Discontinuous Galerkin Methods for Linear Hyperbolic Systems

Abstract

We develop a space-time method with a discontinuous Galerkin discretization in space for linear wave problems. For the ansatz space we use piecewise polynomials in every cell, where the traces on the cell interfaces can be different from the two sides. Therefore, we need to extend the first-order operator A to discontinuous finite element spaces. Here, we introduce the discrete operator A h with upwind flux, where the evaluation of the upwind flux is based on solving Riemann problems, i.e., by construction of piecewise constant solutions in space and time. We start with simple examples for interface and transmission problems, and then consider the general case for waves in heterogeneous media.