On multisymplectic integrators based on Runge–Kutta–Nyström methods for Hamiltonian wave equations

Abstract

Many conservative partial differential equations (PDEs), such as wave equations, Schrödinger equations, KdV equations, Maxwell equations and so on, allow for a multisymplectic formulation which can be regarded as a generalization of the symplectic structure of Hamiltonian ordinary differential equations (ODEs). In this note, for Hamiltonian wave equations, we show the discretization in space and time using two symplectic Runge–Kutta–Nyström (SRKN) methods respectively leads to a multisymplectic integrator which can preserve a discrete multisymplectic conservation law. Moreover, we discuss the energy and momentum conservative properties of the multisymplectic integrator for the wave equations with a quadratic potential.