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

Many conservative PDEs, such as various wave equations, Schrödinger 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.