RBF-FD Formulas and Convergence Properties

The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.This work has been supported by Spanish MECD Grants FIS2007-62673, FIS2008-04921 and by Madrid Autonomous Region Grant S2009-1597

A well-balanced meshless tsunami propagation and inundation model

We present a novel meshless tsunami propagation and inundation model. We discretize the nonlinear shallow-water equations using a well-balanced scheme relying on radial basis function based finite differences. The inundation model relies on radial basis function generated extrapolation from the wet points closest to the wet-dry interface into the dry region. Numerical results against standard one- and two-dimensional benchmarks are presented.Comment: 20 pages, 13 figure

Optimal shape parameter for the solution of elastostatic problems with the RBF method

Radial basis functions (RBFs) have become a popular method for the solution of partial differential equations. In this paper we analyze the applicability of both the global and the local versions of the method for elastostatic problems. We use multiquadrics as RBFs and describe how to select an optimal value of the shape parameter to minimize approximation errors. The selection of the optimal shape parameter is based on analytical approximations to the local error using either the same shape parameter at all nodes or a node-dependent shape parameter. We show through several examples using both equispaced and nonequispaced nodes that significant gains in accuracy result from a proper selection of the shape parameter.This work was supported by Spanish MICINN Grants FIS2011-28838 and CSD2010-00011 and by Madrid Autonomous Region Grant S2009-1597. M.K. acknowledges Fundación Caja Madrid for its financial support

A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

A numerical model based on radial basis functiongenerated finite differences (RBF-FD) is developed for simulating the global electric circuit (GEC) within the Earth's atmosphere, represented by a 3-D variable coefficient linearelliptic partial differential equation (PDE) in a sphericallyshaped volume with the lower boundary being the Earth's topography and the upper boundary a sphere at 60 km. To ourknowledge, this is (1) the first numerical model of the GECto combine the Earth's topography with directly approximating the differential operators in 3-D space and, related to this,(2) the first RBF-FD method to use irregular 3-D stencils fordiscretization to handle the topography. It benefits from themesh-free nature of RBF-FD, which is especially suitable formodeling high-dimensional problems with irregular boundaries. The RBF-FD elliptic solver proposed here makes nolimiting assumptions on the spatial variability of the coefficients in the PDE (i.e., the conductivity profile), the righthand side forcing term of the PDE (i.e., distribution of current sources) or the geometry of the lower boundary.This work was supported by NSF awards AGS-1135446 and DMS-094581. The National Center for Atmospheric Research is sponsored by the NSF.Publicad