National Center for Atmospheric Research

Spectral element formulations of the atmospheric 2-D shallow-water equations on the cubed-sphere are described. The equations are written in generalized curvilinear coordinates using contravariant/covariant components following Rancic, Purser and Mesinger (1996). A semi-implicit time discretization results in a Helmholtz problem for the pressure. The Laplacian operator is approximated by the L_2 pseudo-Laplacian arising in the P_N/P_N-2 spectral element formulation of the incompressible Stoke's problem. The two-level overlapping Schwarz preconditioner of Fischer and Tufo (1998), based on the fast diagonalization method (FDM) and scalable coarse grid solver, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite-element tensor-product approximation within each spectral element, the minimum of the inverse metric tensor and the maximum of its determinant are employed. Convergence rates and parallel CPU timings on an IBM SP are compared against a block-Jacobi preconditioner.