The Overlapped Radial Basis Function-Finite Difference (RBF-FD) Method: A Generalization of RBF-FD

摘要

We present a generalization of the RBF-FD method that computes RBF-FD weightsin finite-sized neighborhoods around the centers of RBF-FD stencils byintroducing an overlap parameter $\delta \in [0,1]$ such that $\delta=1$recovers the standard RBF-FD method and $\delta=0$ results in a full decouplingof stencils. We provide experimental evidence to support this generalization,and develop an automatic stabilization procedure based on local Lebesguefunctions for the stable selection of stencil weights over a wide range of$\delta$ values. We provide an a priori estimate for the speedup of our methodover RBF-FD that serves as a good predictor for the true speedup. We apply ourmethod to parabolic partial differential equations with time-dependentinhomogeneous boundary conditions-- Neumann in 2D, and Dirichlet in 3D. Ourresults show that our method can achieve as high as a 60x speedup in 3D overexisting RBF-FD methods in the task of forming differentiation matrices.