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

doi:10.1007/s10915-018-0733-7

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 journal › Article › peer-review

42 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 language	English (US)
Pages (from-to)	689-725
Number of pages	37
Journal	Journal of Scientific Computing
Volume	77
Issue number	2
DOIs	https://doi.org/10.1007/s10915-018-0733-7
State	Published - May 12 2018

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Lucas Friedrich and Andrew Winters were funded by the Deutsche Forschungsgemeinschaft (DFG) Grant TA 2160/1-1. Special thanks goes to the Albertus Magnus Graduate Center (AMGC) of the University of Cologne for funding Lucas Friedrich’s visit to the National Institute of Aerospace, Hampton, VA, USA. Gregor Gassner has been supported by the European Research Council (ERC) under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme, ERC grant agreement no. 714487. This work was partially performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum Köln (RRZK) at the University of Cologne.

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10.1007/s10915-018-0733-7

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

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.",

author = "Lucas Friedrich and Winters, {Andrew R.} and {Del Rey Fern{\'a}ndez}, {David C.} and Gassner, {Gregor J.} and Matteo Parsani and Carpenter, {Mark H.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Lucas Friedrich and Andrew Winters were funded by the Deutsche Forschungsgemeinschaft (DFG) Grant TA 2160/1-1. Special thanks goes to the Albertus Magnus Graduate Center (AMGC) of the University of Cologne for funding Lucas Friedrich{\textquoteright}s visit to the National Institute of Aerospace, Hampton, VA, USA. Gregor Gassner has been supported by the European Research Council (ERC) under the European Union{\textquoteright}s Eights Framework Program Horizon 2020 with the research project Extreme, ERC grant agreement no. 714487. This work was partially performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum K{\"o}ln (RRZK) at the University of Cologne.",

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AU - Gassner, Gregor J.

AU - Parsani, Matteo

AU - Carpenter, Mark H.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: Lucas Friedrich and Andrew Winters were funded by the Deutsche Forschungsgemeinschaft (DFG) Grant TA 2160/1-1. Special thanks goes to the Albertus Magnus Graduate Center (AMGC) of the University of Cologne for funding Lucas Friedrich’s visit to the National Institute of Aerospace, Hampton, VA, USA. Gregor Gassner has been supported by the European Research Council (ERC) under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme, ERC grant agreement no. 714487. This work was partially performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum Köln (RRZK) at the University of Cologne.

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N2 - 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.

AB - 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.

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

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