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Efficient Implementation of Modern Entropy Stable and Kinetic Energy Preserving Discontinuous Galerkin Methods for Conservation Laws

Authors:

Hendrik Ranocha,

Michael Schlottke-Lakemper,

Andrés M. Rueda-Ramírez,

Andrew R. Winters,

Florian Hindenlang,

Gregor J. GassnerAuthors Info & Claims

ACM Transactions on Mathematical Software, Volume 49, Issue 4

Article No.: 37, Pages 1 - 30

https://doi.org/10.1145/3625559

Published: 15 December 2023 Publication History

Abstract

Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG methods. We present several implementation techniques to improve the efficiency of flux differencing DG methods that use tensor product quadrilateral or hexahedral elements, in 2D or 3D, respectively. Focus is mostly given to CPUs and DG methods for the compressible Euler equations, although these techniques are generally also useful for other physical systems, including the compressible Navier-Stokes and magnetohydrodynamics equations. We present results using two open source codes, Trixi.jl written in Julia and FLUXO written in Fortran, to demonstrate that our proposed implementation techniques are applicable to different code bases and programming languages.

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Cited By

Rueda-Ramírez ASikstel AGassner G(2025)An entropy-stable discontinuous Galerkin discretization of the ideal multi-ion magnetohydrodynamics systemJournal of Computational Physics10.1016/j.jcp.2024.113655523(113655)Online publication date: Feb-2025
https://doi.org/10.1016/j.jcp.2024.113655
Ranocha HWinters ASchlottke-Lakemper MÖffner PGlaubitz JGassner G(2025)On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation lawsJournal of Computational Physics10.1016/j.jcp.2024.113471520(113471)Online publication date: Jan-2025
https://doi.org/10.1016/j.jcp.2024.113471
Ranocha HSchütz J(2024)Multiderivative time integration methods preserving nonlinear functionals via relaxationCommunications in Applied Mathematics and Computational Science10.2140/camcos.2024.19.2719:1(27-56)Online publication date: 17-Jun-2024
https://doi.org/10.2140/camcos.2024.19.27
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Efficient Implementation of Modern Entropy Stable and Kinetic Energy Preserving Discontinuous Galerkin Methods for Conservation Laws

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 49, Issue 4

December 2023

226 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/3637452

Editors:
Zhaojun Bai
University of California at Davis, USA
,
Wolfgang Bangerth
Colorado State University, USA

Issue’s Table of Contents

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Publication History

Published: 15 December 2023

Online AM: 27 September 2023

Accepted: 18 September 2023

Revised: 03 April 2023

Received: 20 December 2021

Published in TOMS Volume 49, Issue 4

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Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
Mathematics Münster: Dynamics-Geometry-Structure
Daimler und Benz Stiftung
European Research Council
Vetenskapsrådet, Sweden
United States National Science Foundation

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Rueda-Ramírez ASikstel AGassner G(2025)An entropy-stable discontinuous Galerkin discretization of the ideal multi-ion magnetohydrodynamics systemJournal of Computational Physics10.1016/j.jcp.2024.113655523(113655)Online publication date: Feb-2025
https://doi.org/10.1016/j.jcp.2024.113655
Ranocha HWinters ASchlottke-Lakemper MÖffner PGlaubitz JGassner G(2025)On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation lawsJournal of Computational Physics10.1016/j.jcp.2024.113471520(113471)Online publication date: Jan-2025
https://doi.org/10.1016/j.jcp.2024.113471
Ranocha HSchütz J(2024)Multiderivative time integration methods preserving nonlinear functionals via relaxationCommunications in Applied Mathematics and Computational Science10.2140/camcos.2024.19.2719:1(27-56)Online publication date: 17-Jun-2024
https://doi.org/10.2140/camcos.2024.19.27
Alberti LBassi FCarnevali EColombo ACrivellini ANigro A(2024)A Comparative Study of Different Sets of Variables in a Discontinuous Galerkin Method with Entropy Balance EnforcementInternational Journal of Computational Fluid Dynamics10.1080/10618562.2024.231053737:6(487-508)Online publication date: 28-Apr-2024
https://doi.org/10.1080/10618562.2024.2310537
Montoya TZingg D(2024)Efficient entropy-stable discontinuous spectral-element methods using tensor-product summation-by-parts operators on triangles and tetrahedraJournal of Computational Physics10.1016/j.jcp.2024.113360(113360)Online publication date: Aug-2024
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Worku ZZingg D(2024)Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equationsJournal of Computational Physics10.1016/j.jcp.2024.112821502:COnline publication date: 25-Jun-2024
https://dl.acm.org/doi/10.1016/j.jcp.2024.112821

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