On variational and symplectic time integrators for Hamiltonian systems

Various systems in nature have a Hamiltonian structure and therefore accurate time integrators for those systems are of great practical use. In this paper, a finite element method will be explored to derive symplectic time stepping schemes for (non-)autonomous systems in a systematic way. The technique used is a variational discontinuous Galerkin finite element method in time. This approach provides a unified framework to derive known and new symplectic time integrators. An extended analysis for the new time integrators will be provided. The analysis shows that a novel third order time integrator presented in this paper has excellent dispersion properties. These new time stepping schemes are necessary to get accurate and stable simulations of (forced) water waves and other non-autonomous variational systems, which we illustrate in our numerical results.