Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs

Abstract

Many partial differential equations (PDEs) can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we give several systematic methods for discretizing general multi-symplectic formulations of Hamiltonian PDEs, including a local energy-preserving algorithm, a class of global energy-preserving methods and a local momentum-preserving algorithm. The methods are illustrated by the nonlinear Schrödinger equation and the Korteweg–de Vries equation. Numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods.