An Entropy Stable h / p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property

Lucas Friedrich, Andrew R. Winters, David C. Del Rey Fernández, Gregor J. Gassner, Matteo Parsani, Mark H. Carpenter

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

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.
Original languageEnglish (US)
Pages (from-to)689-725
Number of pages37
JournalJournal of Scientific Computing
Volume77
Issue number2
DOIs
StatePublished - May 12 2018

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