Cubed Sphere Grid

While understanding how to apply the 3D mesh topology to meteorological and climatological simulations, I took inspiration from observing the shape of a waterpolo ball, which is divided in only 6 identical zones, a different look compared to the classical pentagonal and hexagonal patches of the soccer/european football. It was indeed possible to construct a quasi-uniform grid on a sphere, by looking at how the grid on a circonscribed cube would project on the inscribed ball (...sphere). With the help of experts in meteorology and climatology, Corrado Ronchi and Roberto Iacono, we obtained a really efficient method for Finite Difference simulations of spherical problems. Note that a classical latitude longitude mesh creates singularities at the poles, and huge differences of areas between the patches at the poles and near the equator.

Here-below attached the journal paper describing the solution and the previous conference presentation introducing the idea.

C. Ronchi, R. Iacono, and Pier S. Paolucci. 1996. The “cubed sphere”: a new method for the solution of partial differential equations in spherical geometry. J. Comput. Phys. 124, 1 (March 1996), 93-114. DOI=10.1006/jcph.1996.0047 http://dx.doi.org/10.1006/jcph.1996.0047

Abstract: A new gridding technique for the solution of partial differential equations in spherical geometry is presented. The method is based on a decomposition of the sphere into six identical regions, obtained by projecting the sides of a circumscribed cube onto a spherical surface. By choosing the coordinate lines on each region to be arcs of great circles, one obtains six coordinate systems which are free of any singularity and define the same metric. Taking full advantage of the symmetry properties of the decomposition, a variation of the composite mesh finite difference method can be applied to couple the six grids and obtain, with a high degree of efficiency, very accurate numerical solutions of partial differential equations on the sphere. The advantages of this new technique over both spectral and uniform longitude–latitude grid point methods are discussed in the context of applications on serial and parallel architectures. We present results of two test cases for numerical approximations to the shallow water equations in spherical geometry: the linear advection of a cosine bell and the nonlinear evolution of a Rossby–Haurwitz wave. Performance analysis for this latter case indicates that the new method can provide, with substantial savings in execution times, numerical solutions which are as accurate as those obtainable with the spectral transform method.

The first paper introducing the idea was presented at HPCN'95

Corrado Ronchi, Roberto Iacono, and Pier Stanislao Paolucci. “Finite difference approximation to the shallow water equations on a quasi-uniform spherical grid.” In Proceedings of the International Conference and Exhibition on High-Performance Computing and Networking (HPCN Europe '95), Bob Hertzberger and Giuseppe Serazzi (Eds.). Springer-Verlag, London, UK, (1995) 741-747.