are undefined at the poles. These facts prevent a straightforward implementation of finite-difference and spectral A new gridding technique for the solution of partial differential equations in spherical geometry is presented. The method is based methods previously developed in Cartesian coordinates on a decomposition of the sphere into six identical regions, obtained and they have been the main motivation for the developby projecting the sides of a circumscribed cube onto a spherical ment of alternative techniques for solving partial differensurface. By choosing the coordinate lines on each region to be arcs tial equations on the sphere (see Browning et al. [4] for of great circles, one obtains six coordinate systems which are free a detailed discussion).
of any singularity and define the same metric. Taking full advantage of the symmetry properties of the decomposition, a variation of the In one of his pioneering papers on large-scale numerical composite mesh finite difference method can be applied to couple weather prediction, Phillips [16] put forward a list of desirthe six grids and obtain, with a high degree of efficiency, very able features that a mapping of the sphere and the numeriaccurate numerical solutions of partial differential equations on the cal scheme based on it should have in order to be used sphere. The advantages of this new technique over both spectral with success for global forecasting purposes:
and uniform longitude-latitude grid point methods are discussed in the context of applications on serial and parallel architectures.
- The mapping should be free of any singularities.
We present results of two test cases for numerical approximations to the shallow water equations in spherical geometry: the linear 2. The mapping should preserve the general form of advection of a cosine bell and the nonlinear evolution of a Rossbythe hydrodynamic equations.
Haurwitz wave. Performance analysis for this latter case indicates 3. The physical grid defined on the spherical surface that the new method can provide, with substantial savings in execution times, numerical solutions which are as accurate as those obshould be as close as possible to a regular equidistant grid.
tainable with the spectral transform method.