Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method, III: Unstructured Meshes

Abstract

In [J. Comput. Phys. 193:115---135, 2004] and [Comput. Fluids 34:642---663, 2005], Qiu and Shu developed a class of high order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving nonlinear hyperbolic conservation law systems, and applied them as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods on structured meshes. In this continuation paper, we extend the method to solve two dimensional problems on unstructured meshes. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers' equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.