An implicit discontinuous Galerkin finite element method for watet waves

An over-view is given of a discontinuous Galerkin finite element method for linear free surface water waves. The method uses an implicit time integration method which is unconditionally stable and does not suffer from the frequently encountered mesh dependent saw-tooth type instability at the free surface. The numerical discretization has minimal dissipation and small dispersion errors in the wave propagation. The algorithm is second order accurate in time and has an optimal rate of convergence O(h(p+1)) in the L-2-norm, both in the potential and wave height, with p the polynomial order and h the mesh size. The numerical discretization is demonstrated with the simulation of water waves in a basin with a bump at the bottom.