On multi-symplectic partitioned Runge-Kutta methods for Hamiltonian wave equations

Many conservative PDEs, such as various wave equations, Schrodinger equations, KdV equations and so on, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. In this note, we show the discretization to Hamiltonian wave equations in space and time using two symplectic partitioned Runge-Kutta methods respectively leads to multi-symplectic integrators which preserve a symplectic conservation law. Under some conditions, we discuss the energy and momentum conservative property of partitioned Runge-Kutta methods for the wave equations with a quadratic potential. (c) 2005 Elsevier Inc. All rights reserved.