On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy

Radial basis function-generated finite difference (RBF-FD) approximations generalize classical grid-based finite differences (FD) from lattice-based to scattered node layouts. This greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas. Many different types of radial functions have been considered in this RBF-FD context. In this study, we find that (i) polyharmonic splines (PHS) in conjunction with supplementary polynomials provide a very simple way to defeat stagnation (also known as saturation) error and (ii) give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter. In follow-up studies, we will focus on how to best use these hybrid RBF polynomial bases for FD approximations in the contexts of solving elliptic and hyperbolic type PDEs.The presented research was supported by the NSF grants DMS-0934317, OCI-0904599 and by Shell International Exploration and Production, Inc. Victor Bayona was a post-doctoral fellow funded by the Advanced Study Program at the National Center for Atmospheric Research (NCAR) during the development of this research. NCAR is sponsored by the National Science Foundation

Exact polynomial reproduction for oscillatory radial basis functions on infinite lattices

AbstractUntil now, only nonoscillatory radial basis functions (RBFs) have been considered in the literature. It has recently been shown that a certain family of oscillatory RBFs based on J-Bessel functions gives rise to nonsingular interpolation problems and seems to be the only class of functions not to diverge in the limit of flat basis functions for any node layout. This paper proves another interesting feature of these functions: exact polynomial reproduction of arbitrary order on an infinite lattice in ℝn. First, a closed form expression is derived for calculating the expansion coefficients for any order polynomial in any dimension. Then, a proof is given showing that the resulting interpolant, using this class of oscillatory RBFs, will give exact polynomial reproduction. Examples in one and two dimensions are presented. It is specifically noted that such closed form expressions cannot be derived for other classes of RBFs due to the fact that J-Bessel RBFS reproduce polynomials via a different mechanism

A Robust RBF-FD Formulation based on Polyharmonic Splines and Polynomials

We introduce a local method based on radial basis function-generated finite differences (RBF-FD) for interpolation and the numerical solution of partial differential equations (PDEs). The method uses polyharmonic spline (PHS) RBFs together with polynomials to derive differentiation weights on different node configurations. The formulation is explored in three directions: (i) Interpolation and approximation of differential operators, (ii) Elliptic PDEs, and (iii) Hyperbolic PDEs. In particular, the novel RBF-FD methodology is applied to standard test cases in numerical weather prediction, modeled by the compressible Navier-Stokes equations in 2D. Furthermore, the evaluation of the method on different node layouts, Cartesian, hexagonal, and scattered, is studied. The RBF-FD implementation acts as an extension of conventional finite-differences, achieving high accuracy on scattered nodes with no need for a computational mesh