An entropy stable spacetime discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations

Abstract

In this paper, we present an entropy stable scheme for solving the compressible Navier-Stokes equations in two space dimensions. Our scheme uses entropy variables as degrees of freedom. It is an extension 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 ST-SDSC-NIPG. The streamline diffusion and shock-capturing terms from the original scheme have been kept, but have been adjusted appropriately. This guarantees that the new scheme essentially reduces 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. We also present numerical results confirming the accuracy and robustness of our schemes.