Communications in Mathematical Sciences

Volume 16 (2018)

Number 8

Entropy stable spacetime discontinuous Galerkin methods for the two-dimensional compressible Navier–Stokes equations

Pages: 2095 – 2124



Andreas Hiltebrand (Seminar for Applied Mathematics, Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland)

Sandra May (Department of Mathematics, Technische Universität Dortmund, Germany)


In this paper, we present entropy stable schemes for solving the compressible Navier–Stokes equations in two space dimensions. Our schemes use entropy variables as degrees of freedom. They are extensions of an existing spacetime discontinuous Galerkin method for solving the compressible Euler equations. The physical diffusion terms are incorporated by means of the symmetric (SIPG) or nonsymmetric (NIPG) interior penalty method, resulting in the two versions ST-SDSC-SIPG and STSDSC-NIPG. The streamline diffusion (SD) and shock-capturing (SC) terms from the original scheme have been kept, but have been adjusted appropriately. This guarantees that the new schemes essentially reduce to the original scheme for the compressible Euler equations in regions with underresolved physical diffusion. We show entropy stability for both versions under suitable assumptions for the case of adiabatic solid wall boundary conditions. We also present numerical results confirming the accuracy and robustness of our schemes.


discontinuous Galerkin method, compressible Navier–Stokes equations, entropy stability, entropy variables, interior penalty method, wall boundary conditions

2010 Mathematics Subject Classification

65M12, 65M60

Received 26 January 2018

Accepted 13 August 2018

Published 18 April 2019