Abstract

In this study, the performance of four numerical techniques for solving Initial Value Problems (IVPs) in ordinary differential equations: Euler’s method, Runge-Kutta fourth-order (RK4), Heun’s method, and Milne’s method is evaluated. Based on the analysis, RK4 and Milne’s methods provide excellent accuracy and sustain low absolute errors over time, thus making them perfect for high-precision tasks. Heun’s method offers a balance between accuracy and efficiency, making it a moderate advance over Euler’s. On the other hand, Euler’s method exhibits the biggest absolute errors, particularly when step sizes are bigger, which makes it less appropriate for applications that demand a high level of precision. The findings show that while choosing a numerical approach for IVPs, accuracy and computational efficiency must be balanced.

Keywords: Initial value problem, Heun’s method, Milne’s method, RK4 method, Error analysis, Stability