Discontinuous Galerkin methods with nodal and hybrid modalodal triangular, quadrilateral, and polygonal elements for nonlinear shallow water flow

摘要

We present a comprehensive assessment of nodal and hybrid modalodal discontinuous Galerkin (DG) finite element solutions on a range of unstructured meshes to nonlinear shallow water flow with smooth solutions. The nodal DG methods on triangles and a tensor-product nodal basis on quadrilaterals are considered. The hybrid modalodal DG methods utilize two different synergistic 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. These are implemented on triangular, quadrilateral, and polygonal elements. In addition, we discuss aspects to be considered in order to achieve the so-called well-balanced property that preserves steady state at rest with a spatially varying bed. The performance in terms of accuracy and computational cost is demonstrated using h and p convergence studies on a nonlinear problem with a manufactured solution and the nonlinear Stommel problem with flat and non-flat beds. To assess the performance of quadrilateral and polygonal elements in comparison to triangular 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 using the quadrilateral elements in terms of computational cost per accuracy and computing time. More importantly, the numerical results clearly show that high order schemes significantly improve the cost performance for a given level of accuracy, with cubic or bi-cubic interpolants particularly achieving dramatic improvements in accuracy as compared to linear and quadratic interpolants, with diminishing benefit as p > 3.