Sub-Cell Shock Capturing for Discontinuous Galerkin Methods

A shock capturing strategy for higher order Discontinuous Galerin approximations of scalar conservation laws is presented. We show how the original explicit artificial viscosity methods proposed over fifty years ago for finite volume methods, can be used very effectively in the context of high order approximations. Rather than relying on the dissipation inherent in Discontinuous Galerkin approximations, we add an artificial viscosity term which is aimed at eliminating the high frequencies in the solution, thus eliminating Gibbs-type oscillations. We note that the amount of viscosity required for stability is determined by the resolution of the approximating space and therefore decreases with the order of the approximating polynomial. Unlike classical finite volume artificial viscosity methods, where the shock is spread over several computational cells, we show that the proposed approach is capable of capturing the shock as a sharp, but smooth profile, which is typically contained within one element. The method is complemented with a shock detection algorithm which is based on the rate of decay of the expansion coefficients of the solution when this is expressed in a hierarchical orthonormal basis. For the Euler equations, we consider and discuss the performance of several forms of the artificial viscosity term. I