RKDG with WENO type limiters

Abstract

The discontinuous Galerkin (DG) method is a spatial discretization procedure for convection dominated equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as RKDG when TVD Runge-Kutta method is applied for time discretization. It has the advantage of flexibility in handling complicated geometry, h∈-∈p adaptivity, and efficiency of parallel implementation and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this chapter, we will describe the procedure of using WENO and Hermite WENO finite volume methodology as limiters for RKDG methods on unstructure meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition for RKDG methods. © 2010 Springer-Verlag Berlin Heidelberg.