Journal of Applied Mathematics
Volume 2012 (2012), Article ID 705179, 22 pages
Research Article

Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

Department of Mathematics, Harbin Institute of Technology, 2 Wenhua West Road, Shandong, Weihai 264209, China

Received 12 August 2012; Accepted 4 October 2012

Academic Editor: Renat Zhdanov

Copyright © 2012 Cuicui Liao and Xiaohua Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 Klein-Gordon 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.