title: A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction-Diffusion Equations on Surfaces.
abstract: In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in ℝd. Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximatedlocallyusing RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.
authors.affiliation: School of Computing, University of Utah, Salt Lake City, UT 84112.
authors.forename: Varun
authors.initials: V
authors.lastname: Shankar
authors.affiliation: Department of Mathematics, Boise State University, Boise, ID 83725-1555.
authors.forename: Grady B
authors.initials: GB
authors.lastname: Wright
authors.affiliation: School of Computing, University of Utah, Salt Lake City, UT 84112.
authors.forename: Robert M
authors.initials: RM
authors.lastname: Kirby
authors.affiliation: Department of Mathematics, University of Utah, Salt Lake City, UT 84112.
authors.forename: Aaron L
authors.initials: AL
authors.lastname: Fogelson
keywords: RBF-FD; finite differences; manifolds; mesh-free; method-of-lines; radial basis functions; reaction-diffusion