Comparative Analysis of Explicit Runge-Kutta Methods for Solving First-Order Linear Ordinary Differential Equations

This study analyzes and compares the explicit Runge-Kutta methods for solving first-order linear differential equations. The methods examined include the Forward Euler method, Heun's method, and fourth-order Runge-Kutta method. The research methodology involved stability analysis and convergence studies to evaluate the performance of these methods. The results indicate that the fourth-order Runge-Kutta method offers the best performance because of its high convergence rate and wider stability regions, whereas the Forward Euler method has the smallest stability region and lower convergence accuracy. Heun's method also demonstrates satisfactory performance in terms of stability and convergence but is less effective compared to the fourth-order Runge-Kutta method. This study is particularly significant in enhancing the understanding of the accuracy and stability of explicit numerical methods for solving differential equations, showing that the fourth-order Runge-Kutta method is more efficient, in terms of both convergence and stability, for the problems under consideration.