Analytical and Numerical Advances in Radial Basis Functions

Radial basis function (RBF) approximations have been used for some time to in-terpolate data on a sphere (as well as on many other types of domains). Theirability to solve, to spectral accuracy, convection-type PDEs over a sphere has beendemonstrated only very recently. In such applications, there are two main choicesthat have to be made: (i) which type of radial function to use, and (ii) what valueto 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-QRalgorithm has made it practical to compute stably also for small values of ε. Resultsfrom solving a convective-type PDE on a sphere are compared here for many choicesof radial functions over the complete range of ε-values (from very large down to thelimit of ε → 0). The results are analyzed with a methodology that has similaritiesto 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 thanspherical harmonics (since the latter arise as an often non-optimal special case of theformer). Anticipated future application areas for RBF-based methods in sphericalgeometries include weather and climate modeling.