An In-Depth Stability and Convergence Analysis of the Runge-Kutta 4th Order Method for Nonlinear Ordinary Differential Equations

Abstract

The Runge-Kutta 4th Order (RK4) method is one of the most widely used numerical techniques for solving ordinary differential equations (ODEs), particularly in cases where analytical solutions are difficult or impossible to obtain. This paper presents an in-depth analysis of the stability and convergence properties of the RK4 method when applied to nonlinear ODEs. We begin by reviewing the theoretical foundation of the RK4 algorithm, including its derivation, local and global error characteristics, and conditions for convergence. A formal definition of numerical stability is provided, and the method’s behavior is evaluated using the linear test equation as well as nonlinear systems such as the logistic equation, the Van der Pol oscillator, and the Lorenz attractor. Through numerical experiments, we analyze how step size influences accuracy and stability, particularly in stiff and chaotic systems. Our results confirm that RK4 achieves fourth-order convergence for a wide range of nonlinear problems but also highlight limitations in stability, especially for stiff equations or long-term simulations of sensitive dynamical systems. The findings offer practical insights into when RK4 can be reliably used and when more advanced or adaptive methods may be necessary.