High-order shock capturing discontinuous Galerkin schemes

A shock capturing strategy for high-order accurate Discontinuous Galerkin (DG) approximations of the compressible Euler equations and the ideal magnetohydrodynamic equations is presented. Capturing of discontinuities without numerical oscillations is achieved by applying filters to the computed solution after each time step. The filter dissipative terms are based on Artificial Compression Method (ACM), which is applied in the DG finite-element framework. Application of these filters is demonstrated for high order accurate numerical solutions of classical problems with strong hydrodynamic and magnetohydrodynamic discontinuities. It was found that these filters perform equally well with slope limiters and in addition allow shock-capturing with high order spatial discretizations