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

In this paper, a family of two- and three-stage explicit multiquadric (MQ) and inverse multiquadric (IMQ) radial basis functions (RBFs) Runge–Kutta methods is introduced for solving ordinary differential equations. These methods are developed by utilizing MQ- and IMQ-radial basis function (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 methods. 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 Runge–Kutta methods.