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An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
Mathematisches Institut, Universität zu Köln, Köln, Germany.
Mathematisches Institut, Universität zu Köln, Köln, Germany.ORCID iD: 0000-0002-5902-1522
Mathematisches Institut, Universität zu Köln, Köln, Germany.
Department of Mathematics, Virginia Tech, Blacksburg, USA.
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 447-480Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier, 2018. Vol. 375, p. 447-480
Keywords [en]
Shallow water equations, Discontinuous Galerkin spectral element method, Shock capturing, Positivity preservation, GPUs, OCCA
National Category
Computational Mathematics Oceanography, Hydrology and Water Resources
Identifiers
URN: urn:nbn:se:liu:diva-156850DOI: 10.1016/j.jcp.2018.08.038ISI: 000450907600022Scopus ID: 2-s2.0-85052988975OAI: oai:DiVA.org:liu-156850DiVA, id: diva2:1315775
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved

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The full text will be freely available from 2020-12-15 08:00
Available from 2020-12-15 08:00

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Winters, Andrew Ross

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