Comparative analysis of Euler's and Runge-Kutta's methods for solving differential equations

Abstract

Ordinary differential equations (ODEs) are among the most vital mathematical tools utilized in modeling numerous scientific and engineering phenomena. This research aims to conduct a comparative analysis between Euler's method and the fourth-order Runge-Kutta (RK4) method for solving ODEs, by investigating numerical accuracy, numerical stability, and computational efficiency, while demonstrating the impact of step size on the quality of the numerical solution. The comparative results reveal that the Runge-Kutta method achieves a higher level of accuracy and stability compared to Euler's method. Conversely, Euler's method is characterized by ease of implementation and low computational cost, indicating that selecting the appropriate method depends on the nature of the problem and the required level of accuracy.