Entropy-stable space–time DG schemes for non-conservative hyperbolic systems

¹ ANSYS Switzerland GmbH, Zürich 8005, Switzerland
² Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG G 57.2, Zürich 8092, Switzerland
³ Center of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, Oslo 0316, Norway
⁴ Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

^* Corresponding author: pares@uma.es

Received: 13 June 2017
Revised: 28 September 2017
Accepted: 9 November 2017

Abstract

We propose a space–time discontinuous Galerkin (DG) method to approximate multi-dimensional non-conservative hyperbolic systems. The scheme is based on a particular choice of interface fluctuations. The key difference with existing space–time DG methods lies in the fact that our scheme is formulated in entropy variables, allowing us to prove entropy stability for the method. Additional numerical stabilization in the form of streamline diffusion and shock-capturing terms are added. The resulting method is entropy stable, arbitrary high-order accurate, fully discrete, and able to handle complex domain geometries discretized with unstructured grids. We illustrate the method with representative numerical examples.