Friedrich, Lucas, Winters, Andrew R., Fernandez, David C. Del Rey, Gassner, Gregor J., Parsani, Matteo ORCID: 0000-0001-7300-1280 and Carpenter, Mark H. (2018). An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property. J. Sci. Comput., 77 (2). S. 689 - 726. NEW YORK: SPRINGER/PLENUM PUBLISHERS. ISSN 1573-7691

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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: Journal Article
Creators:
CreatorsEmailORCIDORCID Put Code
Friedrich, LucasUNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Winters, Andrew R.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Fernandez, David C. Del ReyUNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Gassner, Gregor J.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Parsani, MatteoUNSPECIFIEDorcid.org/0000-0001-7300-1280UNSPECIFIED
Carpenter, Mark H.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-168689
DOI: 10.1007/s10915-018-0733-7
Journal or Publication Title: J. Sci. Comput.
Volume: 77
Number: 2
Page Range: S. 689 - 726
Date: 2018
Publisher: SPRINGER/PLENUM PUBLISHERS
Place of Publication: NEW YORK
ISSN: 1573-7691
Language: English
Faculty: Unspecified
Divisions: Unspecified
Subjects: no entry
Uncontrolled Keywords:
KeywordsLanguage
FINITE-DIFFERENCE METHODS; NAVIER-STOKES EQUATIONS; NONLINEAR CONSERVATION-LAWS; SHALLOW-WATER EQUATIONS; SCHEMES; ORDER; FORM; OPERATORS; SYSTEMS; GRIDSMultiple languages
Mathematics, AppliedMultiple languages
Refereed: Yes
URI: http://kups.ub.uni-koeln.de/id/eprint/16868

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