The Cartesian method for solving partial differential equations in spherical geometry

Cartesian coordinates are used to solve the nonlinear shallow-water equations on the sphere. The two-dimensional equations, in spherical coordinates, are first embedded in a three-dimensional system in a manner that preserves solutions of the two-dimensional system. That is, solutions of the three-dimensional system, with appropriate initial conditions, also solve the two-dimensional system on the surface of the sphere. The higher dimensional system is then transformed to Cartesian coordinates, Computations are limited to the surface of the sphere by projecting the equations, gradients, and solution onto the surface. The projected gradients are approximated by a weighted sum of function values on a neighboring stencil. The weights are determined by collocation using the spherical harmonics in trivariate polynomial form. That is, the weights are computed from the requirement that the projected gradients are near exact for a small set of spherical harmonics. The method is applicable to any distribution of points and two test cases are implemented on an icosahedral geodesic grid. The method is both vectorizable and parallelizable. (C) 1997 Elsevier Science B.V.