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We design an arbitrary high-order accurate nodal discontinuous Galerkinspectral element approximation for the nonlinear two dimensional shallow waterequations with non-constant, possibly discontinuous, bathymetry onunstructured, possibly curved, quadrilateral meshes. The scheme is derived froman equivalent flux differencing formulation of the split form of the equations.We prove that this discretisation exactly preserves the local mass andmomentum. Furthermore, combined with a special numerical interface fluxfunction, the method exactly preserves the mathematical entropy, which is thetotal energy for the shallow water equations. By adding a specific form ofinterface dissipation to the baseline entropy conserving scheme we create aprovably entropy stable scheme. That is, the numerical scheme discretelysatisfies the second law of thermodynamics. Finally, with a particulardiscretisation of the bathymetry source term we prove that the numericalapproximation is well-balanced. We provide numerical examples that verify thetheoretical findings and furthermore provide an application of the scheme for apartial break of a curved dam test problem.

An Entropy Stable Nodal Discontinuous Galerkin Method for the Two  Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with  Discontinuous Bathymetry