An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

We design an arbitrary high order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. This entropy conserving scheme is the baseline for a provably entropy stable scheme. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. The proofs and derivations use skew-symmetric reformulations of the problem to remove aliasing errors. However, as many additional terms are introduced, the resulting skew-symmetric scheme is not computationally tractable. Therefore, we provide an equivalent reformulation of the skew-symmetric scheme, which restores computational efficiency. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.