing, 1990.
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation q Jinsong Hu a, Kelong Zheng b,⇑, Maobo Zheng c a
School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China School of Science, Southwest University of Science and Technology, Mianyang 621010, China c Chengdu Technological University, Chengdu 611730, China b
a r t i c l e
i n f o
Article history: Received 17 June 2013 Received in revised form 21 February 2014 Accepted 29 April 2014 Available online xxxx Keywords: SRLW equation Conservative difference scheme Richardson extrapolation Stability Convergence
a b s t r a c t Coupled with the Richardson extrapolation, a new conservative Crank–Nicolson finite dif4 ference scheme, which has the accuracy of Oðs2 þ h Þ without refined mesh for the symmetric regularized long wave equation is proposed. The corresponding conservative quantities are discussed, and the existence of numerical solution is proved by the Browder fixed point theorem. The convergence, unconditional stability and uniqueness of the scheme are also derived using the energy method. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Consider the following initial boundary value problem of the Symmetric Regularized Long Wave (SRLW) equation
ut þ qx þ uux uxxt ¼ 0;
ð1:1Þ
qt þ ux ¼ 0; x 2 ðxL ; xR Þ; t 2 ð0; T; uðx; 0Þ ¼ u0 ðxÞ; qðx; 0Þ ¼ q0 ðxÞ; x 2 ½xL ; xR ; uðxL ; tÞ ¼ uðxR ; tÞ ¼ 0; qðxL ; tÞ ¼ qðxR ; tÞ ¼ 0; t 2 ½0; T;
ð1:3Þ
ð1:2Þ ð1:4Þ
where u0 ðxÞ and q0 ðxÞ are two known smooth functions. SRLW equation has been shown to describe weakly nonlinear ð1 þ 1Þ-dimensional ion-acoustic and space-charge waves, where q and u are the dimensionless electron charge density and the fluid velocity, respectively [1]. Eliminating q from (1.1) and (1.2), we can get another form of SRLW equation,
1 utt uxx þ ðu2 Þxt uxxxt ¼ 0; 2
ð1:5Þ
which is explicitly symmetric about the x and t derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves [2–6]. The solitary wave solutions of Eqs. (1.1) and (1.2) are (v 2 > 1 is velocity), q This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department (No. 11ZB009), the Doctoral Program Research Fund of Southwest University of Science and Technology (No. 11zx7129) and the Applied Basic Research Project of Sichuan Province (No. 2013JY0096). ⇑ Corresponding author. Tel.: +86 13547116902. E-mail address: [email protected] (K. Zheng).
http://dx.doi.org/10.1016/j.apm.2014.04.062 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v2 1 ðx v tÞ; 2 v v2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðv 2 1Þ v2 1 21 qðx; tÞ ¼ sech ðx v tÞ; 2 2 v v2 uðx; tÞ ¼
3ðv 2 1Þ
2
sech
and physical boundary requires
uðx; tÞ ! 0;
qðx; tÞ ! 0; as jxj ! 1; t > 0:
ð1:6Þ
Hence, if xL 0 and xR 0, problem (1.1)–(1.4) is in accordance with the Cauchy problem of Eq. (1.1) and (1.2). It can be proved easily that SRLW equation has the following conserved laws,
Q 1 ðtÞ ¼
Z
xR
uðx; tÞdx ¼
xL xR
Q 2 ðtÞ ¼
Z
Z
xR
u0 ðxÞdx ¼ Q 1 ð0Þ;
ð1:7Þ
q0 ðxÞdx ¼ Q 2 ð0Þ;
ð1:8Þ
x
qðx; tÞdx ¼
xL
Z L xR xL
and
EðtÞ ¼ kuk2L2 þ kux k2L2 þ kqk2L2 ¼ Eð0Þ;
ð1:9Þ
where Q 1 ð0Þ; Q 2 ð0Þ and Eð0Þ are three positive constants which relate to the initial condition. SRLW equation (1.1) and (1.2) or (1.5) arises also in many other areas of mathematical physics [7]. Numerical investigation indicates that interactions of solitary waves are inelastic[8], thus the solitary wave of SRLW equation is not a solution. Research on the well-posedness for its solution and numerical methods has aroused more and more interest. In [9], Guo studied the existence, uniqueness and regularity of numerical solutions for the periodic initial value problem of generalized SRLWE by spectral method. In Ref. [10], Zheng et al. presented a Fourier pseudospectral method with a restraint operator for SRLW equation, and proved the stability and the optimum error estimates. In Ref. [11], Shang and Guo analyzed a chebyshev pseudospectral scheme for multi-dimensional generalized SRLW equations and other results[12–14]. In Ref. [15], an Euler midpoint scheme in time and a Fourier pseudospectral method in space are used to the multisymplectic formulation of SRLW equation, in which a multisymplectic Fourier pseudospectral scheme is proposed, and several discrete conservation laws of this scheme are proved. Wang and Zhang[16] studied SRLW equation by some conservative finite difference methods. For the dissipative SRLW equations with damping term, a Crank–Nicolson scheme is proposed by Hu et al. [17] and a mixed finite element method is presented by Xu et al. [18], respectively. In [19], Li and Vu-Quoc pointed out that ‘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’. Thus, the main purpose of this paper is to study the conservative schemes[21] for SRLW equation. Coupled with the Richardson extrapolation, a two-level nonlinear finite 4 difference scheme which has the accuracy of Oðs2 þ h Þ without refined mesh is proposed. The scheme well approximates three conserved quantities (1.7), (1.8) and (1.9). Moreover, the existence and the uniqueness of numerical solutions are discussed. Priori estimate, convergence and unconditional stability of the scheme are also proved. The remainder of this paper is arranged as follows. In Section 2, a nonlinear conservative difference scheme is proposed. In Section 3, we prove the existence of scheme by the Browder fixed point theorem. Priori estimate, convergence and stability are proved in Section 4, and numerical experiments to verify the theoretical analysis are reported in Section 5. 2. Nonlinear finite difference scheme L Let h ¼ xR x be the step size for the spatial grid such J that xj ¼ xL þ jh; ðj ¼ 1; 0; 1; . . . ; J; J þ 1Þ. Let s be the step size for the temporal direction, t n ¼ ns; ðn ¼ 0; 1; 2; . . . ; NÞ; N ¼ Ts . Denote unj uðxj ; tn Þ; qnj qðxj ; tn Þ and
Z 0h ¼ u ¼ ðuj Þju1 ¼ u0 ¼ uJ ¼ uJþ1 ¼ 0;
j ¼ 1; 0; 1; 2; . . . ; J; J þ 1 :
Define
ðunj Þx ¼ ðunj Þ€x ¼
unjþ1 unj
ðunj Þx ¼
;
h unjþ2 unj2
;
4h J1 X n n hu ; v i ¼ h unj v nj ;
unj unj1
ðunj Þt ¼
j¼1
h unþ1 j
unj
s
ðunj Þ^x ¼
; ;
kun k2 ¼ hun ; un i;
nþ12
uj
unjþ1 unj1
¼
2h þ unj
unþ1 j
2
; ;
kun k1 ¼ max junj j: 16j6J1
In this paper, C denotes a general positive constant which may have different values in different occurrences. Lemma 2.1. [20] For a mesh function u 2 Z 0h , by Cauchy–Schwarz inequality, we have
ku€x k2 6 ku^x k2 6 kux k2 : Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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The following difference scheme for problem (1.1)–(1.4) is considered,
n nþ1 2 4 n 1 n 4 nþ12 1 nþ12 4 nþ12 nþ12 1 nþ1 nþ1 uj unj uj þ uj þ q q þ uj þ ðuj 2 Þ uj 2 uj 2 j j ^x^xt ^x €x ^x €x t xxt 3 3 3 3 9 ^x 9 2 nþ12 þ uj ¼ 0; j ¼ 1; 2; . . . ; J 1; n ¼ 1; 2; . . . ; N 1;
4 nþ12 1 nþ12 uj u ¼ 0; ^x €x t 3 3 j u0j ¼ u0 ðxj Þ; q0j ¼ q0 ðxj Þ; 0 6 j 6 J;
qnj þ
n
u 2
ð2:1Þ
€x
Z 0h ;
n
q 2
Z 0h ;
ð2:2Þ ð2:3Þ
n ¼ 0; 1; 2; . . . ; N:
ð2:4Þ
From boundary condition (1.4) and physical boundary (1.6), discrete boundary condition (2.4) is reasonable. Based on scheme (2.1)–(2.4), the discrete versions of (1.7)–(1.9) are obtained as follows. Theorem 2.1. Scheme (2.1)–(2.4) is conservative in the senses
Q n1 ¼ h
J1 X unj ¼ Q n1 ¼ ¼ Q 01 ; 1
ð2:5Þ
j¼1
Q n2 ¼ h
J1 X
qnj ¼ Q n1 ¼ ¼ Q 02 ; 2
ð2:6Þ
j¼1
4 1 En ¼ kun k2 þ kunx k2 ku^nx k2 þ kqn k2 ¼ En1 ¼ ¼ E0 ; 3 3
ð2:7Þ
for n ¼ 1; 2; . . . ; N. Proof. Multiplying (2.1) and (2.2) with h respectively, then summing up for j from 1 to J 1, we have J1 X h ðunj Þt ¼ 0;
h
J1 X ðqnj Þt ¼ 0:
j¼1
j¼1
Q n1
By the definition of and Q n2 , Eqs. (2.5) and (2.6) are obtained from formula above. 1 Taking the inner product of (2.1) with 2unþ2 , according to boundary condition (2.4), we get
D E D E 4 1 8 D nþ12 nþ1 E 2 D nþ12 nþ1 E 1 1 1 1 kun k2t þ kunx k2t ku^nx k2t þ q^x ; u 2 q€x ; u 2 þ 2 uðunþ2 Þ; unþ2 2 jðunþ2 Þ; unþ2 ¼ 0; 3 3 3 3
where u
nþ1 uj 2
¼ 49
nþ1 uj 2
nþ1 nþ1 2 uj 2 þ uj 2 ^ x
and
^ x
j
nþ1 uj 2
¼ 19
nþ1 uj 2
nþ1 nþ1 2 uj 2 þ uj 2 € x
ð2:8Þ
.
€ x
Since
D
nþ1
1
E
D
1
nþ12
q^x 2 ; unþ2 ¼ qnþ2 ; u^x
E
;
D
nþ1
1
E
D
1
nþ12
q€x 2 ; unþ2 ¼ qnþ2 ; u€x
E
;
ð2:9Þ
J1 J1 J1 D 1 E 4 X nþ1 2 nþ1 4 X 2 nþ1 4 X nþ12 2 1 nþ1 nþ1 nþ1 nþ1 uj 2 ¼ h u unþ2 ; unþ2 ¼ h uj 2 uj 2 þ uj 2 uj 2 uj 2 þ h uj uj 2 ^x ^ x 9 j¼1 9 9 ^x ^x j¼1 j¼1 ¼
J1 J1 4 X nþ12 2 nþ12 4 X nþ12 2 nþ12 h uj uj h u uj ¼0 ^x ^x 9 j¼1 9 j¼1 j
ð2:10Þ
and
J1 J1 J1 D 1 E 1 X nþ1 2 nþ1 1 X 2 nþ1 1 X nþ12 2 1 nþ1 nþ1 nþ1 nþ1 uj 2 ¼ h j unþ2 ; unþ2 ¼ h uj 2 uj 2 þ uj 2 uj 2 uj 2 þ h uj uj 2 €x € x 9 j¼1 9 9 €x €x j¼1 j¼1 ¼
J1 J1 1 X nþ12 2 nþ12 1 X nþ12 2 nþ12 h uj uj h u uj ¼ 0; €x €x 9 j¼1 9 j¼1 j
ð2:11Þ
from (2.8)–(2.11), we have
4 1 8 D nþ1 nþ12 E 2 D nþ1 nþ12 E kun k2t þ kunx k2t ku^nx k2t q 2 ; u^x þ q 2 ; u€x ¼ 0: 3 3 3 3
ð2:12Þ
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Then taking the inner product of (2.2) with 2qnþ2 , we have
kqn k2t þ
8 D nþ12 nþ1 E 2 D nþ12 nþ1 E u ;q 2 u ; q 2 ¼ 0: 3 ^x 3 €x
ð2:13Þ
Adding 2.12,2.13, we have
4 1 kunþ1 k2 kun k2 þ kunþ1 k2 kunx k2 kunþ1 k2 ku^nx k2 þ kqnþ1 k2 kqn k2 ¼ 0: ^x x 3 3 By the definition of En , (2.7) is obtained from formula above. h
3. Existence To prove the existence of solution for scheme (2.1)–(2.4), the following Browder fixed point theorem should be introduced. For the proof, see[22]. Lemma 3.1. Let H be a finite dimensional inner product space. Suppose that g : H ! H is continuous and there exists an a > 0 such that hgðxÞ; Then there exists x 2 H such that gðx Þ ¼ 0 and jjx jj 6 a. n xi > 0 for all x 2 H with jjxjj ¼ a. o 0 Let Z D ¼ v ¼ ðv 1 ; v 2 Þ ¼ ðv 1;j ; v 2;j Þjv 1 ; v 2 2 Z h and define
hv ; v 0 i ¼ hðv 1 ; v 2 Þ;
ðv 01 ; v 02 Þi ¼ hv 1 ; v 01 i þ hv 2 ; v 02 i;
kv k2 ¼ kv 1 k2 þ kv 2 k2 :
Theorem 3.1. There exists ðun ; qn Þ 2 Z 0h satisfying difference scheme (2.1)–(2.4). Proof. Consider the mathematical induction for the proof. Obviously, with the condition (2.3), the solution exists for n ¼ 0. Suppose that for n 6 N 1, ðu0 ; q0 Þ; ðu1 ; q1 Þ; . . . ; ðun ; qn Þ 2 Z D satisfy (2.1)–(2.4), then we prove that there exists ðunþ1 ; qnþ1 Þ satisfying (2.1)–(2.4). Define an operator g ¼ ðg 1 ; g 2 Þ as follows,
8 8 2 2 4 1 v 1xx þ unxx v 1^x^x þ un^x^x þ sv 2^x sv 2€x þ suðv 1 Þ sjðv 1 Þ; 3 3 3 3 3 3 4 1 n g 2 ðv Þ ¼ 2v 2 2q þ sv 1^x sv 1€x ; 8v ¼ ðv 1 ; v 2 Þ 2 Z D : 3 3
g 1 ðv Þ ¼ 2v 1 2un
ð3:1Þ ð3:2Þ
v ¼ ðv 1 ; v 2 Þ, similar to (2.9)–(2.11), we get hv 2€x ; v 1 i ¼ hv 2 ; v 1€x i; huðv 1 Þ; v 1 i ¼ 0; hjðv 1 Þ; v 1 i ¼ 0:
Taking the inner product of (3.1) and (3.2) with
hv 2^x ; v 1 i ¼ hv 2 ; v 1^x i;
From Lemma 2.1 and Cauchy–Schwarz inequality, we get
hgðv Þ; v i ¼ hg 1 ðv Þ; v 1 i þ hg 2 ðv Þ; v 2 i 8 8 2 2 ¼ 2kv 1 k2 2hun ; v 1 i þ kv 1x k2 hunx ; v 1x i kv 1^x k2 þ un^x ; v 1^x þ 2kv 2 k2 2hqn ; v 2 i 2kv 1 k2 3 3 3 3 8 2 4 n 2 1 n 2 kuk2 þ kv 1 k2 þ kv 1x k2 kux k þ kv 1x k2 kv 1^x k2 ku^x k þ kv 1^x k2 þ 2kv 2 k2 3 3 3 3 1 4 n 2 1 n 2 2 2 2 2 n 2 n 2 kq k þ kv 2 k P kv 1 k þ kv 2 k ku k þ kv 1x k kux k ku^x k kqn k2 P kv k2 3 3 3 4 n 2 1 n 2 n 2 n 2 ku k þ kux k þ ku^x k þ kq k : 3 3
Hence, for 8v 2 Z D ; hgðv Þ; v i P 0 when kv k2 ¼ kun k2 þ 43 kunx k2 þ 13 kun^x k2 þ kqn k2 þ 1. By Lemma 3.1, there exists v 2 Z D satisfying gðv Þ ¼ 0. Let unþ1 ¼ 2v 1 un ; qnþ1 ¼ 2v 2 qn , and it can be proved easily that ðunþ1 ; qnþ1 Þ is the solution of scheme (2.1)–(2.4). h
4. Convergence and unconditional stability Let v ðx; tÞ and /ðx; tÞ be the solutions of problem (1.1)–(1.4) and v nj ¼ uðxj ; tn Þ; /nj ¼ qðxj ; t n Þ, then the truncation errors of scheme (2.1)–(2.4) are obtained as follows:
1 nþ1 nþ1 nþ1 1 4 n 1 4 2 v j þ v nj ^^ þ /nþ /j 2 þ u v j 2 j v j 2 ; j ¼ 1;2;...;J 1; n ¼ 1;2;...;N 1; j ^x €x t xxt xxt 3 3 3 3 4 nþ1 1 nþ1 n n 2 2 vj ; j ¼ 1;2;...;J 1;n ¼ 1;2;...;N 1; sj ¼ /j þ v j ^x €x t 3 3
rnj ¼
v nj
ð4:1Þ ð4:2Þ
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v 0j ¼ u0 ðxj Þ; /0j ¼ q0 ðxj Þ; 0 6 j 6 J; v n 2 Z0h ; /n 2 Z 0h ; n ¼ 0;1;2;...;N:
5
ð4:3Þ ð4:4Þ
4
Theorem 4.1. jrnj j þ jsnj j ¼ Oðs2 þ h Þ holds as s; h ! 0. Proof. Since v ðx; tÞ and /ðx; tÞ are the solutions of problem (1.1)–(1.4), we have
v t þ /x þ vv x v xxt ¼ 0; /t þ v x ¼ 0; x 2 ðxL ; xR Þ;
t 2 ð0; T:
v t , by Taylor expansion at the point ðxj ; tnþ Þ, we get 1 s2 v nþ1 ¼ vj þ v t jj þ v tt jnþ þ Oðs3 Þ; j j 2! 2 2 s nþ 1 s2 v nj ¼ v nþ þ v t jj þ v tt jnþ þ Oðs3 Þ: j j
Firstly, considering the term
s
nþ12
1 2
nþ12
1 2
1 2
1 2
1 2
2!
2
2
ð4:5Þ ð4:6Þ
It follows from (4.5) and (4.6) that 1 2
v t jnþ j
¼
v nþ1 v nj j
þ Oðs2 Þ ¼ ðv nj Þt þ Oðs2 Þ:
s
ð4:7Þ
Similarly, by Taylor expansion, we can obtain the following results, respectively, nþ12
/t jj
¼
/nþ1 /nj j
s nþ1 /jþ12
þ Oðs2 Þ ¼ ð/nj Þt þ Oðs2 Þ:
ð4:8Þ
nþ1 /j12
nþ1 1 2 1 2 nþ1 nþ1 4 4 h ð/xxx Þjj 2 þ Oðh Þ ¼ /j 2 h ð/xxx Þjj 2 þ Oðh Þ; ^x 6 6 2h nþ1 nþ1 nþ1 /jþ22 /j22 2 2 2 2 nþ1 nþ1 4 4 ¼ h ð/xxx Þjj 2 þ Oðh Þ ¼ /j 2 h ð/xxx Þjj 2 þ Oðh Þ; €x 3 3 4h
nþ12
ð/x Þjj
¼
nþ12
ð/x Þjj
ð4:9Þ ð4:10Þ
and nþ12
ðv xx Þjj
nþ12
ðv xx Þjj
1 2 nþ1 4 h ðv xxxx Þjj 2 þ Oðh Þ; 12 nþ1 1 2 nþ1 4 ¼ vj 2 h ðv xxxx Þjj 2 þ Oðh Þ: ^x^x 3 ¼
1 2
v nþ j
xx
ð4:11Þ ð4:12Þ
Thus, it follows from (4.9) and (4.10) that
4 nþ12 1 nþ12 nþ1 4 /j / ¼ ð/x Þjj 2 þ Oðh Þ: ^x €x 3 3 j
ð4:13Þ
By (4.11) and (4.12), we have
1 4 nþ12 1 nþ1 4 2 v j v nþ ¼ ðv xx Þjj 2 þ Oðh Þ: j ^x^x xx 3 3
ð4:14Þ
Moreover,
nþ12
u vj
nþ1 4 nþ1 nþ1 1 nþ1 nþ1 2 1 2 nþ12 2 2 2 v j 2 v j 2 ^ þ v nþ v v þ v j vj 2 ¼ j j j j €x x 9 9 ^x €x
2 2 1 nþ12 4 nþ12 1 nþ12 1 4 1 nþ12 nþ12 nþ1 4 þ ¼ ðuux Þjj 2 þ Oðh Þ: u u uj uj ¼ vj ^x €x 3 3 j 3 j 3 3 3 ^x €x
ð4:15Þ
4
Apparently, it follows from results above that jrnj j þ jsnj j ¼ Oðs2 þ h Þ holds. h Lemma 4.1. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of initial-boundary value problem (1.1)–(1.4) satisfy
kukL2 6 C;
kux kL2 6 C;
kqkL2 6 C;
kukL1 6 C:
Proof. It follows from (1.9) that
EðtÞ ¼ kuk2L2 þ kux k2L2 þ kqk2L2 ¼ Eð0Þ ¼ C;
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which yields
kukL2 6 C;
kqkL2 6 C:
kux kL2 6 C;
By Sobolev inequality, kukL1 6 C holds. h Lemma 4.2. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of scheme (2.1)–(2.4) satisfy
kun k 6 C;
kunx k 6 C;
kqn k 6 C;
kun k1 6 C;
for n ¼ 1; . . . ; N. Proof. It follows from Theorem 2.1 and Lemma 2.1 that
kun k2 þ kunx k2 þ kqn k2 6 En ¼ E0 ¼ C; that is,
kun k 6 C;
kunx k 6 C;
kqn k 6 C:
By discrete Sobolev inequality[23], we have kun k1 6 C. h In fact, Lemma 4.2 indicates the stability of the solutions of scheme (2.1)–(2.4). Theorem 4.2. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of scheme (2.1)–(2.4) are stable by the L1 norm for un and by the L2 norm for qn . Theorem 4.3. Suppose that u0 2 H10 ½xL ; xR ; q0 2 L2 ½xL ; xR , then the solutions of difference scheme (2.1)–(2.4) converge to the 4 solutions of problem (1.1)–(1.4) with order Oðs2 þ h Þ by the L1 norm for un , and by the L2 norm for qn . Proof. Letting enj ¼ v nj unj ; gnj ¼ /nj qnj and subtracting (2.1)–(2.4) from (4.1)–(4.4), respectively, we have
nþ1 nþ1 nþ1 nþ1 4 1 4 nþ12 1 nþ1 rnj ¼ ðenj Þt ðenj Þxxt þ ðenj Þ^x^xt þ gj ^ gj 2 € þ u v j 2 u uj 2 j v j 2 þ j uj 2 ; x x 3 3 3 3 j ¼ 1; 2; . . . ; J 1; n ¼ 1; 2; . . . ; N 1;
snj ¼ gnj
e0j ¼ 0; n
e 2
4 nþ12 1 nþ12 þ ej e ; ^x €x t 3 3 j
ð4:16Þ
j ¼ 1; 2; . . . ; J 1; n ¼ 1; 2; . . . ; N 1;
gnj ¼ 0; 0 6 j 6 J;
Z 0h ;
n
g 2
Z 0h ;
ð4:17Þ ð4:18Þ
n ¼ 0; 1; 2; . . . ; N:
ð4:19Þ nþ12
Computing the inner product of (4.16) with 2e
, and using boundary condition (4.19), we get
D E D 1 E 4 1 8 D nþ12 nþ1 E 2 D nþ12 nþ1 E 1 1 1 r n ; 2enþ2 ¼ ken k2t þ kenx k2t ke^nx k2t þ g^x ; e 2 g€x ; e 2 þ 2 u v nþ2 u unþ2 ; enþ2 3 3 3 3 D 1 E nþ12 nþ2 nþ12 2 j v j u ;e :
ð4:20Þ
Similar to (2.9), we have
D
nþ1
1
E
D
1
nþ12
g^x 2 ; enþ2 ¼ gnþ2 ; e^x
E
;
D
nþ1
1
E
D
1
nþ12
g€x 2 ; enþ2 ¼ gnþ2 ; e€x
E
:
ð4:21Þ
According to Lemma 4.1, Lemma 4.2, Theorem 2.1 and Cauchy-Schwartz inequality, we get
D
1
1
1
J1 J1 nþ1 i nþ1 4 X 2 nþ1 2 nþ1 1 4 Xh nþ12 nþ12 nþ12 2 2 2 v j v j ^ unþ u þ h v uj 2 ej 2 e j j j j ^x x 9 j¼1 9 j¼1 ^x
E
u v nþ2 u unþ2 ; enþ2 ¼ h
J1 J1 h nþ1 i nþ1 4 X nþ1 i nþ1 1 4 Xh nþ12 nþ12 nþ1 nþ1 2 h v j ej ^ þ enþ u j 2 ej 2 h ej 2 uj 2 þ v j 2 ej 2 j ^x ^x x 9 j¼1 9 j¼1 6 C kenþ1 k2 þ ken k2 þ ke^xnþ1 k2 þ ken^x k2 6 C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 ; x
¼
ð4:22Þ
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D
1
1
1
7
J1 J1 nþ1 i nþ1 1 X 2 nþ1 2 nþ1 1 1 Xh nþ12 nþ12 nþ12 2 v j v j € unþ u j 2 ej 2 þ h v uj 2 ej 2 j j €x x 9 j¼1 9 j¼1 €x
E
j v nþ2 j unþ2 ; enþ2 ¼ h
J1 J1 h nþ1 i nþ1 1 X nþ1 i nþ1 1 1 Xh nþ12 nþ12 nþ1 nþ1 2 h v j ej € þ enþ u j 2 ej 2 h ej 2 uj 2 þ v j 2 ej 2 j € €x x x 9 j¼1 9 j¼1 6 C kenþ1 k2 þ ken k2 þ ke€nþ1 k2 þ ken€x k2 6 C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 ; x x
¼
and
D
E 1 rn ; 2enþ2 ¼ r n ; enþ1 þ en 6 krn k2 þ kenþ1 k2 þ ken k2 :
ð4:23Þ
ð4:24Þ
Substituting (4.21)–(4.24) into (4.20), we get
4 1 8 D nþ1 nþ12 E 2 D nþ1 nþ12 E ken k2t þ kenx k2t ken^x k2t g 2 ; e^x þ g 2 ; e€x 6 krn k2 þ C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 : x 3 3 3 3
ð4:25Þ
1
Then computing the inner product of (4.17) with 2gnþ2 , we get
kgn k2t þ
8 D nþ12 nþ1 E 2 D nþ12 nþ1 E e ;g 2 e ; g 2 6 ksn k2 þ kgnþ1 k2 þ kgn k2 : 3 ^x 3 €x
ð4:26Þ
Adding (4.25) and (4.26), we have
4 1 ken k2t þ kenx k2t ken^x k2t þ kgn k2t 6 kr n k2 þ ksn k2 þ C kenþ1 k2 þ ken k2 þ kenþ1 k2 þ kenx k2 þ kgnþ1 k2 þ kgn k2 : x 3 3
ð4:27Þ
Letting
4 1 Bn ¼ ken k2 þ kenx k2 ken^x k2 þ kgn k2 ; 3 3 and summing up (4.27) from 0 to n 1, we have
Bn 6 B0 þ C s
n1 n1 n X X X krl k2 þ C s ksl k2 þ C s kel k2 þ kelx k2 þ kgl k2 : l¼0
l¼0
l¼0
Noticing that
s
n1 X 4 2 krl k2 6 ns max kr l k2 6 T Oðs2 þ h Þ ; 06l6n1
l¼0
s
n1 X
4 2
ksl k2 6 ns max ksl k2 6 T Oðs2 þ h Þ ; 06l6n1
l¼0 0
4 2
and B ¼ Oðs2 þ h Þ , from Lemma 2.1, we get 4 2
ken k2 þ kenx k2 þ kgn k2 6 Bn 6 Oðs2 þ h Þ þ C s
n X kel k2 þ kelx k2 þ kgl k2 : l¼0
By discrete Gronwall inequality [23], we have 4
ken k 6 Oðs2 þ h Þ;
4
kenx k 6 Oðs2 þ h Þ;
4
kgn k 6 Oðs2 þ h Þ:
Finally, by discrete Sobolev inequality [23], we get 4
ken k1 6 Oðs2 þ h Þ: This completes the proof of Theorem 4.3. h Theorem 4.4. The solutions un ; qn of scheme (2.1)–(2.4) is unique. ~n ; q ~ n are the different solutions of scheme (2.1)–(2.4), and let Proof. Suppose that u
~en ¼ u ~ n un ;
g~ n ¼ q~ n qn :
Similar to the proof of Theorem 4.3, we obtain
k~en k ¼ 0;
~ n k ¼ 0; kg
~ n ¼ un ; q ~ n ¼ qn . h that is, u Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062
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J. Hu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
5. Numerical experiments Take
v ¼ 1:5, then the initial functions of problem (1.1)–(1.4) can be derived as follows,
u0 ðx; tÞ ¼
pffiffiffi 5 5 2 sech x; 2 6
5 3
q0 ðx; tÞ ¼ sech2
pffiffiffi 5 x: 6
ð5:1Þ
Scheme (2.1)–(2.4) is a nonlinear system of equations which can be solved by the Newton iteration. In numerical experiments, we take xL ¼ 50; xR ¼ 50 and T ¼ 10. The errors in the sense of L1 -norm of the numerical solutions are listed in Table 1 under different mesh steps h and s. Table 2 shows that the computational and the theoretical orders of the scheme are very close to each other. Furthermore, since we have shown in Theorem 2.1 that the numerical solution un satisfies invariants (2.5), (2.6) and (2.7), respectively, Table 3 is also presented to show the conservative laws Q n1 ; Q n2 and En . To better verify the unconditional stability, we also present error estimates for a lager s in Table 4.
Table 1 Error estimates of numerical solution with various h and s. ken k1
t=2 t=4 t=6 t=8 t = 10
kgn k1
s ¼ 0:2
s ¼ 0:05
h ¼ 0:1
h ¼ 0:05
s ¼ 0:0125 h ¼ 0:025
s ¼ 0:2 h ¼ 0:1
s ¼ 0:05 h ¼ 0:05
s ¼ 0:0125 h ¼ 0:025
1.0223604e2 1.8365924e2 2.4788018e2 3.0337987e2 3.5448953e2
6.4667574e4 1.1620124e3 1.5703416e3 1.9234880e3 2.2488626e3
4.0445600e5 7.2681438e5 9.8229106e5 1.2032503e4 1.4068627e4
7.8628740e3 1.4279499e2 1.8826206e2 2.2662662e2 2.6132566e2
4.97196559e4 9.0356974e3 1.1932482e3 1.4368754e3 1.6581380e3
3.1104326e5 5.6516632e5 7.4656508e5 8.9893676e5 1.0374639e4
Table 2 4 Numerical verification of theoretical accuracy Oðs2 þ h Þ. ken ðh; sÞk1 =ke4n
t=2 t=4 t=6 t=8 t = 10
h s 2 ; 4 k1
kgn ðh; sÞk1 =kg4n
h s 2 ; 4 k1
s ¼ 0:2
s ¼ 0:05
s ¼ 0:0125
h ¼ 0:1
h ¼ 0:05
h ¼ 0:025
s ¼ 0:2 h ¼ 0:1
s ¼ 0:05 h ¼ 0:05
s ¼ 0:0125 h ¼ 0:025
– – – – –
15.809475 15.805273 15.785112 15.772381 15.763058
15.988778 15.987746 15.986520 15.985767 15.984946
– – – – –
15.814416 15.803427 15.777275 15.772183 15.760187
15.984805 15.987678 15.983178 15.984165 15.982609
Table 3 Discrete mass and discrete energy with various h and s.
s ¼ 0:05; h ¼ 0:05 t=2 t=4 t=6 t=8 t = 10
s ¼ 0:0125; h ¼ 0:0125
Q n1
Q n2
En
Q n1
Q n2
En
13.4164052631 13.4164046882 13.4164046613 13.4164046443 13.4164037680
8.9442688298 8.9442687851 8.9442688177 8.9442688167 8.9442688106
34.7832795967 34.7832795967 34.7832795967 34.7832795967 34.7832795967
13.4164051942 13.4164046000 13.4164046122 13.4164047256 13.4164044001
8.9442687468 8.9442687048 8.9442687383 8.9442687335 8.9442687196
34.7832796466 34.7832796466 34.7832796466 34.7832796466 34.7832796466
Table 4 Error estimates of numerical solution with various h when
s ¼ 0:5.
n
kgn k1
ke k1
t=2 t=4 t=6 t=8 t = 10
s ¼ 0:5
s ¼ 0:5
s ¼ 0:5
h ¼ 0:2
h ¼ 0:1
h ¼ 0:05
s ¼ 0:5 h ¼ 0:2
s ¼ 0:5 h ¼ 0:1
s ¼ 0:5 h ¼ 0:5
5.9460011e2 1.0835575e1 1.5004266e1 1.8617672e1 2.1546069e1
5.9530277e2 1.0734165e1 1.5178814e1 1.8605148e1 2.1532498e1
5.9525561e2 1.0744797e1 1.5171686e1 1.8611669e1 2.1531646e1
5.0365296e2 8.5067424e2 1.1212972e1 1.3735943e1 1.6765103e1
5.0351574e2 8.4983794e2 1.1256341e1 1.3637629e1 1.6660025e1
5.0367228e2 8.4978534e2 1.1250638e1 1.3631451e1 1.6653422e1
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From these results, the stability and convergence of the scheme are verified and it shows that our proposed algorithm is effective and reliable. 6. Conclusions In this paper, with the help of the Richardson extrapolation, we propose a new conservative Crank–Nicolson finite difference scheme for the SRLW equation with second order temporal accuracy and four order partial accuracy. The discrete conservative quantities are discussed in detail and a local truncation error is derived by the Taylor expansion. Moreover, the existence of numerical solution is proved by the Browder fixed point theorem and the detailed analysis for the unconditional stability and convergence of the proposed scheme are also presented by the energy method. Some numerical experiments also verify this convergence in the maximum norm, second order accurate in time and four order accurate in space. In particular, such effective difference scheme applied to certain more complicated PDEs will be reported in our future works. Acknowledgments The authors are very grateful to both reviewers for carefully reading this paper and their comments, and thank Dr. Cheng Wang (University of Massachusetts, Dartmouth) for his valuable discussion. References [1] C.E. Seyler, D.C. Fenstermacler, A symmetric regularized long wave equation, Phys. Fluids 27 (1) (1984) 4–7. [2] C.J. Amick, J.L. Bona, M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differ. Equ. 81 (1) (1989) 1–49. [3] 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. [4] D. Bhardwaj, R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl. 40 (2000) 1397–1404. [5] V.G. Makhankov, Dynamics of classical solitons (in non-integrable systems), Phys. Rep. 35 (1) (1978) 1–128. A review section of physics letters (section C). [6] A.A. Soliman, Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method, Math. Comput. Simul. 70 (2) (2005) 119–124. [7] P.A. Clarkson, New similarity reductions and Painleve analysis for the symmetric regularized long wave and modified Benjamin–Bona–Mahoney equations, J. Phys. A: Math. Gen. 22 (18) (1989) 3821–3848. [8] J.L. Bogolubsky, Some examples of inelastic soliton interaction, Comput. Phys. Commun. 13 (1) (1977) 149–155. [9] B. Guo, The spectral method for symmetric regularized wave equations, J. Comput. Math. 5 (4) (1987) 297–306. [10] J. Zheng, R. Zhang, B. Guo, The Fourier pseudospectral method for the SRLW equation, Appl. Math. Mech. 10 (9) (1989) 801–810. [11] Y. Shang, B. Guo, Analysis of chebyshev pseudospectral method for multi-dimentional generalized SRLW equations, Appl. Math. Mech. 24 (10) (2003) 1035–1048. [12] Y. Shang, B. Guo, S. Fang, Long time behavior of the dissipative generalized symmetric regularized long wave equations, J. Partial Differ. Equ. 15 (2002) 35–45. [13] S. Fang, B. Guo, H. Qiu, The existence of global attractors for a system of multi-dimensional symmetric regularized long wave equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 61–68. [14] B. Guo, Y. Shang, Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term, Acta Math. Appl. Sin. 19 (2) (2003) 191–204. [15] L. Kong, W. Zeng, R. Liu, A multisymplectic Fourier pseudo-spectral scheme for the SRLW equation and conservation laws, Chin. J. Comput. Phys. 23 (1) (2006) 25–31. [16] T. Wang, L. Zhang, F. Chen, Conservative schemes for the symmetric regularized long wave equations, Appl. Math. Comput. 190 (2007) 1063–1080. [17] J. Hu, B. Hu, Y. Xu, C-N difference schemes for dissipative symmetric regularized long wave equations with damping term, Math. Prob. Eng. 2011 (2011) (Article ID 651642), 16 pp. [18] Y. Xu, B. Hu, X. Xie, J. Hu, Mixed finite element analysis for dissipative SRLW equations with damping term, Appl. Math. Comput. 218 (2012) 4788– 4797. [19] 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 (6) (1995) 1839–1875. [20] K. Zheng, J. Hu, High-order conservative Crank–Nicolson scheme for regularized long wave equation, Adv. Differ. Equ. 2013 (2013) 287. [21] F. Zhang, M.P. Victor, V. Luis, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. Math. Comput. 71 (2–3) (1995) 165–177. [22] F.E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc. Sympos. Appl. Math. 17 (1965) 24–49. [23] Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Method, Inter. Acad. Publishers, Beijing, 1990.
Please cite this article in press as: J. Hu et al., Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.04.062