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

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.