The $p$-Weighted Limiter for Nodal Discontinuous Galerkin Methods on Two Dimensional Mixed Grids

Abstract

Mixed grids offer advantages in grid generation, calculation efficiency and numerical accuracy. The $p$-weighted limiter works effectively on 1D, triangular and tetrahedral grids for the discontinuous Galerkin (DG) method. Whereas the generalization to unstructured mixed grids for nodal DG methods is not straightforward. On quadrilateral grids, the tensor-product of solution points for nodal DG methods makes it complex and time-consuming to construct bi-linear candidate polynomials and perform weighted summation with the tensor-product polynomials. A simple yet effective way is proposed to address this issue. The linear candidate polynomial is constructed by interpolating values on the face and performing a forward difference in the central troubled cell. The truncation strategy on the linear weights is also improved. The weighted summation is carried out dimension-by-dimension on local coordinates as the 1D case. Since the limiting only depends on the central cell type, the final implementation for nodal DG methods on mixed grids is straightforward. A parameter-free shock detector is proposed to identify the troubled cells near shock waves. Several inviscid, laminar and turbulent flows with shocks are employed to showcase the efficiency of the improved $p$-weighted limiter and shock detector on unstructured mixed grids for nodal DG schemes.