Appl. Math. Mech. -Engl. Ed.     2014, Vol. 35 Issue (3) : 369–380     PDF       
http: //dx. doi. org/10.1007/s10483-014-1797-6
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Article Information

Hao-chen LI, Jian-qiang SUN, Meng-zhao QIN 2014.
New explicit multi-symplectic scheme for nonlinear wave equation
Appl. Math. Mech. -Engl. Ed., 35 (3) : 369–380
http: //dx. doi. org/10.1007/s10483-014-1797-6

Article History

Received 2012-10-15;
in final form 2013-06-05
Citation
Hao-chen LI, Jian-qiang SUN, Meng-zhao QIN 2014. New explicit multi-symplectic scheme for nonlinear wave equation[J]. Appl. Math. Mech. -Engl. Ed., 35 (3) : 369–380.
New explicit multi-symplectic scheme for nonlinear wave equation
Hao-chen LI1, Jian-qiang SUN1 , Meng-zhao QIN2       
1 Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, Hainan Province, P. R. China;
2 State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China
Received 2012-10-15; in final form 2013-06-05
Fund program：Supported by the National Natural Science Foundation of China (Nos. 11161017, 11071251, and 10871099), the National Basic Research Program of China (973 Program) (No. 2007CB209603), the Natural Science Foundation of Hainan Province (No. 110002), and the Scientific Research Foundation of Hainan University (No. kyqd1053)
Corresponding author: Jian-qiang SUN, Professor, Ph. D., E-mail: sunjq123@qq.com
ABSTRACT：Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the KleinGordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.
Keywords： nonlinear wave equation        multi-symplectic method        backward error analysis       

1 Introduction

The nonlinear Klein-Gordon equation

where f(u) is a nonlinear smooth function,is a class of important nonlinear wave equations, describing vibration and wave propagation phenomena. It also plays an important role in quantum mechanics. The Sine-Gordon equation is a special case of the nonlinear Klein-Gordon equation with f(u) = sin(u)[1]. The main exact solution methods of the nonlinear KleinGordon equation are the method of separation of variables and the method of characteristics, by which the periodic solitary wave solutions of the equation can be derived[2]. Besides,a lot of computation methods have been proposed to solve the nonlinear Klein-Gordon equation[3]. Based on the leapfrog scheme,Perring and Skyme[4] proposed two integral methods. Strauss and Vazquez[5] proposed an energy conservation three-level time method. Guo et al.[6] proposed two difference schemes which could preserve the discrete energy. Fei and Vazquez[7] extended this technology and proposed two explicit conservation schemes. Argyris and Hasse[8] developed the finite element method to solve the Klein-Gordon equation. Ablowitz et al.[9] proposed the Fourier pseudo-spectral method of the Klein-Gordon equation.

rier pseudo-spectral method of the Klein-Gordon equation. Recently,the multi-symplectic method,which has a long accurately computing capability and approximately preserves the energy conservation characteristic,is a class of important methods in solving nonlinear evolution equations[10, 11, 12, 13, 14, 15, 16, 17]. Bridges and Reich[13] and Reich[14] proposed the multi-symplectic Preissman scheme and the multi-symplectic Runge-Kutta scheme to solve the Klein-Gordon equation. Wang and Wang[15] applied the combination method to construct a high order implicit multi-symplectic scheme of the Klein-Gordon equation. However, these methods are implicit and require a lot of computational cost. Hong et al.[18] and Hu et al.[19] proposed the explicit multi-symplectic scheme to solve the nonlinear Klein-GordonSchr¨odinger equation and the regularized long-wave (RLW) equation. In this paper,a new explicit multi-symplectic scheme is proposed to solve the nonlinear Klein-Gordon equation.

licit multi-symplectic scheme is proposed to solve the nonlinear Klein-Gordon equation. The paper is organized as follows. In Section 2,the multi-symplectic structure for the nonlinear Klein-Gordon equation is introduced,a new explicit multi-symplectic scheme for the nonlinear Klein-Gordon equation is proposed,and the corresponding discrete multi-symplectic conservation law is proved. In Section 3,the backward error analysis of the new multi-symplectic scheme of the nonlinear Klein-Gordon equation is investigated. In Section 4,the solitary wave behaviors of the nonlinear wave equations are simulated by the new multi-symplectic scheme. We finish the paper with conclusion remarks in Section 5.

2 New explicit scheme of Klein-Gordon equation

The nonlinear Klein-Gordon equation (1) is equivalent to

which can be stated explicitly in the multi-symplectic form

as

where

and ∇zS(z) is the gradient of S(z) with respect to the standard inner product on R3[13, 14]. Equation (3) is a Hamiltonian formulation of the nonlinear wave on a multi-symplectic structure, where K,L ∈ R3× 3 are skew-symmetric matrices,and S(z) : R3 → R is a smooth function of z(x,t).

One of the most important characteristics in Eq. (3) is that it satisfies the multi-symplectic conservation law. Therefore,when a numerical scheme is developed,it is expected that the multi-symplectic conservation law is preserved. Bridges and Reich defined a numerical scheme as the multi-symplectic scheme which can preserve a discrete multi-symplectic conservation law. Specifically,if we discretize the Hamiltonian partial differential equation (PDE) (3) as follows:

where zji = z(xi,tj),and ∂xi,j and ∂ti,j are the discretizations of the derivatives ∂x and ∂t, respectively. Then,the scheme is multi-symplectic,which provides that it can preserve the following discrete conservation law:

where

Here,we propose a new multi-symplectic scheme. Let

be the regular grids of the integral domain

zji is an approximation to z(xi,tj,∆t is the time step,∆x is the spatial step,and

We propose a new scheme for Eq. (3). Then,it can be rewritten as

where K+,K− and L+,L− are matrices splitting for the matrices K and L,respectively,s.t.,

Theorem 1 The new scheme (7) is a multi-symplectic scheme with the following discrete multi-symplectic conservation law:

where

Proof Consider the variational equation of Eq. (7) as follows:

Taking the wedge product with dzji+ 1/2 and the variation equation (10),since

we have

Considering the items containing δt+ or δt− in Eq. (11),we have

Considering the items containing δx/2+ or δx/2- in Eq. (11),we have

Taking Eqs. (12) and (13) into Eq. (11),we have

The proof is completed.

Note that the matrix splitting (8) is not unique. Different splitting may produce different schemes. We take K+ and L+ as the upper triangle matrices,i.e.,

Substituting the above matrices into the new multi-symplectic scheme (7),we have

Equations (15)-(17) can be rewritten as

If we replace the time index i by i − 1 in Eqs. (18)-(20),we obtain

From Eqs. (18) and (21),we can get

From Eqs. (19) and (22),we can get

If we replace the space index j by j + 1 in Eq. (25),we obtain

Taking Eqs. (20),(23),(25),and (26) into Eq. (24),we can obtain a new explicit multisymplectic scheme

The new multi-symplectic scheme (27) satisfies the discrete multi-symplectic conservation law

3 Backward error analysis of new scheme

Theorem 2 The perturbation of the new explicit multi-symplectic scheme (27) is

Proof We now assume that z is a sufficiently smooth function which satisfies Eq. (7) when evaluated at the lattice points. Expanding z in the Taylor series about xi+1/2,we obtain

where z = z(xi+1/2,tj). Then,we have

From Eq. (29),we have

Expanding z in a Taylor series about tj,we obtain

From Eqs. (32)-(34),we have

Then,we can get

In the same way,we can get

Substituting Eqs. (30)-(32) and (35)-(36) into Eq. (7) yields the modified PDE

Substituting K,L,and z into the modified equation (37) gives

Taking Eqs. (39) and (40) into Eq. (38),we can get

From Eq. (41),we can obtain the following approximate equation:

which is an O(∆t2 + ∆x2) perturbation of the nonlinear Klein-Gordon equation (1).

The modified equation (37) can be written in the form of a standard multi-symplectic PDE

for

with the skew-symmetric matrices

where

4 Numerical simulations 4.1 Numerical example 1

Considering the potential function

we can get the sine-Gordon equation

The usual solution to Eq. (44) is

We consider the collision of two solitons,one with the + sign (kink) and the other with the − sign (antikink). The two solitons have equal but opposite velocities. The two kink and antikink solitons with a distance x0 on each side of the origin point of the x-axis. Therefore, we take the sine-Gordon equation with the periodic boundary condition[15]

and the initial conditions

We test the new derived scheme on this problem over long time intervals with

The discrete energy is taken as

which approximates the important first integral of Eq. (44) with

and the relative energy error is defined as

Figure 1(a) shows the numerical wave forms for the soliton collision with ∆x = 0.2 and ∆t = 0.001 in the range of t ∈ [0, 200]. It can be seen that the new multi-symplectic scheme has good numerical performance. Figure 1(b) shows the relative energy error corresponding to Fig. 1(a). We can get that the new multi-symplectic scheme can approximately preserve the relative energy error.

Fig. 1. Numerical results at t ∈ [0, 200].
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Figure 2 shows the numerical results of the new scheme in the range of t ∈ [10 000,10 200]. The new scheme gives the accurate wave form and approximately preserves the relative energy error in long time. This indicates that the new explicit multi-symplectic scheme of the sineGordon equation is suitable for long time simulation. Figure 3 shows the error of the numerical solution and the exact solution in the range of t ∈ [0, 45]. The error does not obviously increase. The error of the new explicit multi-symplectic scheme matches the accuracy O(∆t2 + ∆x2) of the scheme.

Fig. 2. Numerical results at t ∈ [10 000,10 200].
Figure options
Fig. 3. Error of numerical solution and exact solution in t ∈ [0, 45].
Figure options
4.2 Numerical example 2

Then,we consider the Klein-Gordon equation

with

where D = 1.28,and A is the amplitude. Jim´enez and V´azquez[20] showed that with the increase in A,the difficulty of the numerical simulation of the problem grew. If A is more than 15,the numerical solutions obtained with the three symplectic schemes produce a blow-up. We test the new multi-symplectic scheme with

Figure 4 shows the numerical results of the new scheme with A = 20. We can obtain that the numerical solutions of the new scheme can be simulated very well in the range of t ∈ [0,1.6]. Compared with the three symplectic schemes[20],the solution obtained with the new explicit multi-symplectic scheme is bounded and preserves the spatial symmetry with long time,as it is the underlying continuous solution,and this is consistent with the results[21].

Fig. 4. Numerical results of new scheme with A = 20.
Figure options
5 Conclusions

Based on the splitting multi-symplectic structure,a new multi-symplectic scheme is proposed and applied to the nonlinear wave equations. The accuracy order of the new explicit multi-symplectic scheme is O(∆t2 + ∆x2). The nonlinear wave equations are simulated by the new explicit multi-symplectic scheme. The numerical results show that the new explicit multisymplectic scheme has a nice stability and can well simulate the solitary wave behaviors of the nonlinear wave equations in long time and approximately preserve the relative energy error. Obviously,the new proposed explicit multi-symplectic scheme can greatly reduce the computation cost. The new proposed multi-symplectic scheme can also be applied to other multi-symplectic structure partial differential equations.

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