Hamiltonian Finite Element Discretization for Nonlinear Free Surface Water Waves

A novel finite element discretization for nonlinear potential flow water waves is presented. Starting from Luke’s Lagrangian formulation we prove that an appropriate finite element discretization preserves the Hamiltonian structure of the potential flow water wave equations, even on general time-dependent, deforming and unstructured meshes. For the time-integration we use a modified Störmer–Verlet method, since the Hamiltonian system is non-autonomous due to boundary surfaces with a prescribed motion, such as a wave maker. This results in a stable and accurate numerical discretization, even for large amplitude nonlinear water waves. The numerical algorithm is tested on various wave problems, including a comparison with experiments containing wave interactions resulting in a large amplitude splash.