A Comparison of Three Numerical Methods for Solving Differential Equations on the Sphere

Abstract

We compare three numerical methods for solving vector differential equations on a sphere. A composite mesh finite-difference method using overlapping stereographic coordinate systems is compared to transform methods based on scalar and vector spherical harmonics. The methods are compared in terms of total computer time, memory requirements, and execution rates for relative accuracy requirements of two and four digits in a five-day forecast. The computational requirements of the three methods were well within an order of magnitude of one another. In most of the cases that are examined, the time step was limited by accuracy rather than stability. This problem can be overcome by the use of a higher order time integration scheme, but at the expense of an increase in the memory requirements.