Combining radial basis functions with the partition-of-unity method for numerically solving PDEs on the sphere

We discuss a new collocation-type method for numerically solving partial differential equations (PDEs) on the sphere.  The method uses radial basis function (RBF) approximations in a partition of unity framework for approximating spatial derivatives on the sphere.  High-orders of accuracy are achieved for smooth solutions, while the overall computational cost of the method scales linearly with the number of unknowns.  The discussion will be primarily limited to the transport equation and results will be presented for a few well-known test cases.  We conclude with a preliminary application to the non-linear shallow water wave equations on a rotating sphere.