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Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws
Mathematical Institute, University of Cologne, Cologne, Germany.
Mathematical Institute, University of Cologne, Cologne, Germany.
Mathematical Institute, University of Cologne, Cologne, Germany.ORCID iD: 0000-0002-5902-1522
National Institute of Aerospace and Computational Aero, Sciences Branch, NASA Langley Research Center, Hampton, USA.
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2019 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 80, no 1, p. 175-222Article in journal (Refereed) Published
Abstract [en]

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations.

Place, publisher, year, edition, pages
Springer, 2019. Vol. 80, no 1, p. 175-222
Keywords [en]
Space-Time Discontinuous Galerkin, Summation-by-Parts, Entropy Conservation, Entropy Stability, Kinetic Energy Preservation
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-156874DOI: 10.1007/s10915-019-00933-2Scopus ID: 2-s2.0-85062733957OAI: oai:DiVA.org:liu-156874DiVA, id: diva2:1315790
Funder
EU, European Research Council, 714487Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-24Bibliographically approved

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The full text will be freely available from 2020-03-15 08:00
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Winters, Andrew Ross

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