Lagrange–Galerkin Methods on Spherical Geodesic Grids: The Shallow Water Equations

The weak Lagrange–Galerkin finite element method for the 2D shallow water equations on the sphere is presented. This method offers stable and accurate solutions because the equations are integrated along the characteristics. The equations are written in 3D Cartesian conservation form and the domains are discretized using linear triangular elements. The use of linear triangular elements permits the construction of accurate (by virtue of the second-order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent CFL condition of Lagrangian methods) schemes on unstructured domains. Using linear triangles in 3D Cartesian space allows for the explicit construction of area coordinate basis functions thereby simplifying the calculation of the finite element integrals. The triangular grids are constructed by a generalization of the icosahedral grids that have been typically used in recent papers. An efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations. In addition a high-order scheme for computing the characteristic curves in 3D Cartesian space is presented: a general family of Runge–Kutta schemes. Results for six test cases are reported in order to confirm the accuracy of the scheme.