High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

Abstract

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2 N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129–150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss–Lobatto quadrature points (order 2 N − 1) and a set based on Gauss quadrature points (order 2 N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2 N integration being the most accurate of the four DG methods studied. The strong advection form with 2 N integration performed extremely well even for flows with shock waves. The strong conservation form with 2 N − 1 integration yielded results almost as good as those with 2 N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.