Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter on unstructured meshes

In this paper we generalize a new type of compact Hermite weighted essentially nonoscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) methods, which were recently developed in [34] for structured meshes, to two dimensional unstructured triangular meshes. The main idea of this limiter is to reconstruct the new polynomial using the entire polynomials of the DG solution from the target cell and its neighboring cells in a least square fashion [10] while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.