Multi-Symplectic Runge–Kutta Collocation Methods for Hamiltonian Wave Equations

A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss–Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for Hamiltonian ODEs. We also discuss the issue of conservation of energy and momentum. Since time discretization by a Gauss–Legendre method is computational rather expensive, we suggest several semi-explicit multi-symplectic methods based on Gauss–Legendre collocation in space and explicit or linearly implicit symplectic discretizations in time.