Journal of Scientific Computing

, Volume 80, Issue 1, pp 175–222 | Cite as

Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws

  • Lucas Friedrich
  • Gero Schnücke
  • Andrew R. WintersEmail author
  • David C. Del Rey Fernández
  • Gregor J. Gassner
  • Mark H. Carpenter


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.


Space–time discontinuous Galerkin Summation-by-parts Entropy conservation Entropy stability Kinetic energy preservation 



Gregor Gassner, Lucas Friedrich and Gero Schnücke have 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. Furthermore, we would like to thank the unknown referee for mentioning the general proof of the inequality (2.47).


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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of CologneCologneGermany
  2. 2.National Institute of Aerospace and Computational AeroSciences Branch, NASA Langley Research CenterHamptonUSA
  3. 3.Computational AeroSciences Branch, NASA Langley Research CenterHamptonUSA

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