One-dimensional shock-capturing for high-order discontinuous Galerkin methods - Casoni, E.; Peraire, J.; Huerta, A. (International journal for numerical methods in fluids, 2013)
SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, 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. Copyright © 2012 John Wiley & Sons, Ltd. Figure 1 compares the artificial diffusion technique proposed here with high‐order limiters (described as moments’ in the figures), with the same number of degrees of freedom. The artificial diffusion technique outperforms the high‐order limiting scheme, demonstrating that it seems more beneficial to increase the degree of approximation than to refine the mesh in order to obtain accurate solutions.