One-dimensional shock-capturing for high-order Discontinuous Galerkin methods

Abstract

Discontinuous Galerkin methods have emerged in recent years as a reasonable alternative for nonlinear conservation equations. In particular, their inherent structure (the need of a numerical flux based on a suitable approximate Riemann solver which in practice introduces some stabilization) seem to suggest that they are specially adapted to capture shocks. however, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for high-order discontinuous Galerkin. Thus, slope-limiter methods, which are extensions of finite volume methods, have been proposed for high-order approximations. Here it is shown that these techniques require mesh adaption and a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology.