Multisymplectic Integrators for Coupled Nonlinear Partial Differential Equations

Abstract

The numerical solution of nonlinear partial differential equations (PDEs) using symplectic geometric integrators has been the subject of many studies in recent years. Many nonlinear partial differential equations can be formulated as an infinite dimensional Hamiltonian system. After semi-discretization in the space variable, a system of Hamiltonian ordinary differential equations (ODEs) is obtained, for which various symplectic integrators can be applied. Numerical results show that symplectic schemes have superior performance, especially in long time simulations. The concept of multisymplectic PDEs and multisymplectic schemes can be viewed as the generalization of symplectic schemes. In the last decade, many multisymplectic methods have been proposed and applied to nonlinear PDEs, like to nonlinear wave equation, nonlinear Schr̈odinger equation, Korteweg de Vries equation, Dirac equation, Maxwell equation and sine-Gordon equation. In this review article, recent results of multisymplectic integration on the coupled nonlinear PDEs, the coupled nonlinear Schr̈odinger equation, the modified complex Korteweg de Vries equation and the Zakharov system will be given. The numerical results are discussed with respect to the stability of the schemes, accuracy of the solutions, conservation of the energy and momentum, preservation of dispersion relations. © 2012 Nova Science Publishers, Inc. All rights reserved.