Shock detection and limiting strategies for high order discontinuous Galerkin schemes

Abstract

Discontinuous Galerkin (DG) schemes are a combination of finite volume (FV) and finite element (FE) schemes. While the approximate solution is a continuous polynomial in every grid cell, discontinuities at the grid cell interfaces are allowed which enables the resolution of strong gradients. How to calculate the fluxes between the grid cells and to take into account the jumps is well-known from the finite volume community. Due to their interior grid cell resolution with high order polynomials the DG schemes may use very coarse grids. In this approach the cumbersome reconstruction step of finite volume schemes is avoided, but for every degree of freedom a variational equation has to be solved. The main advantage of DG schemes is that the high order accuracy is preserved even on distorted and irregular grids. In the following we present a DG scheme based on a space-time expansion (STE-DG), which was proposed in [4]. Our scheme features time consistent local time-stepping, where every grid cell runs with its optimal time step.