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An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property
Mathematical Institute, University of Cologne, Cologne, Germany.
Mathematical Institute, University of Cologne, Cologne, Germany.ORCID iD: 0000-0002-5902-1522
National Institute of Aerospace and Computational Aero, Sciences Branch, NASA Langley Research Center, Hampton, USA.
Mathematical Institute, University of Cologne, Cologne, Germany.
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2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, no 2, p. 689-725Article in journal (Refereed) Published
Abstract [en]

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre–Gauss–Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h / p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.

Place, publisher, year, edition, pages
Springer, 2018. Vol. 77, no 2, p. 689-725
Keywords [en]
Summation-by-Parts, Discontinuous Galerkin, Entropy Conservation, Entropy Stability, h/p Non-Conforming Mesh, Non-Linear Hyperbolic Conservation Laws
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-156856DOI: 10.1007/s10915-018-0733-7ISI: 000446594600001Scopus ID: 2-s2.0-85046752892OAI: oai:DiVA.org:liu-156856DiVA, id: diva2:1315784
Funder
German Research Foundation (DFG), TA 1260/1-1EU, European Research Council, 714487Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved

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