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:
CreatorsEmailORCID
Wintermeyer, Niklasnwinterm@math.uni-koeln.deUNSPECIFIED
Winters, Andrew R.UNSPECIFIEDUNSPECIFIED
Gassner, Gregor J.UNSPECIFIEDUNSPECIFIED
Warburton, TimothyUNSPECIFIEDUNSPECIFIED
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
Subjects: Data processing Computer science
Mathematics
Technology (Applied sciences)
Uncontrolled Keywords:
KeywordsLanguage
Shallow water equationsEnglish
Discontinuous Galerkin spectral element methodEnglish
Shock capturingEnglish
Positivity preservationEnglish
GPUsEnglish
OCCAEnglish
Faculty: Central Institutions / Interdisciplinary Research Centers
Divisions: Central Institutions / Interdisciplinary Research Centers > Center for Data and Simulation Science
Language: English
Date: 2 October 2018
URI: http://kups.ub.uni-koeln.de/id/eprint/8656

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