Radial Basis Function-generated Finite Differences: A Mesh-free Method for Computational Geosciences

Radial basis function generated finite differences (RBF-FD) is a mesh-free method for numerically solving partial differential equations (PDEs) that emerged in the last decade and has shown rapid growth in the last few years. From a practical standpoint, RBF-FD sprouted out of global RBF methods, which have shown exceptional numerical qualities in terms of accuracy and time stability for numerically solving PDEs, but are not practical when scaled to very large problem sizes because of their computational cost and memory requirements. RBF-FD bypass these issues by using local approximations for derivatives instead of global ones. Matrices in the RBF-FD methodology go from being completely full to 99% empty. Of course, the sacrifice is the exchange of spectral accuracy from the global RBF methods for high-order algebraic convergence of RBF-FD, assuming smooth data. However, since natural processes are almost never infinitely differentiable, little is lost and much gained in terms of memory and runtime. This article provides a survey of a group of topics relevant to using RBF-FD for a variety of problems that arise in the geosciences. Particular emphasis is given to problems in spherical geometries, both on surfaces and within a volume. Applications discussed include non-linear shallow water equations on a sphere, reaction-diffusion diffusion equations, global electric circuit, and mantle convection in a spherical shell. The results from the last three of these applications are new and have not been presented before for RBF-FD. ∗This work was partially supported by NSF grant DMS-0934317. †This work was partially supported by NSF grants DMS-0934581 and DMS-1160379. ‡This work was partially supported by NSF grant DMS-0914647.