Radial basis functions: Developments and applications to planetary scale flows

Radial basis functions (RBFs) can be seen as a major generalization of pseudospectral (PS) methods, abandoning the orthogonality of the basis functions and in return obtaining much improved simplicity and geometric flexibility. Spectral accuracy becomes now easily available also when using completely unstructured node layouts, permitting local node refinements in critical areas. The first major PDE applications for which RBFs have been shown to compete successfully against the best currently available numerical approaches can be found in the geosciences. Examples that are discussed here include translating vortex roll-ups (cyclogenesis), nonlinear flows on the sphere modeled by the shallow water equations, and 3D convection in the earth’s mantle.