Lagrange-Galerkin methods on spherical geodesic grids: The shallow water equations

The weak Lagrange-Galerkin finite element method fur the 2D shallow water e quations on the sphere is presented. This method offers stable and accurate solutions because the equations are integrated along thr characteristics. The equations are written in 3D Cartesian conservation form and the domains are discretized using linear triangular elements. The use of linear triang ular elements permits the construction of accurate (by virtue of the second -order spatial and temporal accuracies of the scheme) and efficient (by vir tue of the less stringent CFL condition of Lagrangian methods) schemes on u nstructured domains. Using linear triangles in 3D Cartesian space allows fo r the explicit construction of area coordinate basis functions thereby simp lifying the calculation of the finite element integrals. The triangular gri ds are constructed by a generalization of the icosahedral grids that have b een typically used in recent papers. An efficient searching strategy fur th e departure points is also presented for these generalized icosahedral grid s which involves very few Boating point operations. In addition a high-orde r scheme for computing the characteristic curves in 3D Cartesian space is p resented: a general family of Runge-Kutta schemes. Results for six test cas es are reported in order to confirm the accuracy of the scheme.