Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations

A number of conservative PDEs, like various wave equations, allow for a multisymplectic 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 multisymplectic methods based on Gauss-Legendre collocation in space and explicit or linearly implicit symplectic discretizations in time. (C) 2000 Academic Press.