Discontinuous Galerkin method for linear free-surface gravity waves

In this paper, we discuss a discontinuous Galerkin finite element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the $L^2$-norm is in both cases of optimal order and proportional to $O(\triangle t^2+h^{p+1})$, without the need for a separate velocity reconstruction, with $p$ the polynomial order, $h$ the mesh size and $\triangle t$ the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.