Numerical analysis of a multi-symplectic scheme for a strongly coupled Schrödinger system

Applied Mathematics and Computation 203 (2008) 413–431

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Numerical analysis of a multi-symplectic scheme for a strongly coupled Schrödinger system q Tingchun Wang *, Luming Zhang, Fangqi Chen College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

a r t i c l e

i n f o

Keywords: Coupled Schrödinger equations Multi-symplectic method Solvability Convergence

a b s t r a c t In this paper, we analyze a multi-symplectic scheme for a strongly coupled nonlinear Schrödinger equations. We derive a general box scheme which is equivalent to the multi-symplectic scheme by reduction method. Based on the general box scheme, we prove that the multi-symplectic scheme preserves not only the multi-symplectic structure of the equation but also conservation law of mass. In general, the multi-symplectic schemes are not conservative to energy in the nonlinear case, so it is difficult to obtain the estimates of numerical solutions in k  k1 norm. Hence proofs of convergence and stability are difficult for multi-symplectic schemes of nonlinear equations. A deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values, and second order convergent to the exact solutions in k  k2 norm. A fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction From the 1980s to now, the symplectic schemes were introduced and systematically developed for the Hamiltonian systems within the framework of symplectic geometry [13–16]. Numerical results show that symplectic schemes have superior performance, especially in long time simulations. Recently, Marsden et al. [17] and Bridges and Reich [18,19] proposed the concept of multi-symplectic PDEs and multi-symplectic schemes which can be viewed as the generalization of symplectic schemes. And from then on, the multi-symplectic methods have been proposed and investigated for some important Hamiltonian partial differential equations (HPDEs), such as Schrödinger equations, KdV equations, nonlinear Dirac equations, Klein–Gordon–Schrödinger equations, etc. [20–32]. In those papers, some basic results on the methods have been presented, the theoretical framework on generalizing symplectic integrators of Hamiltonian systems to multi-symplectic integrators of HPDEs is established. But up to now, strict analysis of solvability, stability and convergence of the multi-symplectic method is few. In this paper, we take a strongly coupled Schrödinger equation as an example to construct a multi-symplectic scheme, we prove that the scheme preserves the conservation of mass exactly and preserves the conservations of local and global energy approximatively, we also prove that the scheme is uniquely solvable, second order convergent to the exact solution and stable for initial values. In [7], Sonnier et al. proposed a two-level conservative scheme to simulate the strong coupling of the following Schrödinger equations:

q

This work is supported by the National Natural Science Foundation of China, No. 10572057. * Corresponding author. E-mail address: [email protected] (T. Wang).

0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.053

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

iut þ buxx þ ½a1 juj2 þ ða1 þ 2a2 Þjvj2 u þ cu þ Cv ¼ 0; 2

ð1:1Þ

2

ivt þ bvxx þ ½a1 jvj þ ða1 þ 2a2 Þjuj v þ cv þ Cu ¼ 0

ð1:2Þ

with initial conditions

uðx; 0Þ ¼ u0 ðxÞ;

vðx; 0Þ ¼ v0 ðxÞ

ð1:3Þ

and the asymptotic boundary conditions

uðx; tÞ; vðx; tÞ ! 0;

jxj ! 1;

ð1:4Þ

where, it can be seen from [7] and its references that, the linear coupling parameter C accounts for effects that arise from twisting of the fiber and elliptic deformation of the fiber. It is also referred to as linear birefringence or relative propagation constant. The term proportional to a1 , describes the self-focusing of a signal for pulses in birefringent media. The parameter b describes the group velocity dispersion, and ða1 þ 2a2 Þ is the cross-phase modulation, and defines the integrability of Eq. (1.2). Finally, the term c appears as constant ambient potential called normalized birefringence. The coupled nonlinear Schrödinger system has extensive application in many areas of physics, including nonlinear optics and plasma physics [1–4]. In [8–10], Ismail numerically studied the following system:

iut þ buxx þ ½juj2 þ ajvj2 u ¼ 0;

ð1:5Þ

2

ð1:6Þ ð1:7Þ

2

ivt þ bvxx þ ½jvj þ ajuj v ¼ 0; uðx; 0Þ ¼ u0 ðxÞ; vðx; 0Þ ¼ v0 ðxÞ;

where a is a constant, which describes the minimum approximation of the transformation of light wave, to describe the behaviors of Bose–Einstein condensates as well as optic pulse propagation. In [5,6], a multi-symplectic method is constructed for solving problems (1.5)–(1.7) with the asymptotic boundary conditions

uðx; tÞ; vðx; tÞ ! 0;

jxj ! 1:

ð1:8Þ

However, it is lack of proof of the conservation, the unique solvability, unconditional stability and the second-order convergence of the difference solution. It is known that, in practical computation, we just use a scheme to compute on a bounded computational domain. For example, we choose the interval ½xL ; xR  as a computational space domain in this paper, where xL ; xR are large enough such that the initial-boundary problem is consistent to the initial-asymptotic boundary problem. The purpose of this paper is to construct and numerically analyze a multi-symplectic scheme for problems (1.1)–(1.4). The system (1.1)–(1.4) has two standard conserved quantities: mass and energy. i.e., the mass conservative law

Z

QðtÞ ¼

xR

ðjuj2 þ jvj2 Þdx ¼ Q ð0Þ

ð1:9Þ

xL

and the energy conservative law

EðtÞ ¼

1 2

Z

xR

xL

bðu2x þ v2x Þ þ

a1 2

 vgdx ¼ Eð0Þ: ðjuj4 þ jvj4 Þ þ ða1 þ 2a2 Þðjuj2 jvj2 Þ þ cðjuj2 þ jvj2 Þ þ 2C  Refu

ð1:10Þ

Cover the domain ½xL ; xR   ½0; T by Xhs ¼ Xh  Xs , where Xh ¼ fxj ¼ jh; j ¼ 0; 1; 2; . . . ; Jg; Xs ¼ ft n ¼ ns; n ¼ 0; 1; . . . ; Ng, h ¼ ðxR  xL Þ=J and s ¼ T=N are spatial and temporal step sizes, respectively, xj1 ¼ 12 ðxj þ xj1 Þ; tn1 ¼ 12 ðtn þ tn1 Þ. Let fwnj g 2 2 and fvnj g be two net functions on Xhs . Introducing the following notations:

wnj1 ¼

wnj þ wnj1 2

2

n12

wj

¼

kwn k ¼

wn1 þ wnj j 2

2

;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðwn ; wn Þ;

n12

dt wj

wnj  wnj1

1 ; d2x wnj ¼ ðdx wnjþ1  dx wnj1 Þ; 2 2 h h J X wnj  wn1 j n n n ¼ ; ðw ; v Þ ¼ h wj1 vnj1 ;

dx wnj1 ¼

;

s

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 X u 4 kwn k4 ¼ th jwnj1 j4 ; j¼1

kdx wn k ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdx wn ; dx wn Þ;

2

j¼1

2

2

kwn k1 ¼ max jwnj1 j; j

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u J1 u X 1 n jw j1 ¼ th j ðdx wnjþ1 þ dx w2j1 Þj2 ; 2 2 2 j¼1

2

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

where vnj1 is the complex conjugate of vnj1 . In this paper, we always let C denote a general positive constant, which may have 2

2

different values in different occurrences. The remainder of this paper is arranged as follows. In Section 2, we construct a multi-symplectic scheme for coupled Schrödinger system (1.1)–(1.4), give its two equivalent schemes, discuss the properties of the multi-symplectic scheme. In Section 3, a deduction argument and the discrete energy analysis method [33] are used to prove the convergence and stability of the difference scheme. In Section 4, we prove that the multi-symplectic scheme is uniquely solvable by a fixed point theorem. 2. The multi-symplectic scheme The general multi-symplectic formation in one dimensional space variable is

Kzt þ Lzx ¼ rz SðzÞ;

ð2:1Þ

where K and L are n  nðn P 3Þ skew symmetry matrices, and z 2 Rn is a state variable, and S : Rn ! R is a smooth Hamiltonian function. The multi-symplectic conservation law

ot w þ ox j ¼ 0;

ð2:2Þ

where w ¼ dz ^ K dz; j ¼ z ^ L dz, holds for the multi-symplectic formation (2.1). This can be obtained by taking wedge with the variational formation of (2.1) and making use of dz ^ dz ¼ 0. Taking inner product with (2.1) by zt and zx , respectively, we obtain the local energy conservation law and the local momentum conservation law, respectively, i.e.,

ot E þ ox F ¼ 0;

ð2:3Þ

ot G þ ox I ¼ 0;

ð2:4Þ

where E ¼ SðzÞ  12 z> Lzx ; F ¼ 12 z> Lzt ; G ¼ SðzÞ  12 z> Kzt ; F ¼ 12 z> Kzx . Definition 1. A scheme is called multi-symplectic scheme if it satisfies the discete formation of multi-symplectic conservation law (2.2). If Euler mid-point schemes are adopted in both time and spatial directions, we get the following Preissmann scheme:

1

s

1 nþ1 nþ1 nþ1 Kðznþ1  znjþ1 Þ þ Lðzjþ12  zj 2 Þ ¼ rz Sðzjþ12 Þ: jþ12 2 h 2

ð2:5Þ

The Preissmann scheme (2.5) is a multi-symplectic scheme since it satisfies discrete multi-symplectic conservation law [18]. In this section, we propose a multi-symplectic scheme for problems (1.1)–(1.4) and discuss its multi-symplectic properties and conservative properties. 2.1. The derivation of the multi-symplectic scheme Let uðx; tÞ ¼ pðx; tÞ þ qðx; tÞi; vðx; tÞ ¼ lðx; tÞ þ fðx; tÞi, then (1.1) and (1.2) can be written as follows:

iðpt þ iqt Þ þ bðpxx þ iqxx Þ þ ½a1 ðp2 þ q2 Þ þ ða1 þ 2a2 Þðl2 þ f2 Þðp þ iqÞ þ cðp þ iqÞ þ Cðl þ ifÞ ¼ 0;

ð2:6Þ

iðlt þ ift Þ þ bðlxx þ ifxx Þ þ ½a1 ðl2 þ f2 Þ þ ða1 þ 2a2 Þðp2 þ q2 Þðl þ ifÞ þ cðl þ ifÞ þ Cðp þ iqÞ ¼ 0:

ð2:7Þ

Further, (2.6) and (2.7) can be written as the following equivalent form:

pt þ bqxx þ ½a1 ðp2 þ q2 Þ þ ða1 þ 2a2 Þðl2 þ f2 Þq þ cq þ Cf ¼ 0;

ð2:8Þ

2

2

2

2

ð2:9Þ

2

2

2

2

ð2:10Þ

2

2

2

2

ð2:11Þ

qt  bpxx  ½a1 ðp þ q Þ þ ða1 þ 2a2 Þðl þ f Þp  cp  Cl ¼ 0;

lt þ bfxx þ ½a1 ðl þ f Þ þ ða1 þ 2a2 Þðp þ q Þf þ cf þ Cq ¼ 0; ft  blxx  ½a1 ðp þ q Þ þ ða1 þ 2a2 Þðl þ f Þl  cl  Cp ¼ 0:

Introducing the canonical momenta px ¼ b; qx ¼ a; lx ¼ d; fx ¼ c; the multi-symplectic PDEs can be obtained as follows:

Kzt þ Lzx ¼ rz SðzÞ;

ð2:12Þ T

8

where the states variables z ¼ ðp; q; b; a; l; f; d; cÞ 2 R , and Hamiltonian

SðzÞ ¼

  1 1 1 1 2 2 bða2 þ b þ c2 þ d Þ þ a1 ðp2 þ q2 Þ2 þ a1 ðl2 þ f2 Þ2 þ ða1 þ 2a2 Þðp2 þ q2 Þðl2 þ f2 Þ 2 2 2 2 1 2 2 2 2 þ cðp þ q þ l þ f Þ þ Cðlp þ fqÞ: 2

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

So

3 ða1 ðp2 þ q2 Þ þ ða1 þ 2a2 Þðl2 þ f2 ÞÞp þ cp þ Cl 6 ða ðp2 þ q2 Þ þ ða þ 2a Þðl2 þ f2 ÞÞq þ cq þ Cf 7 1 2 7 6 1 7 6 7 6 bb 7 6 7 6 ba 7 rz SðzÞ ¼ 6 6 ða ðl2 þ f2 Þ þ ða þ 2a Þðp2 þ q2 ÞÞl þ cl þ Cp 7 1 2 7 6 1 7 6 6 ða1 ðl2 þ f2 Þ þ ða1 þ 2a2 Þðp2 þ q2 ÞÞf þ cf þ Cq 7 7 6 5 4 bd 2

bc and the pair of the skew symmetric matrices (which can be singular)

2

0 6 1 6 6 6 0 6 6 0 K¼6 6 0 6 6 6 0 6 4 0 0

1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0

3 0 07 7 7 07 7 07 7; 07 7 7 07 7 05 0

2

0 60 6 6 61 6 60 L ¼ b6 60 6 6 60 6 40 0

3 0 1 0 0 0 0 0 0 0 1 0 0 0 0 7 7 7 0 0 0 0 0 0 0 7 7 1 0 0 0 0 0 0 7 7: 0 0 0 0 0 1 0 7 7 7 0 0 0 0 0 0 1 7 7 0 0 0 1 0 0 0 5 0 0 0 0 1 0 0

Using the midpoint difference scheme to discretize the multi-symplectic coupled nonlinear Schrödinger system, it can be obtained the following Preissman scheme:

    nþ1 2  2  nþ1 2  nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 þ ða1 þ 2a2 Þ lj12 þ fj12 pj12 þ cpj12 þ Clj12 ; dt qj12  bdx bj12 ¼ a1 ðpj12 Þ2 þ qj12 2 2 2 2 2 2 2 2 2    2  nþ1 2  2  nþ1 2  nþ1 nþ12 nþ12 nþ12 nþ12 nþ12 nþ1 2 2 2 þ ða1 þ 2a2 Þ lj1 þ fj1 qj1 þ cqj1 þ Cfj12 ;  dt pj1  bdx aj1 ¼ a1 pj1 þ qj1 2 2 2 2 2 2 2 2 2            2 2 2 2 1 1 1 1 1 1 1 1 1 nþ nþ nþ nþ nþ nþ nþ nþ 2 þ ða1 þ 2a2 Þ pj12 þ qj12 lnþ þ clj12 þ Cpj12 ; dt fj12  bdx dj12 ¼ a1 lj12 þ fj12 j12 2 2 2 2 2 2 2 2            2 2 2 2 1 1 1 1 1 1 1 1 nþ nþ nþ nþ nþ nþ nþ nþ nþ1 þ ða1 þ 2a2 Þ pj12 þ qj12 fj12 þ cfj12 þ Cqj12 ;  dt lj12  bdx cj12 ¼ a1 lj12 þ fj12 2

nþ12 j12

dx p

nþ12 j12

dx q

nþ12 j12

dx l

2

2

2

2

2

ð2:15Þ ð2:16Þ

2

;

ð2:17Þ

nþ12 j12

;

ð2:18Þ

nþ12 j12

ð2:19Þ

¼a

¼d

;

nþ12

dx fj1 ¼ cj1 : 2

2

ð2:14Þ

nþ12 j12

¼b

nþ12

2

ð2:13Þ

ð2:20Þ

2

From ð2:13Þ  ð2:14Þ  i; ð2:15Þ  ð2:16Þ  i; ð2:18Þ þ ð2:17Þ  i; ð2:20Þ þ ð2:19Þ  i, we obtain

 1 2  1 nþ1 nþ1 nþ nþ1 nþ1 nþ1 2 nþ idt uj12 þ bdx xj12 þ a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 þ cuj12 þ Cvj12 ¼ 0; 2 2 2 2 2 2 2  1 2  1 nþ12 nþ12 nþ12 nþ12 nþ12 2 nþ2 nþ2 idt vj1 þ bdx -j1 þ a1 vj1 þ ða1 þ 2a2 Þ uj1 vj1 þ cvj1 þ Cuj1 ¼ 0; 2

2

nþ1

2

2

2

2

ð2:22Þ

2

nþ1

dx uj12 ¼ xj12 ; 2

ð2:21Þ

ð2:23Þ

2

nþ12

nþ12

dx vj1 ¼ -j1 ; 2

ð2:24Þ

2

nþ12

nþ12

nþ12

nþ12

nþ12

nþ12

where xj1 ¼ bj1 þ iaj1 ; -j1 ¼ dj1 þ icj1 . 2

2

2

2

2

2

Rewriting (2.21) and (2.23) as follows:

 1 2  1 nþ1 nþ1 nþ nþ1 nþ1 nþ1 2 nþ bdx xj12 ¼ idt uj12  a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12  cuj12  Cvj12 ; 2

nþ1

2

nþ1

xj12 ¼ dx uj12 : 2

2

2

2

2

2

ð2:25Þ

2

ð2:26Þ

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

417

From ð2:26Þ  b þ ð2:25Þ  2h, we obtain nþ12

bxj

nþ1

¼ bdx uj12  2

  1 2  1 h nþ1 nþ nþ1 nþ1 nþ1 2 nþ idt uj12  a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12  cuj12  Cvj12 : 2 2 2 2 2 2 2

ð2:27Þ

From ð2:26Þ  b  ð2:25Þ  2h, we obtain

 1 2  1 nþ1 nþ1 nþ1 nþ nþ1 nþ1 nþ1 2 nþ bxj12 ¼ bdx uj12 þ idt uj12 þ a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 þ cuj12 þ Cvj12 ; 2

2

2

2

2

2

2

i.e., nþ12

bxj

nþ1

¼ bdx ujþ12 þ 2

  1 2  1 h nþ1 nþ nþ1 nþ1 nþ1 2 nþ idt ujþ12 þ a1 ujþ12 þ ða1 þ 2a2 Þ vjþ12 ujþ12 þ cujþ12 þ Cvjþ12 : 2 2 2 2 2 2 2

ð2:28Þ

Subtracting (2.27) from (2.28), we obtain

1 2  1   1  nþ1  1  nþ1 2 1  nþ12 1  nþ1 nþ1 nþ1 nþ1 nþ1 nþ nþ idt uj1 þ ujþ12 þ bd2x uj 2 þ c uj12 þ ujþ12 þ C vj12 þ vjþ12 þ a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 2 2 2 2 2 2 2 2 2 2 2 2 2  1 2  1 1 nþ2 nþ12 2 nþ2 þ a1 ujþ1 þ ða1 þ 2a2 Þ vjþ1 ujþ1 ¼ 0: 2 2 2 2

ð2:29Þ

Similarly, we obtain

1 2  1   1  nþ1  1  nþ1 2 1  nþ12 1  nþ1 nþ1 nþ1 nþ1 nþ1 nþ nþ idt vj1 þ vjþ12 þ bd2x vj 2 þ c vj12 þ vjþ12 þ C uj12 þ ujþ12 þ a1 vj12 þ ða1 þ 2a2 Þ uj12 vj12 2 2 2 2 2 2 2 2 2 2 2 2 2  1 2  1 1 nþ12 2 nþ2 nþ2 þ a1 vjþ1 þ ða1 þ 2a2 Þ ujþ1 vjþ1 ¼ 0: 2 2 2 2

ð2:30Þ

From above discussion, we obtain the following proposition. Proposition 2.1. The three schemes, i.e., the scheme (2.13)–(2.20), the scheme (2.21)–(2.24) and the scheme (2.29) and (2.30), are equivalent. 2.2. The property of the multi-symplectic scheme The Preissman scheme (2.13)–(2.20) can be rewritten as the following matrix formation:

 1  nþ1   nþ1  1  nþ1 nþ1 K zjþ1  znjþ1 þ L zjþ12  zj 2 ¼ r nþ1 S zjþ12 : 2 2 h s 2 z 12 jþ

ð2:31Þ

2

Theorem 2.1. The Preissman scheme (2.31) has the following discretized multi-symplectic conservation:

 1  nþ1  1  nþ1 nþ1 2 wjþ1  wnjþ1 þ j  jj 2 ¼ 0; jþ1 2 2 h s

ð2:32Þ

cn ; u b nj ; u  nj ; i; jnj ¼ hL u  nj ; i, f c  nj g are any two solutions of the discretized variational equation associated with where wnj ¼ hK u unj g; fu j (2.31), h; i is standard Euclidean inner production. Remark 2.1. The discrete form of the local energy conservation law (2.3) and the local momentum conservation law (2.4) are not preserved exactly by the Preissman scheme (2.31). The residual of the local energy conservation law is denoted nþ1

Enþ1  Enjþ1 jþ1

2

s

Ejþ1 2 ¼

2

2

nþ12

þ

F nþ1 jþ1  F j h

;

where

Enjþ1 2

!   2  2 2  2  2 2 2  2  1 1 n n n n n n ¼ a1 pjþ1 þ qjþ1 þ l þ m a þ 2 a Þ p þ q þ ð 1 1 1 1 1 2 jþ2 jþ2 jþ2 jþ2 2 2 2 2   2  2  2  2  2  2  n n  b anjþ1 þ bjþ1 þ cnjþ1 þ djþ1  lnjþ1 þ mnjþ1 2 2 2 2 2 2     1  n 2  n 2  n 2  n 2 þ C lnjþ1 pnjþ1 þ fnjþ1 qnjþ1 ; þ c pjþ1 þ qjþ1 þ ljþ1 þ fjþ1 2 2 2 2 2 2 2 2 2      1 1 n 4 n 4 n 2 n 2 n 2 n 2 a1 ujþ1 þ vjþ1 þ ða1 þ 2a2 Þ ujþ1 vjþ1  b dx ujþ1 þ dx vjþ1 ¼ 2 2 2 2 2 2 2 2

ð2:33Þ

418

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

F njþ1 2

 2 2  n o 1 þ c unjþ1 þ vnjþ1 þ C  Re unjþ1 vnjþ1 ; 2 2 2 2 2 0 1 pnþ1  pnjþ1 qnþ1  qnjþ1 lnþ1  lnjþ1 nþ1 fnþ1  fnjþ1 jþ12 nþ12 jþ12 nþ12 jþ12 nþ12 jþ12 2A 2 2 2 2 @ : ¼ b bj þ aj þ dj þ cj

s

s

s

s

Next, we turn to the discrete mass conservation law firstly. Theorem 2.2. The Preissman scheme (2.31) possesses the discrete mass conservation law, i.e.,

Q nþ1 ¼ kunþ1 k2 þ kvnþ1 k2 ¼ Q n ¼    ¼ Q 0 :

ð2:34Þ

Proof. According to Proposition 2.1, it is sufficient to prove that Theorem 2.2 holds for the solution of the difference scheme nþ1

(2.21)–(2.24). Multiplying (2.21) by huj12 and summing up for j, we obtain 2

ih

J X

nþ1 nþ1

dt uj12 uj12 þ bh 2

j¼1

J X

2

nþ1 nþ1

dx xj12 uj12 þ ch 2

j¼1

2

J X

nþ1 nþ1

uj12 uj12 þ Ch 2

j¼1

J X

2

j¼1

nþ1 nþ1

vj12 uj12 þ h 2

2

J  X j¼1



2



2 

a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 uj12 ¼ 0; nþ1 2

nþ1 2

nþ1 nþ1 2

2

ð2:35Þ i.e.,

i

J J J1  1 2 1 2  1 2 X X X kunþ1 k2  kun k2 1 nþ1 nþ1 nþ1 nþ1 nþ nþ nþ þ bh dx xj12 uj12 þ ckunþ2 k2 þ Ch vj12 uj12 þ h a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 ¼ 0: 2s 2 2 2 2 2 2 2 j¼1 j¼1 j¼1

ð2:36Þ nþ12

Multiplying (2.23) by bhxj1 and summing up for j, we obtain 2

J J X X nþ1 nþ1 nþ12 2 bh dx uj12 xj12 ¼ 0: xj1  bh 2

j¼1

j¼1

2

ð2:37Þ

2

Adding (2.36) and (2.37), then taking the imaginary part of the result, we obtain

( ) ( ) J J X X kunþ1 k2  kun k2 nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 J þ Im bh dx xj12 uj12  bhj¼1 dx uj12 xj12 þ Im Ch vj12 uj12 ¼ 0: 2s 2 2 2 2 2 2 j¼1 j¼1

ð2:38Þ

Using the homogeneous boundary conditions, we obtain

( Im bh

J X

nþ1 nþ1

dx xj12 uj12  bh

j¼1

2

2

J X

nþ1

nþ1

dx uj12 xj12

j¼1

2

(

) ¼ Im

2

) J b X nþ12 nþ12 nþ1 nþ1 ðxj uj  xj12 uj12 Þ ¼ 0: 2 j¼1

ð2:39Þ

It follows from (2.38) and (2.39) that

( ) J X kunþ1 k2  kun k2 nþ12 nþ12 þ Im Ch vj1 uj1 ¼ 0: 2s 2 2 j¼1

ð2:40Þ

Similarly, we obtain

( ) J X kvnþ1 k2  kvn k2 nþ12 nþ12 þ Im Ch uj1 vj1 ¼ 0: 2s 2 2 j¼1

ð2:41Þ

Adding (2.40) and (2.41), we obtain

kvnþ1 k2  kvn k2 kunþ1 k2  kun k2 þ ¼ 0: 2s 2s

ð2:42Þ

Then (2.34) can be gotten from (2.42). This completes the proof of Theorem 2.2.  It can be seen evidently that Theorem 2.2 is consistent with the mass conservation law (1.9). It means that the mass conservation law (1.9) can be preserved exactly by our multi-symplectic scheme. In general, as a quadratic invariant the mass conservation law plays an important part in quantum physics, and this is just one of purposes to introduce multi-symplectic methods. The next result concerns the error estimation of the discrete total energy, and for the reason of the nonlinear terms in (1.1) and (1.2), the multi-symplectic scheme cannot preserve the discrete total energy conservation law exactly.

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

Theorem 2.3. Under the homogeneous boundary conditions, the multi-symplectic scheme (2.29) and (2.30) satisfies the following discrete global conservation law, i.e.,

" ! J X 1 1 1 nþ1 4 nþ1 4 nþ1 2 nþ1 2 nþ1 2 nþ1 2 a1 ðku k4 þ kv k4 Þ þ ða1 þ 2a2 Þh jujþ1 j jvjþ1 j  bðkdx u k þ kdx v k Þ þ cðkunþ1 j2 k þ kvnþ1 k2 Þ 2 2 2 2 2 j¼1 ( )# " ! J J X X 1 1 n 4 n 4 n 2 n 2 n 2 n 2 nþ1 nþ1  ujþ1 vjþ1 a1 ðku k4 þ kv k4 Þ þ ða1 þ 2a2 Þh jujþ1 j jvjþ1 j  bðkdx u k þ kdx v k Þ þ C  Re h 2 2 2 2 2 2 j¼1 j¼1 ( )#   J J  X X 1 s2 nþ12 2 nþ1 2 n 2 unjþ1 vnjþ1 ¼ a1 h þ cðkun j2 k þ kvn k2 Þ þ C  Re h dt uj1 uj1  uj1 2 2 2 2 2 2 8 j¼1 j¼1       J   X nþ1 2 2 s2 nþ1 2 2 n 2 n 2 nþ12 2 nþ1 2 n 2 nþ1 2 þ þ d : þ dt vj12 vnþ1 ð  v a þ 2 a Þh d v u  u u v  v 1 1 1 1 2 t t 1 1 1 1 1 j j j j2 j2 j2 j2 j2 2 2 2 4 2 j¼1 ð2:43Þ Proof. According to Proposition 2.1, it is sufficient to prove that Theorem 2.3 holds for the solution of the difference scheme (2.21)–(2.24). It follows from (2.23) and (2.24) that nþ1

nþ1

dt dx uj12 ¼ dt xj12 ; 2

ð2:44Þ

2

nþ12 j12

nþ12 j12

¼ dt -

dt dx v

ð2:45Þ

: nþ1

Multiplying (2.21) by hdt uj12 and summing up for j, we obtain 2

ih

J J J J X X X X nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 nþ12 2 dx xj12 dt uj12 þ ch uj12 dt uj12 þ Ch vj12 dt uj12 dt uj1 þ bh 2

j¼1

þh

2

j¼1

2

2

j¼1

2

j¼1

2

2

J  X j¼1

1 2 1 2  1 nþ nþ nþ nþ1 a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 dt uj12 ¼ 0: 2

2

2

ð2:46Þ

2

nþ1

Multiplying (2.44) by bhxj12 and summing up for j, we obtain 2

bh

J X

nþ1

nþ1

dt xj12 xj12 þ bh 2

j¼1

2

J X

nþ1

dt dx unj1 xj12 ¼ 0: 2

j¼1

ð2:47Þ

2

Adding (2.46) and (2.47), then taking the real part of the result, we obtain

( Re bh

J X

nþ12

dx xj1 dt uj1 þ bh 2

j¼1

(

þ Re h

nþ12

J  X j¼1

2

J X

nþ12

nþ12

dt dx uj1 xj1 þ ch 2

j¼1

2

J X

nþ12

nþ12

uj1 dt uj1 þ Ch 2

j¼1

2

J X j¼1

nþ12

nþ12

)

vj1 dt uj1 2

2

) ( ) J 1 2 1 2  1 X nþ nþ nþ nþ1 nþ1 nþ1 a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 dt uj12  Re bh dt xj12 xj12 ¼ 0: 2

2

2

2

j¼1

2

ð2:48Þ

2

For every terms of (2.48), according to the homogenous boundary conditions, we have

( Re bh ( Re h

J X

nþ1 2

j¼1 J  X j¼1

(

J X

nþ1

dx xj12 dt uj12 þ bh 2

J X

nþ1

)

dt dx unj1 xj12 2

j¼1

( ¼ Re b

2

1 2 1 2  1 nþ nþ nþ nþ1 a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 dt uj12 2

2

J X

2

2

J X nþ1 nþ1 nþ1 nþ1 ðxj 2 dt uj 2  xj12 dt uj12 Þ

) ¼ 0:

) ¼

J  1 2 1 2  2  1 X nþ nþ n 2 ; h a1 uj12 þ ða1 þ 2a2 Þ vj12 unþ1  u 1 1 j j2 2 2s j¼1 2 2

ð2:50Þ

)

1 ½cðkunþ1 k2  kun k2 Þ  bðkxnþ1 k2  kxn k2 Þ; 2 s j¼1 j¼1 ( ) ( ) J J   X X 1 1 1 nþ2 nþ2 nþ1 nþ1 nþ1 n n n nþ1 n Re Ch vj1 dt uj1 ¼ Re Ch vj1 uj1  vj1 uj1 þ vj1 uj1  vj1 uj1 : 2 2 2 2 2s 2 2 2 2 2 2 j¼1 j¼1 Re ch

nþ1

nþ1

uj12 dt uj12  bh 2

2

nþ1

nþ1

dt xj12 xj12 2

ð2:49Þ

j¼1

¼

ð2:51Þ

2

ð2:52Þ

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

Substituting (2.49)–(2.52) into (2.48), we obtain J  1 2 1 2  2  1 1 X nþ nþ n 2 ½cðkunþ1 k2  kun k2 Þ  bðkxnþ1 k2  kxn k2 Þ þ h a1 uj12 þ ða1 þ 2a2 Þ vj12 unþ1  u 1 1 j j2 2 2s 2s j¼1 2 2 ( ) J   X 1 nþ1 n n n nþ1 nþ1 n þ Re Ch vnþ1 ¼ 0: 1 u 1  vj1 u 1 þ vj1 u 1  vj1 u 1 j j j j j 2 2 2 2 2s 2 2 2 2 j¼1

ð2:53Þ

Similarly, we obtain J  1 2 1 2  2  i 1 X 1 h nþ nþ n 2 cðkvnþ1 k2  kvn k2 Þ  bðk-nþ1 k2  k-n k2 Þ þ h a1 vj12 þ ða1 þ 2a2 Þ uj12 vnþ1  v 1 1 j2 j2 2s 2s j¼1 2 2 ( ) J   X 1 nþ1 n n n nþ1 nþ1 n þ Re Ch unþ1 v  u v þ u v  u v ¼ 0: j12 j1 j12 j1 j12 j12 j12 j12 2s 2 2 j¼1

ð2:54Þ

Adding (2.53) and (2.54), we obtain

1 1 c½ðkunþ1 k2 þ kvnþ1 k2 Þ  ðkun k2 þ kvn k2 Þ  b½ðkxnþ1 k2 þ k-nþ1 k2 Þ  ðkxn k2  k-n k2 Þ 2s 2s J  J  1 2 1 2  2  1 X 1 2 1 2  1 X nþ nþ n 2 nþ2 nþ2 þ a1 uj12 þ ða1 þ 2a2 Þ vj12 unþ1  u h a v þ ð a þ 2 a Þ þ h 1 j1 1 2 uj1 j12 j12 2s j¼1 2s j¼1 2 2 2 2 ( )  J 2 2  1   X n n  vnþ1 vnþ1 unþ1  vnþ1 un þ vnj1 unþ1  vnj1 þ Re Ch 1  vj1 uj1 j12 j12 j12 j12 j12 2 2 j2 2 2s 2 j¼1 ( ) J   X 1 nþ1 n n n nþ1 nþ1 n unþ1 ¼ 0: þ Re Ch 1 v 1  uj1 v 1 þ uj1 v 1  uj1 v 1 j j j j j 2 2 2 2 2s 2 2 2 2 j¼1

ð2:55Þ

For the last four terms in left hand side of (2.55), we have J  1 2 1 2  2  1 X nþ nþ n 2 h a1 uj12 þ ða1 þ 2a2 Þ vj12 unþ1  u 1 1 j2 j2 2s j¼1 2 2 J  1 2 1 2  2  1 X nþ nþ n 2 a1 vj12 þ ða1 þ 2a2 Þ uj12 vnþ1  v þ h j12 j12 2s j¼1 2 2

1 a1 ½ðkunþ1 k44 þ kvnþ1 k44 Þ  ðkun k44 þ kvn k44 Þ 4s J   X 1 nþ1 2 nþ1 2 n 2 n 2 þ ða1 þ 2a2 Þh vj1 uj1  uj1 vj1 2 2 2 2 2s j¼1    J 2 2 2   X 1 nþ1 n 2 nþ1 nþ1 2 n 2 n  a1 h þ v  u  v v  v uj1  unj1 unþ1 j12 j12 j12 j12 j12 j12 2 2 8s j¼1 J  2  2  X 1 nþ1 n 2  ða1 þ 2a2 Þh  u vj1  vnj1 unþ1 1 1 j j2 2 2 2 4s j¼1 2  2 2  ; ð2:56Þ  unj1 vnþ1 þ unþ1  vnj1 j12 j12 2 2 ( ) ( ) J  J    X X 1 1 nþ1 nþ1 nþ1 n n n nþ1 nþ1 nþ1 n n n nþ1 nþ1 n n Re Ch vj1 uj1  vj1 uj1 þ vj1 uj1  vj1 uj1 uj1 vj1  uj1 vj1 þ uj1 vj1  uj1 vj1 þ Re Ch 2 2 2 2 2 2 2 2 2s 2s 2 2 2 2 2 2 2 2 j¼1 j¼1 ( ) J X 1 ðvnþ1 unþ1  vnj1 unj1 Þ : ð2:57Þ ¼ Re Ch j1 j1 ¼

s

j¼1

2

2

2

2

Then (2.43) can be gotten from (2.55)–(2.57). This completes the proof of Theorem 2.3.  According to Theorem 2.3, if we set the corresponding discrete total energy of the multi-symplectic scheme (2.29) and (2.30) at time t n as

! J   X 1 1 1 n 4 n 4 n 2 n 2 n 2 n 2 E ¼ a1 ðku k4 þ kv k4 Þ þ ða1 þ 2a2 Þh jujþ1 j jvjþ1 j  b kdx u k þ kdx v k þ cðkun j2 k þ kvn k2 Þ 2 2 2 2 2 j¼1 ( ) J X unjþ1 vnjþ1 ; þ C  Re h n

j¼1

2

2

ð2:58Þ

421

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

then

Enþ1  En ¼

J     2  ða þ 2a Þs2 a1 s2 X nþ12 2 nþ1 2 n 2 nþ1 2 n 2 1 2 þ  v h h dt uj1 uj1  uj1 þ dt vj12 vnþ1 j1 j1

8



2

2

2

j¼1

2

2

2

4

J  X

   2  nþ12 2 nþ1 2 n 2 nþ1 2 n 2  u dt vj1 uj1  uj1 þ dt vj12 unþ1 1 1 j j 2

j¼1

2

2

2

2

2

ð2:59Þ

and (2.58) can be considered as the discrete version of the total energy conservation law (1.9) at time tn , when applying the multi-symplectic scheme (2.29) and (2.30) to system (1.1) and (1.2). 3. The stability and convergence of the difference scheme In this section, we use an important inequality and a deduction argument to prove the second-order convergence of the difference solution. Then based on some priori estimates we prove the stability of the multi-symplectic scheme. For the difference solution of the scheme (2.11)–(2.14), we have the following priori estimates: Lemma 3.1. Suppose that u0 ðxÞ; v0 ðxÞ 2 L2 ½xL ; xR , then for n ¼ 0; 1; 2; . . . ; N, the following inequalities:

kun k 6 C;

kvn k 6 C

hold. Proof. It follows from (2.34) that

kun k þ kvn k 6

qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2ðkun k2 þ kvn k2 Þ ¼ 2Q n ¼    ¼ 2Q 0 ¼ 2ðku0 k2 þ kv0 k2 Þ ¼ 2ðkU 0 k2 þ kV 0 k2 Þ:

ð3:1Þ

It follows from (3.1) that there exists a positive constant C such that

kun k 6 C;

kvn k 6 C:

This completes the proof of Lemma 3.1. h In order to prove the convergence of the scheme (2.11)–(2.14), we need the following lemmas. Lemma 3.2. ([33] Gronwall’s inequality). Suppose that the discrete function fwn jn ¼ 0; 1; 2; . . . ; N; N s ¼ Tg satisfies the inequality

wn 6 A þ s

n X

Bl wl ;

l¼1

where A and Bl ðl ¼ 0; 1; 2; . . . ; NÞ are nonnegative constants. Then 2s

n

max jw j 6 Ae

N P l¼1

16n6N

where

Bl

;

s is sufficiently small, such that s  ðmax16l6N Bl Þ 6 12.

Lemma 3.3. ([33] Gronwall’s inequality). Suppose that the discrete function fwn jn ¼ 0; 1; 2; . . . ; N; N s ¼ Tg satisfies the inequality

wn  wn1 6 Aswn þ Bswn1 þ C n s; where A; B and C n ðl ¼ 0; 1; 2; . . . ; NÞ are nonnegative constants. Then

max jwn j 6 ðw0 þ

16n6N

where

N X

C l Þe2ðAþBÞT ;

l¼1

s is sufficiently small, such that ðA þ BÞs 6 N1 ; ðN > 1Þ. 2N 2

Lemma 3.4 [34]. Suppose that a > 0; b > 0; c > 0; b  4ac > 0 and az2 þ bz  c 6 0, then the following inequality:

z6

2c b

or

zP

b 2c  a b

holds. Lemma 3.5 (Sobolev’s estimate). For any discrete function funj jj ¼ 0; 1; . . . ; Jg on the finite interval ½xL ; xR , there is the inequality

kun k1 6 C 0

pffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kun k jun j1 þ kun k;

where C 0 is a constant independent of funj jj ¼ 0; 1; . . . ; Jg and step length h.

ð3:2Þ

422

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

Proof. Denote uj1 ¼ wj1 þ ivj1 , where wj1 is the real part of uj1 and vj1 is the imaginary part of it. For any m; s ¼ 1; 2; . . . ; J, 2 2 2 2 2 2 we have an identity

w2m1  w2s1 ¼ h 2

2

m 1  X

wjþ1 þ wj1 2

j¼s

 dx wjþ1 þ dx wj1 2

2

2

2

6 2kwk  jwj1 :

ð3:3Þ

If jwj1 j P a P 0 for all 1; 2; . . . ; J, then 2

2

2

kwk2 P Jha P la : So there exists such a value of ws that

1 jws1 j 6 pffi kwk: 2 l

ð3:4Þ

Taking this value of ws1 , we get, for j ¼ 0; 1; . . . ; J, 2

pffiffiffi 1 1 1 jwm1 j 6 pffi kwk þ 2kwk2 jwj21 2 l or

kwk 12 Þ: l

1

jwm1 j 6 2kwk2 ðkwj1 þ 2

ð3:5Þ

Similarly we obtain 1

jvm1 j 6 2kvk2 ðkvj1 þ 2

kvk 12 Þ: l

ð3:6Þ

It follows from (3.5) and (3.6) that there exists a positive constant C 0 independent of funj jj ¼ 0; 1; . . . ; Jg and step length h such that

kun k1 6 C 0

pffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kun k jun j1 þ kun k:

This completes the proof of Lemma 3.4.

ð3:7Þ 

Let x ¼ ux ; - ¼ vx , then (1.1) and (1.2) is equivalent to the following system of equations:

iut þ bxx þ ½a1 juj2 þ ða1 þ 2a2 Þjvj2 u þ cu þ Cv ¼ 0;

ð3:8Þ

x  ux ¼ 0; ivt þ b-x þ ½a1 jvj2 þ ða1 þ 2a2 Þjuj2 v þ cv þ Cu ¼ 0; -  vx ¼ 0:

ð3:10Þ

ð3:9Þ ð3:11Þ

For the convenience of readers, we set truncation errors of scheme as

  nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 2 nþ1 2 nþ1 ðe1 Þj12 ¼ idt U j12 þ bdx Uj12 þ a1 U j12 þ ða1 þ 2a2 Þ V j12 U j12 þ cU j12 þ CV j12 ; 2 2 2 2 2 2 2 2   nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 2 nþ12 2 nþ12 ðe2 Þj1 ¼ idt V j1 þ bdx Wj1 þ a1 V j1 þ ða1 þ 2a2 Þ U j1 V j1 þ cV j1 þ CU j1 ; 2

ðe3 Þ

nþ12 j12

ðe4 Þ

nþ12 j12

2

nþ12 j12

¼U

nþ12 j12

¼W

2

 dx U

nþ12 j12

 dx V

nþ12 j12

2

2

2

2

ð3:12Þ ð3:13Þ

2

ð3:14Þ

;

ð3:15Þ

;

where

U nj ¼ uðxj ; t n Þ;

V nj ¼ vðxj ; tn Þ;

Unj ¼ xðxj ; tn Þ;

Wnj ¼ -ðxj ; t n Þ:

~ nj ¼ Unj  xnj ; ~ nj ¼ Wnj  -nj . It can be verified by Taylor expansion that ~ nj ¼ U nj  unj ; v ~nj ¼ V nj  vnj ; x Let u nþ1

2

ðe1 Þj12 ¼ Oðh þ s2 Þ; 2

nþ1

2

ðe2 Þj12 ¼ Oðh þ s2 Þ; 2

nþ1

2

ðe3 Þj12 ¼ Oðh Þ; 2

nþ1

2

ðe4 Þj12 ¼ Oðh Þ; 2

2

i.e., the truncation errors of the multi-symplectic scheme (2.29) and (2.30) are Oðh þ s2 Þ. It should be pointed out that the scheme (2.21)–(2.24) can be also looked as the scheme for the Eqs. (3.8)–(3.11), and the method of construction of scheme is called the method of reduction of order [11,12]. Now we consider the convergence of the scheme (2.29) and (2.30).

423

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

psffiffi are small enough, then the solution of the difference problems (2.29) Theorem 3.1. Suppose that uðx; tÞ; vðx; tÞ 2 C 4;3 x;t , and h; s; h

2

and (2.30) converges to the solution of problems (1.1) and (1.2) with order Oðh þ s2 Þ in the j  j norm. Proof. According to Proposition 2.1, it is sufficient to prove that the solution of the difference scheme (2.21)–(2.24) converges to the solution of problems (3.8)–(3.11). Subtracting (2.21)–(2.24) from (3.12)–(3.15), respectively, we obtain the error equations as follows: nþ1

nþ1

nþ1

nþ1

nþ1

nþ1

nþ1

~ 12 þ cu ~ 12 þ bdx x ~ 12 þ Cv ~ 12 þ G 12  g 12 ; ðe1 Þj12 ¼ idt u j j j j j j 2

2

1

2

1

2

1

2

1

2

1

ð3:16Þ

2

1

1

nþ nþ nþ nþ nþ nþ ~ nþ12 þ cv ~ 12 þ bdx ~ 12 þ Cu ~ 12 þ F 12  f 1 2 ; ðe2 Þj12 ¼ idt v j j j j j j 2

ðe3 Þ

nþ12 j12

2

~ ¼x

nþ12 j12

2

nþ12 j12

~  dx u

2

2

2

ð3:17Þ

2

;

ð3:18Þ

~ 12  dx v ~ 12 ; ðe4 Þj12 ¼ j j

ð3:19Þ

~ 0j ¼ 0; u

ð3:20Þ

nþ1

nþ1

2

nþ1

2

2

~0j ¼ 0; v

where

  nþ1 nþ1 2 nþ1 2 nþ1 Gj12 ¼ a1 U j12 þ ða1 þ 2a2 Þ V j12 U j12 ; 2 2 2 2   nþ12 nþ12 2 nþ12 2 nþ12 F j1 ¼ a1 V j1 þ ða1 þ 2a2 Þ U j1 V j1 ; 2

2

2

 1 2  1 nþ1 nþ nþ1 2 nþ g j12 ¼ a1 uj12 þ ða1 þ 2a2 Þ vj12 uj12 ; 2 2 2 2  1 2  1 nþ12 nþ12 2 nþ2 nþ2 fj1 ¼ a1 vj1 þ ða1 þ 2a2 Þ uj1 vj1 :

2

2

2

2

2

1

~ nþ12 and summing up for j, we obtain Multiplying (3.16) by 2hu j 2

2ih

X

~ 1u ~ 1 þ 2bh dt u j j 2

j

¼ 2h

J X

nþ12 nþ12 2

J X

nþ1 nþ1

~ 12 u ~ 12 þ 2ch dx x j j 2

j¼1

2

J X

1

1

~nþ12 u ~ nþ12 þ 2Ch u j j

j¼1

2

2

J X

1

1

~nþ12 u ~nþ12 þ 2h v j j 2

j¼1

2

J   nþ1 X nþ1 nþ1 ~ 12 Gj12  g j12 u j j¼1

2

2

2

nþ1 nþ1

~ 12 ; ðe1 Þj12 u j 2

j¼1

2

i.e.,

i

~nþ1 k2  ku ~ n k2 ku

s

J X

þ 2bh

nþ1 nþ1

1

~ 12 u ~ nþ2 k2 þ 2Ch ~ 12 þ 2cku dx x j j 2

j¼1

2

J X

nþ1 nþ1

~ 12 u ~ 12 þ 2h v j j

j¼1

2

2

J  nþ1 X  nþ1 X nþ1 nþ1 nþ12 ~ 12 ¼ 2h ~ 1: Gj12  g j12 u ðe1 Þj12 u j j 2

j

2

2

j¼1

2

2

ð3:21Þ 1

~ nþ12 and summing up for j, we obtain Multiplying (3.18) by 2bhx j 2

J J J X X X 1 1 nþ1 nþ12 ~ nþ12 2 ~ nþ12 ¼ 2bh ~ 1: ~nþ12 x 2bh dx u ðe3 Þj12 x xj1  2bh j j j 2

j¼1

2

j¼1

2

2

j¼1

ð3:22Þ

2

Adding (3.21) and (3.22), then taking the imaginary part of the result, we obtain

~ nþ1 k2  ku ~ n k2 ku

s

( ) ( ) ) J  J J   nþ1 X X X nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 2 ~ ~ ~ ~ ~ ~ ~ þ Im 2Ch þ Im 2bh dx xj1 uj1  dx uj1 xj1 Gj1  g j1 uj1 vj1 uj1 þ Im 2h (

2

j¼1

( ¼ Im 2h

2

2

2

j¼1

)

2

2

J J X X nþ1 nþ12 nþ1 nþ12 ~ 1 : ~ 1 þ 2bh ðe1 Þj12 u ðe3 Þj12 x j j 2

j¼1

2

2

j¼1

j¼1

2

2

2

ð3:23Þ

2

According to the homogeneous boundary conditions, we obtain

( Im 2bh

 J  X 1 nþ1 1 1 ~ nþ12 u ~ nþ12 ~ 12  dx u ~ nþ12 x dx x j j j j 2

j¼1

2

2

)

( ¼ Im 2b

2

J  X

nþ1 nþ12

~ j 2u ~j x

nþ1 nþ1

~ j12 u ~ j12 x

)

¼ 0:

ð3:24Þ

j¼1

Then (3.23) can be reduced as

~nþ1 k2  ku ~ n k2 ku

s ( ¼ Im 2h

( þ Im 2Ch

J X j¼1

J X

nþ1 nþ1

~ 12 þ 2bh ðe1 Þj12 u j

j¼1

2

2

1

1

~nþ12 u ~ nþ12 v j j 2

J X j¼1

)

( þ Im 2h

2

nþ1

nþ1

)

~ 12 : ðe3 Þj12 x j 2

2

J  X j¼1

 nþ1 nþ1 nþ1 ~ 12 Gj12  g j12 u j 2

2

)

2

ð3:25Þ

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

Similarly we obtain

(

~nþ1 k2  kv ~n k2 kv

þ Im 2Ch

s ( ¼ Im 2h

J X

nþ12 nþ12

~ 1v ~ 1 u j j

j¼1 J X

nþ1 nþ1

~ 12 þ 2bh ðe2 Þj12 v j 2

j¼1

2

2

2

J X

)

(

J X nþ1 nþ1 nþ12 ~ 1 ðF j12  fj1 2 Þv þ Im 2h j

nþ1

nþ1

s

2

j¼1

¼ Im 2h

nþ1 nþ1

~ 12 þ 2bh ðe1 Þj12 u j 2

j¼1

2

2

ð3:26Þ

(

J X 1 nþ1 nþ1 ~ nþ12 ðGj12  g j12 Þu j 2

j¼1 J X

2

2

~ nþ1 k2 þ kv ~nþ1 k2  ku ~ n k2  kv ~n k2 Þ þ Im 2h ðku (

2

j¼1

~ 12 : ðe4 Þj12 j

Adding (3.25) and (3.26), we obtain

1

)

)

J X

nþ1

nþ1

~ 12 þ2h ðe3 Þj12 x j

j¼1

2

J X

2

2

)

( þ Im 2h

2

nþ1 nþ1

~ 12 þ 2bh ðe2 Þj12 v j

j¼1

2

2

J X nþ1 nþ1 nþ12 ~ 1 ðF j12  fj1 2 Þv j j¼1

J X

nþ1

2

nþ1

2

)

2

)

~ 12 : ðe4 Þj12 j

j¼1

2

ð3:27Þ

2

Using the assumptions of the theorem and Lemma 3.1, we obtain 2

2

krn k1 6 C r ðs2 þ h Þ; krn k1 6 C r ðs2 þ h Þ; kun k 6 C uv ; kun k1 6 C uv ; kvn k 6 C uv ; kvn k1 6 C uv ; kU n k 6 C uv ; kV n k 6 C uv ; 0 6 ns 6 T;

ð3:28Þ

where C r ; C r and C uv are three positive constants. It follows from initial conditions that

~ 0 k ¼ 0; ku

~0 k ¼ 0; kv

ku0 k1 6 C uv ;

kv0 k1 6 C uv :

ð3:29Þ

It follows from Lemma 3.5 that

rffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n 2 ~ n j1 þ ku ~ n k 6 C 0 ku ~n k 6 C 0 ~ k þ ju ~ n k; ~ n k ju ~n k ku þ 1ku ku h h rffiffiffiffiffiffiffiffiffiffiffiffi 2 ~n k1 6 C 0 ~n k; 0 6 ns 6 T kv þ 1kv h

~ n k1 6 C 0 ku

0 6 ns 6 T;

ð3:30Þ ð3:31Þ

and

rffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ n k; 0 6 ns 6 T; þ 1ku ku k1 6 kU k1 þ ku k1 6 C uv þ C 0 h rffiffiffiffiffiffiffiffiffiffiffiffi 2 ~n k; 0 6 ns 6 T: ~n k1 6 C uv þ C 0 þ 1kv kvn k1 6 kV n k1 þ kv h n

n

~n

ð3:32Þ ð3:33Þ

For the second and the third terms of (3.27), we have

( ) J X nþ12 nþ12 nþ12 ~ ðGj1  g j1 Þuj1 Im 2h 2 2 2 j¼1 ( )    J  1 2  1  1 X nþ12 nþ2 nþ2 nþ12 2 nþ12 2 nþ12 2 nþ2 ~ ¼ Im 2h a1 U j1 þ ða1 þ 2a2 Þ V j1 U j1  a1 uj1 þ ða1 þ 2a2 Þ vj1 uj1 uj1 2 2 2 2 2 2 2 ( j¼1  J J  1 2 1 2  1 1 X X nþ nþ nþ nþ nþ1 nþ1 nþ1 nþ1 ¼ Im 2h a1 uj12 þ ða1 þ 2a2 Þ vj12 u~ j12 u~ j12 þ 2h a1 u~ j12 U j12 þ a1 uj12 u~ j12 2 2 2 2 2 2 2 2 j¼1 j¼1   nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 ~ 1 V 1 þ ða1 þ 2a2 Þv 1 v ~ ~ 1 U j1 u þða1 þ 2a2 Þv j2 j2 j2 j12 j2 2 ( )  J  X 1 1 1 nþ1 nþ nþ nþ nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 2 2 2 2 ¼ Im 2h a1 u~ j1 U j1 þ a1 uj1 u~ j1 þ ða1 þ 2a2 Þv~j1 V j1 þ ða1 þ 2a2 Þvj1 v~j1 U j1 u~ j1 2 2 2 2 2 2 2 2 2 2 j¼1 1

1

 ~ n k2 þ kv ~n k2 Þku ~ n k2 þ ½C 1 þ C 2 h2 ðku ~ nþ1 k2 þ kv ~nþ1 k2 Þku ~ nþ1 k2 ; 6 ½C þ C 2 h 2 ðku ( 1 ) J  i  nþ1 h X 1 1 nþ1 nþ1 ~ n k2 þ kv ~n k2 Þ ku ~ n k2 þ ½C 3 þ C 4 h2 ðku ~nþ1 k2 þ kv ~nþ1 k2 Þku ~ nþ1 k2 ; ~ 12 6 C 3 þ C 4 h2 ðku F j12  fj1 2 v Im 2h j 2 2 2 j¼1

ð3:34Þ ð3:35Þ

where C l ; l ¼ 1; 2; 3; 4 are four positive constants independent of n. For the terms in the right-hand side of (3.27), we have

( ) J J J J X X X X nþ1 nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 2 2 n 2 ~ ~ 1 þ 2bh ~ ~ ðe1 Þj12 u ðe Þ x þ 2h ðe Þ þ 2bh ðe Þ v 6 C 5 ðken1 k þ kenþ1 Im 2h 3 j1 2 j1 j1 4 j1 1 1 1 k þ ke2 k j j j 2 2 2 2 2 2 2 2 j¼1 j¼1 j¼1 j¼1 1

1

2 n 2 nþ1 2 n 2 nþ1 2 ~ nþ2 k2 þ k~n 2 ~ nþ1 k2 þ kv ~ n k2 þ kv ~nþ1 k2 þ kx ~ nþ2 k2 Þ; þ kenþ1 2 k þ ke3 k þ ke3 k þ ke4 k þ ke4 k Þ þ C 6 ðku k þ ku

ð3:36Þ where C 5 and C 6 are two positive constants independent of n.

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

Subtracting (3.21) from (3.22), then taking the real part of the result, we obtain

( 1

1

~ nþ2 k2  2cku ~ nþ2 k2  Re 2bh 2bkx (  Re 2h

J X

1

1

1

nþ nþ ~ nþ12 ðGj12  g j12 Þu j

j¼1

2

2

)

J X 1 nþ1 nþ1 nþ12 ~ nþ12 u ~ 1Þ ~ 12 þ dx u ~ 12 x ðdx x j j j j j¼1

2

(

¼ Re 2bh

2

2

2

J X

nþ1

)

2

j¼1

J X

nþ1 nþ1

nþ1 2

J X

nþ1 nþ1

)

~ 12 u ~ 12 v j j

j¼1

~ 12  2h ðe3 Þj12 x j 2

(  Re 2Ch

2

2

)

~ 12 : ðe1 Þj12 u j 2

j¼1

ð3:37Þ

2

According to homogeneous boundary conditions, we obtain

( Re 2bh

( ) ) J  J  X X 1 nþ1 1 1 1 nþ1 nþ12 nþ12 2~ 2 ~ nþ12 u ~ nþ12 ~ nþ ~ ~ 12 þ dx u ~nþ12 x ~ ¼ Re 2b ¼ 0: u u dx x x  x j j j1 j1 j j j j j¼1

2

2

2

2

ð3:38Þ

j¼1

By the similar discussion of (3.34), we obtain 1

1

2 n 2 nþ1 2 2 ~ nþ2 k2 6 C 7 ðken1 k2 þ kenþ1 ~n 2 ~n 2 ~ n 2 2bkx 1 k þ ke3 k þ ke3 k Þ þ ½C 8 þ C 9 h ðku k þ kv k Þku k 1

1

~ nþ2 k2 ; ~ nþ1 k2 þ kv ~nþ1 k2 Þku ~ nþ1 k2 þ bkx þ ½C 8 þ C 9 h 2 ðku i.e., 1

1

~ nþ2 k2 6 C 7 ðku ~n k2 þ ku ~ nþ1 k2 þ kv ~ n k2 þ kv ~nþ1 k2 Þ þ ½C 8 þ C 9 h 2 ðku ~ n k2 þ kv ~n k2 Þku ~ n k2 bkx 1

~ nþ1 k2 þ kv ~nþ1 k2 Þku ~ nþ1 k2 ; þ ½C 8 þ C 9 h 2 ðku

ð3:39Þ

where C 7 ; C 8 and C 9 are three positive constants independent of n. Similarly we obtain 1

1

~ nþ2 k2 6 C 10 ðku ~ n k2 þ ku ~ nþ1 k2 þ kv ~n k2 þ kv ~nþ1 k2 Þ þ ½C 11 þ C 12 h 2 ðku ~ n k2 þ kv ~n k2 Þkv ~ n k2 bk1

~ nþ1 k2 þ kv ~nþ1 k2 Þkv ~nþ1 k2 ; þ ½C 11 þ C 12 h 2 ðku

ð3:40Þ

where C 10 ; C 11 and C 12 are three positive constants independent of n. Substituting (3.39) and (3.40) into (3.36), we obtain

( ) J J J J X X X X nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 nþ12 ~ ~ ~ ~ ðe1 Þj1 uj1 þ 2bh ðe3 Þj1 xj1 þ2h ðe2 Þj1 vj1 þ 2bh ðe4 Þj1 -j1 Im 2h 2 2 2 2 2 2 2 2 j¼1 j¼1 j¼1 j¼1 2 n 2 nþ1 2 n 2 nþ1 2 n 2 nþ1 2 6 C 13 ðken1 k2 þ kenþ1 1 k þ ke2 k þ ke2 k þ ke3 k þ ke3 k þ ke4 k þ ke4 k Þ h i h i 1 1  ~ n k2 þ kv ~n k2 Þ ðku ~ n k2 þ kv ~n k2 Þ þ C 14 þ C 15 h2 ðku ~ nþ1 k2 þ kv ~nþ1 k2 Þ ðku ~ nþ1 k2 þ kv ~nþ1 k2 Þ; þ C 14 þ C 15 h 2 ðku

ð3:41Þ

where C 13 ; C 14 and C 15 are three positive constants independent of n. Substituting (3.34)–(3.41) into (3.27), we obtain

1

s

2 n 2 nþ1 2 n 2 nþ1 2 n 2 ~nþ1 k2 þ kv ~nþ1 k2  ku ~ n k2  kv ~n k2 Þ 6 C 16 ðken1 k2 þ kenþ1 ðku 1 k þ ke2 k þ ke2 k þ ke3 k þ ke3 k þ ke4 k h i 12 2 ~n 2 ~n 2 ~n 2 ~n 2 þ kenþ1 4 k Þ þ C 17 þ C 18 h ðku k þ kv k Þ ðku k þ kv k Þ h i 1 ~ nþ1 k2 þ kv ~nþ1 k2 Þ ðku ~nþ1 k2 þ kv ~nþ1 k2 Þ; þ C 17 þ C 18 h 2 ðku

ð3:42Þ

where C 16 ; C 17 and C 18 are three positive constants independent of n. ~ n k2 þ kv ~n k2 , we obtain Let W n ¼ ku 1

1

2

ð1  C 17 s  C 18 sh 2 W nþ1 ÞW nþ1 6 ð1 þ C 17 s þ C 18 sh 2 W n ÞW n þ C 19 s½Oðh þ s2 Þ2 ;

ð3:43Þ

where C 19 is a positive constant independent of n. 1 Let Y n ¼ ð1  C 17 s  C 18 sh 2 W n ÞW n , then it follows from (3.43) that 1

Y nþ1 6

1 þ C 17 s þ C 18 sh 2 W n 1

1  C 17 s  C 18 sh 2 W n

2

Y n þ C 19 s½Oðh þ s2 Þ2 :

ð3:44Þ

It follows from Lemma 3.4 and (3.44) that 1

W nþ1 6

2 1 þ C 17 s þ C 18 sh 2 W n n 2 ½ Y þ C 19 s½Oðh þ s2 Þ2 1  C 17 s 1  C 17 s  C 18 sh12 W n

ð3:45Þ

426

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

or

W nþ1 P

" # pffiffiffi 1 2 1 þ C 17 s þ C 18 sh 2 W n n hð1  C 17 sÞ 2 2 2  Y þ C s ðOðh þ s ÞÞ : 19 C 18 s 1  C 17 s 1  C 17 s  C 18 sh1 W n

ð3:46Þ

If s; h are small enough such that 1  C 17 s > 12 then for n ¼ 0, and psffiffih is small enough, we have W 0 ¼ 0, Y 0 ¼ 0 and

W1 P

pffiffiffi h 2  4C 19 sðOðh þ s2 ÞÞ2 > 8C 2uv ; 2C 18 s

ð3:47Þ

this is a contradiction to

W n ¼ ken k2 þ kgn k2 ¼ kun  U n k2 þ kvn  V n k2 6 8C 2uv :

ð3:48Þ

Hence (3.45) holds. It follows from (3.45) that 2

W 1 6 4C 19 sðs2 þ h Þ2 ;

ð3:49Þ

Let C 20 is a large constant independent of n such that

C 19 2 ½ð1 þ C 20 sÞl  1sðs2 þ h Þ2 ; C 20 C 19 C 20 T 2 2 Wl 6 4 e ðs þ h Þ2 C 20

Yl 6

ð3:50Þ ð3:51Þ

for l 6 n. Then for n þ 1, we have 2

Y nþ1 6 ð1 þ C 20 sÞY n þ C 19 sðs2 þ h Þ2 C 19 2 6 ½ð1 þ C 20 sÞnþ1  ð1 þ C 20 sÞ þ C 20 sðs2 þ h Þ2 C 20 C 19 2 6 ½ð1 þ C 20 sÞnþ1  1sðs2 þ h Þ2 C 20 C 19 C20 T 2 2 6 e ðs þ h Þ2 ; C 20

ð3:52Þ

2

W nþ1 6 4½Y n þ C 19 sðs2 þ h Þ2  C 19 2 ½ð1 þ C 20 sÞnþ1  1 þ C 20 sðs2 þ h Þ2 64 C 20 C 19 C 20 T 2 2 64 e ðs þ h Þ2 : C 20

ð3:53Þ

It follows from (3.52) and (3.53) that 2

ken k 6 C c ðh þ s2 Þ;

2

kgn k 6 C c ðh þ s2 Þ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C c  2 CC 19 eC 20 T is a constants independent of n. This completes the proof of Theorem 3.1. h 20

ð3:54Þ

psffiffi Lemma 3.6. Suppose that u0 ðxÞ; v0 ðxÞ 2 L2 ½xL ; xR ; uðx; tÞ; vðx; tÞ 2 C 4;3 x;t , if h; s; h are small enough, then for n ¼ 0; 1; 2;    ; N, the following inequalities:

kun k1 6 C;

kvn k1 6 C

ð3:55Þ

hold. Proof. It follows from Theorem 3.1 that

rffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ n k 6 C uv þ Ch1=2 ðh2 þ s2 Þ 6 C; þ 1ku h rffiffiffiffiffiffiffiffiffiffiffiffi 2 ~n k 6 C uv þ Ch1=2 ðh2 þ s2 Þ 6 C; þ 1kv kvn k1 6 C uv þ C 0 h

kun k1 6 C uv þ C 0

This completes the proof of Lemma 3.6.

0 6 ns 6 T;

ð3:56Þ

0 6 ns 6 T:

ð3:57Þ

h

Based on the estimates (3.56) and (3.57) of numerical solutions, we can prove the stability of the multi-symplectic scheme, i.e., Theorem 3.2. Suppose That the conditions of Theorem 3.1 are satisfied, then the solutions of the difference problems (2.11)–(2.14) are stable for the initial values.

T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

427

4. The unique solvability of the difference scheme In order to analyze the solvability of the multi-symplectic scheme, we should introduce the following fixed point theorem on finite dimensional space. Define a mapping T k : H 7! H, i.e.,

T k ðzÞ ¼ u 2 H;

z 2 H;

where H is a finite dimensional Euclidean space, and k 2 ½0; 1. Lemma 4.1 [33]. If the following nonlinear system of equations:

u ¼ T k ðuÞ

ð4:1Þ

defined on a finite dimensional Euclidean space H satisfies the following conditions: (1) The functions of T k ðuÞ are continuous for any u 2 H and 0 6 k 6 1. (2) As k ¼ 0, For any u 2 H, there is a fixed point u0 2 H, such that

T k ðu0 Þ ¼ u0 for any u 2 H. (3) All possible solutions of the system (4.1) are uniformly bounded with respect to the parameter k 2 ½0; 1. Then the nonlinear system of Eq. (4.1) has at least one solution u 2 H for any k 2 ½0; 1 and hence for k ¼ 1, i.e., the nonlinear system of equations

u ¼ T 1 ðuÞ

ð4:2Þ

has at least one solution u 2 H. Now we discuss the solvability of the difference system (2.29) and (2.30), i.e., Theorem 4.1. For any fixed positive integers J; N, the difference scheme (2.29) and (2.30) has at least one solution, i.e., if ðun ; vn Þ satisfies the scheme (2.29) and (2.30), then there exists ðunþ1 ; vnþ1 Þ also satisfies the scheme (2.29) and (2.30).  Þ ¼ T k ðh; gÞ as h; g Proof. Denote H ¼ fðh; gÞjðhl ; gj Þ; l; j ¼ 0; 1; 2; . . . ; J; h0 ¼ g0 ¼ hJ ¼ gJ ¼ 0g. Suppose ðh; gÞ 2 H, we define ð follows:

i  i 1 1  j1 þ g  jþ1 Þ ðh 1 þ h 1 Þ  ðunj1 þ unjþ1 Þ þ sbd2x hj þ cðhj1 þ hjþ1 Þ þ Cðg 2 2 2 2 2 2 2 j2  jþ2 2 2 2  2 s 2 2 2 þ k ða1 hj1 þ ða1 þ 2a2 Þ gj1 Þhj1 þ ða1 hjþ1 þ ða1 þ 2a2 Þ gjþ1 Þhjþ1 ¼ 0; 2 2 2 2 2 2 2  i   1   i 1 g 1 þ g 1  ðvnj1 þ vnjþ1 Þ þ sbd2x g j þ c g j12 þ g jþ12 þ C hj12 þ hjþ12 2 2 2 j2  jþ2 2 2  2  2 2  2 2  s þk a1 gj12 þ ða1 þ 2a2 Þ hj12 gj12 þ a1 gjþ12 þ ða1 þ 2a2 Þ hjþ12 gjþ12 ¼ 0; 2

ð4:3Þ

ð4:4Þ

where 0 6 k 6 1. Now we prove that the mapping T k satisfies the three conditions of Lemma 4.1 with three steps, and then Theorem 4.1 can be proved.  ð1Þ Þ ¼ T k ðhð1Þ ; gð1Þ Þ, ð  ð2Þ Þ ¼ T k ðhð2Þ ; gð2Þ Þ, we have Step 1. Suppose ð hð1Þ ; g hð2Þ ; g

 i   1 i  ð1Þ 1      h j1 þ hð1Þ jþ1  unj1 þ unjþ1 þ sbd2x hð1Þ j þ c hð1Þ j1 þ hð1Þ jþ1 þ Cðgð1Þ j1 þ gð1Þ jþ1 Þ 2 2 2 2 2 2 2 2 2 2 2 2      s ð1Þ 2 ð1Þ 2 ð1Þ ð1Þ 2 ð1Þ 2 ð1Þ þk a1 hj1 þ ða1 þ 2a2 Þ gj1 hj1 þ a1 hjþ1 þ ða1 þ 2a2 Þ gjþ1 hjþ1 ¼ 0; 2 2 2 2 2 2 2  i    1   i  ð1Þ 1  2 ð1Þ n n     g j12 þ gð1Þ jþ12  vj1 þ vjþ1 þ sbdx gð1Þ j þ c gð1Þ j12 þ gð1Þ jþ12 þ C h j12 þ hð1Þ jþ12 2 2 2 2 2    2   s ð1Þ 2 ð1Þ 2 ð1Þ 2 ð1Þ 2 ð1Þ þk a1 gj1 þ ða1 þ 2a2 Þ hj1 gj12 þ a1 gjþ1 þ ða1 þ 2a2 Þ hjþ1 gjþ1 ¼ 0; 2 2 2 2 2 2

ð4:5Þ

ð4:6Þ

and

 i   1   i  ð2Þ 1      h j1 þ hð2Þ jþ1  unj1 þ unjþ1 þ sbd2x hð2Þ j þ c hð2Þ j1 þ hð2Þ jþ1 þ C gð2Þ j1 þ gð2Þ jþ1 2 2 2 2 2 2 2 2 2 2 2    2   s ð2Þ 2 ð2Þ 2 ð2Þ ð2Þ 2 ð2Þ 2 ð2Þ þk a1 hj1 þ ða1 þ 2a2 Þ gj1 hj1 þ a1 hjþ1 þ ða1 þ 2a2 Þ gjþ1 hjþ1 ¼ 0; 2 2 2 2 2 2 2

ð4:7Þ

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

 i   1   i  ð2Þ 1  g j12 þ gð2Þ jþ12  vnj1 þ vnjþ1 þ sbd2x gð2Þ j þ c gð2Þ j12 þ gð2Þ jþ12 þ C hð2Þ j12 þ hð2Þ jþ12 2 2 2 2 2 2   s ð2Þ 2 ð2Þ 2 ð2Þ 2 ð2Þ 2 ð2Þ þ k ða1 gj1 þ ða1 þ 2a2 Þ hj1 Þgj1 þ ða1 gjþ1 þ ða1 þ 2a2 Þ hjþ1 Þgjþ1 ¼ 0: 2 2 2 2 2 2 2

ð4:8Þ

Denote ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð1Þ

ð1Þ

ð2Þ

ð2Þ

 Þ ¼ fðhl ; g  j Þjðhl ; g  j Þ ¼ ðhl ; g  j Þ  ðhl ; g  j Þ; l; j ¼ 0; 1; 2; . . . ; Jg; ðh; g ðh; gÞ ¼ fðhl ; gj Þjðhl ; gj Þ ¼ ðhl ; gj Þ  ðhl ; gj Þ; l; j ¼ 0; 1; 2; . . . ; Jg: Subtracting (4.7) and (4.8) from (4.5) and (4.6), respectively, we obtain

    1   i  1  s ð1Þ 2 ð1Þ 2 ð1Þ  jþ1 þ k  j1 þ g hj1 hj1 þ hjþ1 þ sbd2x hj þ c hj1 þ hjþ1 þ C g a 1 hj1 þ ða1 þ 2a2 Þ gj1 2 2 2 2 2 2 2 2 2 2 2 2 2     2 2  2 2 s ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ hj1 a1 hj þ a1 hjþ1 þ ða1 þ 2a2 Þ gjþ1 hjþ1  k 1 þ ða1 þ 2a2 Þ gj1 2 2 2 2 2 2 2    ð2Þ 2 ð2Þ 2 ð2Þ þ a1 hjþ1 þ ða1 þ 2a2 Þ gjþ1 hjþ1 ¼ 0; 2 2 2  2 2  1   i 1  s 2  j1 þ g  jþ1 Þ þ sbdx g  jþ1 þ C hj1 þ hjþ1 þ k ða1 gð1Þ1 þ ða1 þ 2a2 Þ hð1Þ1 Þgj1 j þ c g  j1 þ g ðg j j 2 2 2 2 2 2 2 2 2 2 2 2 2     2 2  2 2 s ð1Þ ð1Þ ð2Þ ð1Þ a1 gð2Þ þ a1 gjþ1 þ ða1 þ 2a2 Þ hjþ1 gjþ1  k þ ða1 þ 2a2 Þ hj1 gj12 j12 2 2 2 2 2    ð2Þ 2 ð2Þ 2 ð2Þ þ a1 gjþ1 þ ða1 þ 2a2 Þ hjþ1 gjþ1 ¼ 0: 2

2

ð4:9Þ

ð4:10Þ

2

Let

~hj ¼ dx h 1 þ sb H 1 ; jþ2 h jþ2

g~ j ¼ dx g jþ12 þ

sb h

ð4:11Þ

 jþ1 ; 2

where



 2   ð2Þ 2 ð2Þ ð1Þ ð2Þ hjþ1  ks a1 jhjþ1 j2 þ ða1 þ 2a2 Þ gjþ1 hjþ1 ; 2 2 2 2 2 2  2 2 2 2  ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ  jþ1 þ Chjþ1 þ ksða1 g 1 þ ða1 þ 2a2 Þ h 1 Þg 1  ks a1 g 1 þ ða1 þ 2a2 Þ h 1 g 1 :  jþ1 þ cg ¼ ig jþ jþ jþ jþ jþ jþ 2 2 2

2



 jþ1 þ ks a1 hð1Þ1 þ ða1 þ 2a2 Þ gð1Þ1 Hjþ12 ¼ ihjþ12 þ chjþ12 þ Cg jþ jþ 2  jþ1 2

2

2

2

2

2

2

Eqs. (4.9) and (4.10) can be written as the following forms:

sb sb dx hjþ1 þ Hjþ1 ¼ dx hj1  Hj1 ; 2 2 2 2 h h

 jþ1 þ dx g 2

sb h

 j1   jþ1 ¼ dx g 2

2

sb h

 j1 : 2

ð4:12Þ

It follows from (4.11) and (4.12) that

~hj ¼ dx h 1  sb H 1 ; j2 h j2

g~ j ¼ dx g j12 

sb h

ð4:13Þ

 j1 : 2

Eq. (4.11) can be written as the following form:

~hj1 ¼ dx h 1 þ sb H 1 ; j2 h j2 It follows from

ð4:13Þþð4:14Þ 2

~h 1 ¼ dx h 1 ; j j 2

It follows from

2

g~ j1 ¼ dx g j12 þ

sb h

 j1 : 2

that

g~ j12 ¼ dx g j12 :

ð4:13Þð4:14Þ 2

2 H 1 ¼ dx hj1 ; 2 sb j2

ð4:14Þ

ð4:15Þ

that

2

sb

 j1 :  j1 ¼ dx g 2

2

ð4:16Þ

Eqs. (4.15) and (4.16) just are the following equations:

  2 2  2 2  s  j1 þ ks a1 hð1Þ1 þ ða1 þ 2a2 Þ gð1Þ1 hð1Þ1  ks a1 hð2Þ1 þ ða1 þ 2a2 Þ gð2Þ1 hð2Þ1 ¼ 0; ihj1  bdx ~hj1 þ chj1 þ Cg j2 j2 j2 j2 j2 j2 2 2 2 2 2 ð4:17Þ ~h 1 ¼ dx h 1 ; j j 2

2

ð4:18Þ

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

  2 2  2 2  s ~  j1  bdx g  j1 þ Chj1 þ ks a1 gð1Þ1 þ ða1 þ 2a2 Þ hð1Þ1 gð1Þ1  ks a1 gð2Þ1 þ ða1 þ 2a2 Þ hð2Þ1 gð2Þ1 ¼ 0; ig 1 þ cg j j2 j2 j2 j2 j2 j2 2 2 2 2 2 ð4:19Þ

g~ j1 ¼ dx g j1 : 2

ð4:20Þ

2

Multiplying (4.17) by h hj1 and summing up for j, we obtain 2

s

ikhk2  bh 2 

J X

dx ~hj1 hj1 þ ckhk2 þ Ch 2

j¼1

2

J X j¼1

g j12 hj12 þ ksh

J  X

2



2

j¼1

2 



ð1Þ ð1Þ a1 hð1Þ þ ða1 þ 2a2 Þ gj1 hj1 hj12  ksh j1 2

2

J  X

2  ð2Þ 2 ð2Þ  ð2Þ hj1 hj1 ¼ 0: a1 hj 1 þ ða1 þ 2a2 Þ gj1 2

j¼1

2

ð4:21Þ

2

2

hj1 and summing up for j, we obtain Multiplying (4.18) by 2s bh~ 2

s 2

bh

J X

s ~2 dx hj1 ~hj1  bkhk ¼ 0: 2

j¼1

ð4:22Þ

2

2

Summing up (4.15) and (4.16), then taking the imaginary part of the result, we obtain

( ) J X 2    khk þ Im Ch gj1 hj1 2

j¼1

( þ Im ksh

J  X

2



2

2 



J  X

ð1Þ ð1Þ a1 hð1Þ þ ða1 þ 2a2 Þ gj1 hj1 hj12 ksh j1 2

j¼1

2

2



2

2 



2

j¼1



ð2Þ ð2Þ ð2Þ hj1 hj1 ¼ 0: a1 hj 1 þ ða1 þ 2a2 Þ gj1 2 2

ð4:23Þ

2

Similarly, we obtain

(  k2 þ Im Ch kg

J X

) h 1 g  j j1 2

j¼1

( þ Im ksh

J  X

2

2



2 



ð1Þ ð1Þ  j1 ksh a1 gð1Þ þ ða1 þ 2a2 Þ hj1 gj1 g j1 2 2

j¼1

2

J  X

2

2 





ð2Þ ð2Þ  j1 ¼ 0; a1 jgð2Þ j2 þ ða1 þ 2a2 Þ hj1 gj1 g j1 2 2

j¼1

2

ð4:24Þ

2

Summing up (4.23) and (4.24), we obtain

( J  J  2   X X ð1Þ 2 ð1Þ  ð2Þ 2 ð2Þ 2 ð2Þ   k2 þ Im ksh k h h khk2 þ kg a1 hð1Þ þ ð a þ 2 a Þ g s h a h þ ð a þ 2 a Þ g 1 hj1 hj1 g 1 2 1 1 2 j1 j1 j1 j j1 j1 j¼1

( þ Im ksh

J  X j¼1



2

2

2

2

2 



ð1Þ ð1Þ a1 gj gð1Þ g 1 ksh 1 þ ða1 þ 2a2 Þ hj1 j1 j2 2

2

2

2

J  X j¼1

2

j¼1



2

2

2 



2



ð2Þ ð2Þ  j1 ¼ 0: a1 gð2Þ þ ða1 þ 2a2 Þ hj1 gj1 g j1 2 2

2

2

ð4:25Þ

2

According to

( ) J   1    X s2 ð1Þ ð1Þ ð2Þ ð2Þ  G hj1 ; gj1  G hj1 ; gj1 hj1 6 khk2 þ kðGðhð1Þ ; gð1Þ Þ  Gðhð2Þ ; gð2Þ ÞÞk2 ; Im ksh 2 2 2 2 2 2 2 j¼1 ( ) J   1    X s2 ð1Þ ð1Þ ð2Þ ð2Þ  k2 þ kðGðgð1Þ ; hð1Þ Þ  Gðgð2Þ ; hð2Þ ÞÞk2 ; G hj1 ; gj1  G hj1 ; gj1 hj1 6 kg Im ksh 2 2 2 2 2 2 2 j¼1

ð4:26Þ

ð4:27Þ

      ðlÞ 2 ðlÞ 2 ðlÞ ðlÞ 2 ðlÞ 2 ðlÞ ðlÞ ðlÞ ðlÞ ðlÞ where G hj1 ; gj1 ¼ ða1 hj1 þ ða1 þ 2a2 Þ gj1 Þhj1 ; G gj1 ; hj1 ¼ a1 gj1 þ ða1 þ 2a2 Þ hj1 gj1 ; l ¼ 1; 2. 2

2

2

2

2

2

2

2

2

2

It follows from (4.23)–(4.27) that

s s  k2 6 kðGðgð1Þ ; hð1Þ Þ  Gðgð2Þ ; hð2Þ ÞÞk2 þ kðGðgð1Þ ; hð1Þ Þ  Gðgð2Þ ; hð2Þ ÞÞk2 : khk2 þ kg 2 2 2

2

ð4:28Þ

 ! 0; as h ! 0; g ! 0, i.e., ð  Þ ¼ T k ðh; gÞ is a continuous mapping. Because Gðu; vÞ is continuous, so  h ! 0; g h; g  Þ ¼ T k ðh; gÞ is Step 2. When k ¼ 0, for any ðh; gÞ 2 H, the mapping ð h; g

 i   1   i  1   jþ1 ¼ 0;  j1 þ g hj1 þ hjþ1  unj1 þ unjþ1 þ sbd2x hj þ c hj1 þ hjþ1 þ C g 2 2 2 2 2 2 2 2 2 2 2 2 j ¼ 0; 1; . . . ; ðJ  1Þ; 0 6 n 6 N  1;

ð4:29Þ

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T. Wang et al. / Applied Mathematics and Computation 203 (2008) 413–431

 i   1   i 1  g 1 þ g jþ12  vnj1 þ vnjþ1 þ sbd2x g j þ c g j12 þ g jþ12 þ C hj12 þ hjþ12 ¼ 0; 2 2 2 j2 2 2 2 j ¼ 0; 1; . . . ; ðJ  1Þ; 0 6 n 6 N  1:

ð4:30Þ

Obviously, (4.29) and (4.30) is a linear system for ðh; gÞ 2 H, hence, in order to prove that (4.29) and (4.30) has unique solution, it is sufficient to prove that the following homogeneous system of equations:

  1   i  1   jþ1 ¼ 0; j ¼ 0; 1; . . . ; ðJ  1Þ; 0 6 n 6 N  1;  j1 þ g hj1 þ hjþ1 þ sbd2x hj þ c hj1 þ hjþ1 þ C g 2 2 2 2 2 2 2 2 2   1   i 1  2      gj12 þ gjþ12 þ sbdx gj þ c gj12 þ gjþ12 þ C hj12 þ hjþ12 ¼ 0; j ¼ 0; 1; . . . ; ðJ  1Þ; 0 6 n 6 N  1 2 2 2

ð4:31Þ ð4:32Þ

has and only has zero solution. By similar proof of step 1, we can obtain

 k2 ¼ 0; khk2 þ kg

ð4:33Þ

i.e., the solution of (4.29) and (4.30) is unique. Step 3. Suppose ðh; gÞ 2 H is a possible solution of ðh; gÞ ¼ T k ðh; gÞ, then we have

 i   1   i 1  h 1þh 1  un 1 þ unjþ1 þ sbd2x hj þ c hj1 þ hjþ1 þ C gj1 þ gjþ1 2 2 2 2 2 2 j2 jþ2 2 j2 2 2   2 2  2 2  s þk a1 hj12 þ ða1 þ 2a2 Þ gj12 hj12 þ a1 hjþ12 þ ða1 þ 2a2 Þ gjþ12 hjþ12 ¼ 0; 2  i   1   i 1  g 1 þ g 1  vnj1 þ vnjþ1 þ sbd2x gj þ c gj12 þ gjþ12 þ C hj12 þ hjþ12 2 2 2 j2  jþ2 2 2 2  2 2  2 2 s þk a1 gj12 þ ða1 þ 2a2 Þ hj12 gj12 þ ða1 gjþ12 þ ða1 þ 2a2 Þ hjþ12 Þgjþ12 ¼ 0: 2

ð4:34Þ

ð4:35Þ

Let

~h 1 ¼ dx h 1 ; j j 2

g~ j12 ¼ dx gj12 ;

2

then (4.34) and (4.35) can be written as the following equivalent formation:

  2 2  2 2  s ihj1  bdx ~hj1 þ chj1 þ Cgj1 þ ks a1 hj1 þ ða1 þ 2a2 Þ gj1 hj1  ks a1 hj1 þ ða1 þ 2a2 Þ gj1 hj1 ¼ 0; 2 2 2 2 2 2 2 2 2 2 2 ð4:36Þ ~h 1 ¼ dx h 1 ; j j 2

2

s

~ j1 þ cgj1 þ Chj1 þ ks igj1  bdx g 2 2 2 2 2

ð4:37Þ  2 2  2 2  a1 gj1 þ ða1 þ 2a2 Þ hj12 gj1  ks a1 gj1 þ ða1 þ 2a2 Þ hj12 gj1 ¼ 0;



2

g~ j1 ¼ dx gj1 : 2

2

2

2

2

ð4:38Þ ð4:39Þ

By the similar proof of Theorem 2.2 and Theorem 3.1, we obtain

khk1 6 C;

kgk1 6 C;

ð4:40Þ

i.e., all of the possible solution of (4.3) and (4.4) are uniformly bounded with the parameter 0 6 k 6 1. Thus by Lemma 4.1 we know that the mapping (4.3) and (4.4) has at least one solution ðh ; g Þ ¼ T k ðh ; g Þ. Let hj ¼ 2unþ1  unj ; gj ¼ 2vnþ1  vnj , we j j obtain that if ðun ; vn Þ satisfies the scheme (2.29) and (2.30), then ðunþ1 ; vnþ1 Þ also satisfies the scheme (2.29) and (2.30). h Using Lemma 3.6, we can prove the following theorem. Theorem 4.2. The difference solution of system (2.29) and (2.30) is unique. References [1] [2] [3] [4] [5] [6] [7]

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