Adaptive Gaussian radial basis function methods for initial value problems: Construction and comparison with adaptive multiquadric radial basis function methods

Abstract

Adaptive radial basis function (RBF) methods have been developed recently in Gu and Jung (2020) based on the multiquadric (MQ) RBFs for solving initial value problems (IVPs). The proposed adaptive RBF methods utilize the free parameter in order to adaptively enhance the local convergence of the numerical solution. Methods pertaining to the polynomial interpolation yield only fixed rate of convergence regardless of the solution smoothness while the proposed methods use the smoothness of the solution, given in derivatives of the solution, to control the rate of convergence. In this paper, for the completion of the development of the adaptive RBF methods, we develop various adaptive Gaussian RBF methods for solving IVPs by modifying the classical solvers such as the Euler's method, midpoint method, Adams-Bashforth method and Adams-Moulton method by replacing the polynomial basis with the Gaussian RBFs. For each development, we compare the performance with the adaptive MQ-RBF methods and explain when and why the adaptive Gaussian methods are better or not than the MQ-RBF ones. We provide the collection of modifications with the MQ and Gaussian RBFs. We also provide the stability regions for the adaptive Gaussian methods. Numerical results confirm that the adaptivity enhances accuracy and convergence and also show the differences and similarities between MQ and Gaussian RBFs in their performance - we found that the adaptive MQ-RBF method has larger stability region than the Gaussian RBF method. Both MQ and Gaussian RBF methods yield the desired order of convergence while the superiority of one method to the other depends on the method and the problem considered. (C) 2020 Elsevier B.V. All rights reserved.