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We extend the entropy stable high order nodal discontinuous Galerkin spectralelement approximation for the non-linear two dimensional shallow waterequations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkinmethod for the two dimensional shallow water equations on unstructuredcurvilinear meshes with discontinuous bathymetry. Journal of ComputationalPhysics, 340:200-242, 2017] with a shock capturing technique and a positivitypreservation capability to handle dry areas. The scheme preserves the entropyinequality, is well-balanced and works on unstructured, possibly curved,quadrilateral meshes. For the shock capturing, we introduce an artificialviscosity to the equations and prove that the numerical scheme remains entropystable. We add a positivity preserving limiter to guarantee non-negative waterheights as long as the mean water height is non-negative. We prove thatnon-negative mean water heights are guaranteed under a certain additional timestep restriction for the entropy stable numerical interface flux. We implementthe method on GPU architectures using the abstract language OCCA, a unifiedapproach to multi-threading languages. We show that the entropy stable schemeis well suited to GPUs as the necessary extra calculations do not negativelyimpact the runtime up to reasonably high polynomial degrees (around $N=7$). Weprovide numerical examples that challenge the shock capturing and positivityproperties of our scheme to verify our theoretical findings.

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs