Anti-diffusive Flux Corrections For High Order Discontinuous Galerkin Method

Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge-Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax-Friedrich numerical flux is used for its simplicity, although it usually smears the contact discontinuity due to its large numerical viscosity. In this paper, we generalize the technique of anti-diffusive flux corrections, recently introduced by Despres and Lagoutiere for first-order schemes, to discontinuous Galerkin method. The objective is to obtain sharp resolution for contact discontinuities, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical tests are performed for one and two dimensional problems to demonstrate the efficiency of this method.