Explicit Runge-Kutta Methods with Multiquadric and Inverse Multiquadric Radial Basis Functions

TL;DR

This paper introduces new explicit Runge-Kutta methods based on multiquadric and inverse multiquadric radial basis functions that achieve higher accuracy without increasing stages, validated through stability analysis and numerical experiments.

Contribution

The paper develops RBF-based Runge-Kutta methods that improve accuracy over classical schemes using RBF corrections and optimal shape parameters.

Findings

Achieve one-order higher accuracy than classical methods

Maintain stability and convergence comparable to standard schemes

Numerical experiments confirm improved performance

Abstract

In this article, a family of two- and three-stage explicit multiquadric (MQ) and inverse multiquadric (IMQ) radial basis functions (RBFs) Runge-Kutta methods are introduced for solving ordinary differential equations. These methods are developed by utilizing MQ- and IMQ-RBF Euler methods. The main advantage of these RBF-based methods lies in their ability to achieve a one-order higher accuracy than their classical Runge-Kutta counterparts without increasing the number of stages. This improvement is made possible by incorporating RBF corrections, where the optimal shape parameter is determined through the local truncation error analysis of the proposed schemes. Convergence and stability analyses, including the study of stability regions, are presented to illustrate how these methods compare with standard Runge-Kutta schemes. Numerical experiments on five benchmark problems further confirm predicted accuracy and stability, demonstrating that MQ- and IMQ-based RBF Runge-Kutta methods provide an alternative to conventional low-stage explicit RungeKutta schemes.