On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere

Radial basis function (RBF) approximations have been used for some time to interpolate data on a sphere (as well as on many other types of domains). Their ability to solve, to spectral accuracy, convection-type PDEs over a sphere has been demonstrated only very recently. In such applications, there are two main choices that have to be made: (i) which type of radial function to use, and (ii) what value to choose for their shape parameter (denoted by ε, and with flat basis functions -stretched out in the radial directioncorresponding to ε = 0). The recent RBF-QR algorithm has made it practical to compute stably also for small values of ε. Results from solving a convective-type PDE on a sphere are compared here for many choices of radial functions over the complete range of ε-values (from very large down to the limit of ε → 0). The results are analyzed with a methodology that has similarities to the customary Fourier analysis in equispaced 1-D periodic settings. In particular, we find that high accuracy can be maintained also over very long time integrations. We furthermore gain insights into why RBFs sometimes offer higher accuracy than spherical harmonics (since the latter arise as an often non-optimal special case of the former). Anticipated future application areas for RBF-based methods in spherical geometries include weather and climate modeling.