Shock capturing for discontinuous Galerkin methods

Abstract

This work is devoted to solve scalar hyperbolic conservation laws in the presence of strong shocks with discontinuous Galerkin methods (DGM). A standard approach is to use limiting strategies in order to avoid oscillations in the vicinity of the shock. Basically, these techniques reconstruct the solution with a lower order polynomial in those elements where discontinuities lie. These limiting procedures degrade the accuracy of the method and introduce an excessive amount of dissipation to the solution, in particular for high-order approximations. The aim of the present work is to use artificial diffusion instead of limiters to capture the shocks. We show preliminary results with the inviscid's Burgers equation and also with a convection-diffusion problem