Discontinuous Galerkin Methods with Nodal and Hybrid Modal/Nodal Triangular, Quadrilateral, and Polygonal Elements for Nonlinear Shallow Water Flow

We present a comprehensive assessment of nodal and hybrid modal/nodal dis-continuous Galerkin (DG) finite element solutions on a range of unstructured meshes for nonlinear shallow water flow. The nodal DG methods on triangles and a tensor-product nodal basis on quadrilaterals are considered. The hybrid modal/nodal DG methods utilize two different polynomial bases on polygons in realizing the DG discretization; orthogonal basis functions constructed by the Gram-Schmidt process are used as trial and test functions in a DG weak formulation and a nodal basis is used as an efficient means for area integration. The performance of these methods in terms of accuracy and computational cost is demonstrated using h and p convergence studies on a two-dimensional nonlinear shallow water problem with a manufactured solution. To assess the performance of quadrilateral and polygonal elements in comparison to trian-gular elements, we consider a setting in which a quadrilateral mesh, a mixed triangular-quadrilateral mesh, and polygonal mesh are derived from a given triangular mesh and vice versa. The tests conducted reveal the merit of us-ing the quadrilateral elements in terms of computational cost per accuracy an