A hybrid Hermite-WENO/slope limiter for reconstructed discontinuous Galerkin methods on unstructured grids

In this paper, we propose a new limiter for reconstructed discontinuous Galerkin or hybrid discontinuous Galerkin and finite volume (Hybrid DG/FV) methods for solving Euler equations on two-dimensional unstructured triangular grids. The concept of hierarchical limiter which consists of two steps is introduced. The limiter firstly reconstructs a Hermite WENO polynomial by taking the advantage of the underlying DG method where the exact low-order (first-order) derivatives are available. The linear weights for the bias polynomials are optimized by a Lagrangian interpolation. To maintain the accuracy of the original methods in the smoothing region, a correction is introduced by using the second-order derivatives which are already computed in the original hybrid DG/FV methods. Then a slope limiter is used for the limitation of all the second-order derivatives. The present limiter is compact as only the von Neumann neighborhoods are required. Numerical results for both smoothing and shock contained problems are provided to validate the good performance of the present limiter.