Multidomain Hybrid RKDG and WENO Methods for Hyperbolic Conservation Laws

Abstract

In this paper, we develop two versions of multidomain hybrid methods by combining Runge--Kutta discontinuous Galerkin (RKDG) methods and weighted essentially nonoscillatory (WENO) schemes. One is a conservative version based on a third order RKDG method and a fifth order finite volume WENO (WENO-FV) scheme, the other is a nonconservative version based on a third order RKDG method and a fifth order finite difference WENO (WENO-FD) scheme. At the artificial interface of coupling RKDG and WENO, special treatments are used to tackle with discontinuities such as shock waves and preserve high order accuracy for smooth solution as well. We extend the nonconservative multidomain hybrid RKDG and WENO-FD (RKDG+WENO-FD) method to one- and two-dimensional systems of conservation laws for consideration of computational efficiency. Theoretical analysis shows the hybrid RKDG+WENO-FD method has high order accuracy for smooth solution, and numerical results also demonstrate that the hybrid solver is robust for slow and strong shock simulations. Several applications are used to demonstrate the flexibility of the hybrid RKDG+WENO-FD method in handling complex geometries and the capability of saving computational cost in comparison to the traditional RKDG method.