A new vertex-based limiting approach for nodal discontinuous Galerkin methods on arbitrary unstructured meshes

The nodal discontinuous Galerkin (DG) methods possess many good properties that make them very attractive for numerically solving the shallow water equations, but it is necessary to maintain numerical monotonicity by applying a slope-limiting approach to eliminate spurious oscillations. In this study, a new vertex-based slope limiter is developed for the nodal DG method on arbitrary unstructured meshes. This new limiting approach modifies the vertex-based limiter by a weighted reconstruction function, which satisfies the maximum principle and is totally free of tuning parameters. Three different weighting functions are proposed by modifying the stencil reconstruction methods from other limiters, and their performance and accuracy are compared through two shock flow problems and a laboratory-scale tsunami problem on triangular and quadrilateral meshes. The results indicate that the proposed limiter can eliminate the nonphysical oscillations efficiently and provide accurate and robust results on both triangular and quadrilateral unstructured meshes.