Friedrich, Lucas, Winters, Andrew R., Del Rey Fernandez, David C., Gassner, Gregor J., Parsani, Matteo and Carpenter, Mark H. (2019). An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property. Technical Report.

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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.

Item Type: Preprints, Working Papers or Reports (Technical Report)
Creators:
CreatorsEmailORCID
Friedrich, Lucaslfriedri@math.uni-koeln.deUNSPECIFIED
Winters, Andrew R.UNSPECIFIEDUNSPECIFIED
Del Rey Fernandez, David C.UNSPECIFIEDUNSPECIFIED
Gassner, Gregor J.UNSPECIFIEDUNSPECIFIED
Parsani, MatteoUNSPECIFIEDUNSPECIFIED
Carpenter, Mark H.UNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-92759
Series Name at the University of Cologne: Technical report series. Center for Data and Simulation Science
Volume: 2019, 4
Subjects: Data processing Computer science
Mathematics
Technology (Applied sciences)
Uncontrolled Keywords:
KeywordsLanguage
Summation-by-PartsEnglish
Discontinuous GalerkinEnglish
Entropy ConservationEnglish
Entropy StabilityEnglish
h/p Non-Conforming MeshEnglish
Non-Linear Hyperbolic Conservation LawsEnglish
Faculty: Central Institutions / Interdisciplinary Research Centers
Divisions: Central Institutions / Interdisciplinary Research Centers > Center for Data and Simulation Science
Language: English
Date: 25 January 2019
URI: http://kups.ub.uni-koeln.de/id/eprint/9275

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