Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations (Q1971413)

The author considers partial differential equations of the form \[ M\partial_tx+ K\partial_xz= \nabla_z S(z), \] where \(z\in\mathbb{R}^d\) and \(M,K\in \mathbb{R}^{d\times d}\) are skew-symmetric matrices, and shows that Gauss-Legendre collocation in space and time leads to numerical methods that preserve symplectic conservation laws. He suggests several semi-explicit symplectic methods that use explicit or linearly implicit discretizations in time.