Time finite element methods: A unified framework for numerical discretizations of ODEs

We present a unified framework for the numerical discretization of ODEs based on time finite element methods. We relate time finite element methods to Runge-Kutta methods with infinitely many stages. By means of the corresponding numerical quadrature, we establish the relation between time finite element methods and (partitioned) Runge-Kutta methods. We also provide order estimates and superconvergence of the corresponding numerical methods in use of the simplifying assumptions. We apply time finite methods to Hamiltonian systems and investigate the conservation of energy and symplectic structure for the resulting numerical discretizations. For Hamiltonian systems, we also construct new classes of symplectic integrators by combining different time finite element methods and verify the results by performing some numerical experiments.