Shock Detection and Limiting with Discontinuous Galerkin Methods for Hyperbolic Conservation Laws

We describe a strategy for detecting discontinuities and for limiting spurious oscillations near such discontinuities when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The approach is based on a strong superconvergence at the outflow boundary of each element in smooth regions of the flow. By detecting discontinuities in such variables as density or entropy, limiting may be applied only in these regions; thereby, preserving a high order of accuracy in regions where solutions are smooth. Several oneand two-dimensional flow problems illustrate the performance of these approaches.