An efficient method for a finite-difference solution of the poisson equation on the surface of a sphere

An efficient shooting method is presented for the numerical solution of a discrete Poisson equation on the surface of the sphere. The solution is computed via two-dimensional shooting in the physical domain while the “missing initial conditions” needed to start the shooting are obtained in a one-dimensional setting in the Fourier domain. For a computational grid with J × J grid points, the operational count of this method is of O(( J 2/ m)log 2 J), where m is the number of grid points within each shooting subrange. The actual count is, for most practical purposes, less than 26 arithmetic operations per grid point. Stability of the method is a problem; this problem, however, can be overcome by the use of the multiple shooting technique. Numerical examples are given to demonstrate the applicability of the procedure.