Quadrature-free discontinuous Galerkin method with code generation features for shallow water equations on automatically generated block-structured meshes

Abstract

Although discretizations of the shallow water equations (SWE) based on the discontinuous Galerkin (DG) method are well established, their computational performance still generally lags behind that of the finite volume discretizations. In explicit and semi-implicit time stepping schemes commonly used in connection with the SWE models, the most computationally expensive parts of a DG algorithm are the element and edge integrals computed via loops over quadrature points. We propose a quadrature-free DG formulation for the SWE that replaces quadrature integrations by analytical evaluations. The method is implemented within the code generation framework of the ExaStencils project using the SymPy Python library. The new formulation uses block-structured triangular meshes automatically generated for a given number of blocks.