Adaptive Multiquadratic Radial Basis Function-based Explicit Runge--Kutta Methods

TL;DR

This paper introduces an adaptive multiquadratic RBF-based explicit Runge--Kutta method that achieves higher convergence order and improved accuracy over classical methods through parameter optimization and stability analysis.

Contribution

It develops a novel adaptive RBF-based explicit RK method with optimized parameters for enhanced convergence and accuracy, validated by stability analysis and numerical comparisons.

Findings

Higher order of convergence achieved

Improved accuracy over classical RK methods

Numerical examples validate superiority

Abstract

Runge--Kutta (RK) methods are widely used techniques for solving a class of initial value problems. In this article, we introduce an adaptive multiquadratic (MQ) radial basis function (RBF)-based method to develop enhanced explicit RK methods. These methods achieve a higher order of convergence than the corresponding classical RK methods. To improve the local convergence of the numerical solution, we optimize the free parameters (shape functions) involved in the RBFs by forcing the local truncation errors to vanish. We also present a convergence and stability analysis of the proposed methods. To demonstrate the advantages of these methods in terms of accuracy and convergence, we consider several numerical examples and compare the performance of our methods with that of the classical RK methods. The Tables and Figures presented in this article clearly validate the superiority of the proposed methods.