We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear KleinGordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.
It is a fundamental approach to develop the discrete multisymplectic numerical methods based on the discrete Hamilton’s principle, because it leads in a natural way to multisymplectic integrators [
By the Hamilton’s principle [
If
The details of this conclusion could be referred to papers [
Vankerschaver et al. [
When we use the discrete variational principle, we need to make a approximation of the Lagrangian. Here, in our paper, we would use nonstandard finite difference methods, instead of standard finite difference, to approximate the Lagrangian function, and derive the corresponding discrete variational integrators.
The nonstandard finite difference schemes are well developed by Mickens [
The initial foundation of nonstandard finite difference methods is formed by the exact finite difference schemes [
(1) The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives appearing in the differential equations.
In our paper, we combine the advantages of nonstandard finite difference methods and discrete variational principles to construct nonstandard finite difference variational integrators, for two multisymplectic PDEs. These integrators are multysimplectic and their multysimplecticity are presented by their discrete multisymplectic form formulas, respectively.
In Section
We first consider a simple linear wave equation,
This linear wave equation is actually a multisymplectic PDE. As a classical and simple multisymplectic example, wave equation and its multisymplectic structure have been studied from both Hamiltonian [
Assume that we have a uniform quadrangular mesh in the base space, with mesh lengths
Now we use nonstandard finite difference to define the discrete Lagrangian
We have followed the rules of constructing nonstandard finite difference schemes in Mickens’ papers [
The discrete firstderivative is represented by
Nonlocal representation on the discrete computational lattice are used here by
So, for the linear wave equation (
By the discrete Hamilton’s principle [
After some simple calculations, the discrete EulerLagrange equation (
We could find that this scheme is symmetric in
As we mentioned in Section
Since we consider triangulation discretization here, we focus on three adjacent triangles around
Taking twice exterior derivative of both sides, we have, by the fact that
Now we have the first conclusion.
The nonstandard finite difference variational integrator (
We now discuss the convergence of this variational integrator (
By Taylor series expansion, we have
Similarly,
The above two equations show that the scheme is consistent and the truncation error for the integrator (
To explore the stability of the nonstandard finite difference variational integrator (
Then the threelevel explicit integrator (
By using the Von Neumann method [
When
With the consistence and stability conditions, we have following conclusion.
The nonstandard finite difference variational integrator (
We have shown the idea of using the nonstandard finite difference method to get the discrete variational integrator and the corresponding discrete multisymplectic form formula. In the next section, we will consider the discrete variational integrators for a more complicated example, the nonlinear KleinGordon equation.
In this section, we consider the nonlinear KleinGordon equation [
As known, this equation can be obtained by EulerLagrange equation (
Now we consider the triangle discretization and square discretization, respectively, to get the nonstandard finite difference variational integrators.
Following the steps in last section and using the idea of nonstandard finite difference, we define the discrete Lagrangian
The discrete first derivative is represented by
Nonlocal representations for
Similarly, we define discrete Lagrangians on other two adjoint triangles,
Now, the discrete variational integrator with nonstandard finite difference methods could be obtained by discrete EulerLagrange equation (
Substituting
Using the definition of discrete Lagrangian functions, one can find that this scheme is symmetric with respect to
Its corresponding discrete multisymplectic form formula can be obtained from (
It shows the multisymplectic structure of scheme (
We now analyze the truncation error of integrator (
Combining the above two equations and (
The above results are summarized in the following theorem.
The nonstandard finite difference variational integrator (
In this case, we denote a square at
Following the philosophy of the nonstandard finite difference method, we define the discrete Lagrangian
In this case,
the discrete firstderivative is represented by
nonlocal representations for
Similarly, we have the definitions of
Taking derivate of action functional with respect to
Substituting the discrete Lagrangian
After simple calculations, it becomes
It is multisymplectic and symmetric in
To study the truncation error of the integrator, we do the Taylor expansion which leads to
Combing these equations, we can readily observe that the nonstandard finite difference variational integrator (
We summarize our conclusion in the following theorem.
The nonstandard finite difference variational integrator (
In this section, we report the performance of the nonstandard finite difference variational integrator (
For linear wave equation (
The nonstandard finite difference variational integrator (
From Figure
The waveforms of linear wave equation (
We now consider the nonlinear KleinGordon equation (
We use nonstandard finite difference variational integrator (
As depicted in Figure
Waveforms of nonlinear KleinGordon equation by the nonstandard finite difference variational integrator (
Waveforms at the beginning from
Waveforms from
To further investigate the numerical convergence of the proposed schemes, we conduct a series of numerical tests of our nonlinear integrators. In this example, we consider the nonlinear KleinGordon equation (
The nonstandard finite difference integrators (
The error and convergence orders of (
Mesh size 






Error 



Order  —  —  — 
 


Error 



Order  1.9773  1.9522  1.9341 
 


Error 



Order  1.9419  1.9811  1.9988 
 


Error 



Order  1.9905  1.9943  1.9959 
The error and convergence orders of (
Mesh size 






Error 



Order  —  —  — 
 


Error 



Order  1.9556  1.9763  1.9809 
 


Error 



Order  1.9836  1.9386  2.0255 
 


Error 



Order  1.9911  2.0263  1.9984 
Overall, it is clear that the error decreases as the mesh size goes to zero, indicating the convergence of our nonlinear integrators (
In all, the numerical tests verify that the nonstandard finite difference variational integrators that we developed are capable of preserving the characteristics of original equations. They are all accurate, effective, and suitable for solving multisymplectic systems.
In this paper, we have considered a linear wave equation and a nonlinear Kleingordon equation. We have derived the nonstandard finite difference variational integrators and the corresponding multisymplectic form formulas from these two multisymplectic PDEs. We have shown that the nonstandard finite difference methods are flexible in constructing numerical schemes and can be employed to derive multisymiplectic schemes for multisymplectic systems. The convergence of our methods has been discussed. The numerical experiments have shown effectiveness and efficiency of these nonstandard finite difference variational integrators.
The authors are grateful to the editor and anonymous reviewers for their careful reading and many constructive suggestions which lead to a great improvement of this paper. This work is supported by the NNSF of China (no. 11271101) and the NNSF of Shandong Province (no. ZR2010AQ021).