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

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...

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Bibliographic Details
Published in:ACM transactions on mathematical software 2023-12, Vol.49 (4), p.1-30, Article 37
Main Authors: Ranocha, Hendrik, Schlottke-Lakemper, Michael, Chan, Jesse, Rueda-Ramírez, Andrés M., Winters, Andrew R., Hindenlang, Florian, Gassner, Gregor J.
Format: Article
Language:English
Subjects:
Applied computing
Computing methodologies
conservation laws
Continuous simulation
discontinuous Galerkin
entropy stability
flux differencing
Mathematical software performance
Mathematics of computing
Physical sciences and engineering
Solvers
summation-by-parts
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Summary: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.
ISSN:0098-3500
1557-7295
1557-7295
DOI:10.1145/3625559