A Radial Basis Function Partition of Unity Method for Transport on the Sphere

The transport phenomena dominates geophysical fluid motions on all scales making the numerical solution of the transport problem fundamentally important for the overall accuracy of any fluid solver. In this thesis, we describe a new high-order, computationally efficient method for numerically solving the transport equation on the sphere. This method combines radial basis functions (RBFs) and a partition of unity method (PUM). The method is mesh-free, allowing near optimal discretization of the surface of the sphere, and is free of any coordinate singularities. The basic idea of the method is to start with a set of nodes that are quasi-uniformly distributed on the sphere. Next, the surface of the sphere is partitioned into overlapping spherical caps so that each cap contains roughly the same number of nodes. All spatial derivatives of the PDE are approximated locally within the caps using RBFs. The approximations from each cap are then aggregated into one global approximation of the spatial derivatives using an appropriate weight function in the PUM. Finally, we use a method-of-lines approach to advance the system in time. We analyze the computational complexity of this method as compared to global methods based on RBFs and present results for several well-known test cases that probe the suitability of numerical methods for modeling transport in spherical geometries. We conclude with possible future directions of the work