Runge--Kutta Discontinuous Galerkin Method Using WENO Limiters

Abstract

The Runge--Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws. It uses ideas from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, total variation diminishing (TVD) Runge--Kutta time discretizations, and limiters. 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 nonoscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper we investigate using WENO finite volume methodology as limiters for RKDG methods, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. The traditional finite volume WENO framework based on cell averages is used to reconstruct point values of the solution at Gaussian-type points in those cells where limiting is deemed necessary, and the polynomial solutions in those cells are then rebuilt through numerical integration using these Gaussian points. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure.