Radial basis functions for solving differential equations: Ill-conditioned matrices and numerical stability

High-order numerical methods for solving differential equations are, in general, fairly sensitive to perturbations in their data. A previously proposed radial basis function (RBF) method, namely an integrated multiquadric scheme (IMQ), is applied to two-point boundary value problems whose solutions exhibit thin boundary layers. As frequently observed among RBF methods, the matrices arising are ill-conditioned, in this paper to the point of numerical singularity. The sensitivity of the method to perturbations and round-off error is investigated, and evidence is provided that perturbations are not nearly as strongly amplified as suggested by the large condition numbers of the matrices used in the computation. (C) 2015 Elsevier Ltd. All rights reserved.