A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes

A well-balanced, spatially arbitrary high order accurate semi-implicit discontinuous Galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non-orthogonal grids. The semi-implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime. For p=0 the resulting scheme becomes a generalization of the classical semi-implicit finite-volume/finite difference scheme of Casulli and Walters (2000) [25], but with less conditions on the grid geometry. The method proposed in this paper allows large time steps with respect to the surface wave speed gH and is thus particularly suitable for low Froude number flows. The approach is validated on some typical academic benchmark problems using polynomial degrees up to p=6.