Spacetime discontinuous Galerkin methods for solving convection-diffusion systems

Abstract

In this paper, we present two new approaches for solving systems of hyperbolic conservation laws with correct physical viscosity and heat conduction terms such as the compressible Navier–Stokes equations. Our methods are extensions of the spacetime discontinuous Galerkin method for hyperbolic conservation laws developed by Hiltebrand and Mishra [26]. Following this work, we use entropy variables as degrees of freedom and entropy stable fluxes. For the discretization of the diffusion term, we consider two different approaches: the interior penalty approach, resulting in the ST-SIPG and the ST-NIPG method, and a variant of the local discontinuous Galerkin method, resulting in the ST-LDG method. We show entropy stability of the ST-NIPG and the ST-LDG method when applied to the compressible Navier–Stokes equations. For the ST-SIPG method, this result holds under an assumption on the computed solution. All schemes incorporate shock capturing terms. Therefore, the schemes can handle both regimes of underresolved and fully resolved physical diffusion. We present a numerical comparison of the three methods in one space dimension.