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Wintermeyer, Niklas, Winters, Andrew R., Gassner, Gregor J. and Warburton, Timothy (2018). An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. Technical Report.

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Abstract

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N = 7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.

Item Type:	Preprints, Working Papers or Reports (Technical Report)
Creators:	Creators Email ORCID ORCID Put Code Wintermeyer, Niklas nwinterm@math.uni-koeln.de UNSPECIFIED UNSPECIFIED Winters, Andrew R. UNSPECIFIED UNSPECIFIED UNSPECIFIED Gassner, Gregor J. UNSPECIFIED UNSPECIFIED UNSPECIFIED Warburton, Timothy UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-86564
Series Name at the University of Cologne:	Technical report series. Center for Data and Simulation Science
Volume:	2018,4
Date:	2 October 2018
Language:	English
Faculty:	Central Institutions / Interdisciplinary Research Centers
Divisions:	Weitere Institute, Arbeits- und Forschungsgruppen > Center for Data and Simulation Science (CDS)
Subjects:	Data processing Computer science Mathematics Technology (Applied sciences)
Uncontrolled Keywords:	Keywords Language Shallow water equations English Discontinuous Galerkin spectral element method English Shock capturing English Positivity preservation English GPUs English OCCA English
URI:	http://kups.ub.uni-koeln.de/id/eprint/8656