Author(s)

V. Bayona, M. Moscoso & M. Kindelan

Abstract

In this work we derive analytical expressions for the weights of Gaussian RBFFD formulas for some differential operators. These weights are used to derive analytical expressions for the leading order approximations to the local truncation error in powers of the internode distance h and the shape parameter
. We show that for each differential operator, there is a range of values of the shape parameter for which RBF-FD formulas are significantly more accurate than the corresponding standard FD formulas. In fact, very often there is an optimal value of the shape parameter
+ for which the local error is zero to leading order. This value can be easily computed from the analytical expressions for the leading order approximations to the local error. Contrary to what is generally believed, this value is, to leading order, independent of the internodal distance and only dependent on the value of the function and its derivatives at the node. Keywords: Gaussian RBF, RBF-FD formulas, optimal shape parameter. 1 Introduction Radial basis functions (RBFs) were first used as an efficient technique for interpolation of multidimensional scattered data [7]. Later, it became popular as a truly mesh-free method for the solution of partial differential equations (PDEs) on irregular domains. This application of RBFs was first proposed by Edward Kansa [12, 13] and it is based on collocation in a set of scattered nodes. The main advantages of themethod are ease of programming and potential spectral accuracy, but its main drawback is ill-conditioning of the resulting linear system. To overcome this drawback a local RBF method was independently proposed by several authors [15–17]. The method is based on approximating the solution as a linear combination of a set of identical RBFs translated to a set of RBF centers.

Keywords

Gaussian RBF, RBF-FD formulas, optimal shape parameter.