Balancing Precision and Efficiency: Comparative Analysis of Numerical Methods for Initial Value Problems

Numerical methods are essential tools for solving Ordinary Differential Equations (ODEs), especially when obtaining analytical solutions is challenging or impossible.The Euler, Runge-Kutta, Heun, and Milne methods are some of the most popular numerical techniques for solving Initial Value Problems (IVPs) of ODEs among the many that have been developed.These methods are particularly effective for approximating the solutions of first-order differential equations, which are common in scientific and engineering problems such as physics simulations, population dynamics, and chemical reactions.A considerable body of work exists that compares the effectiveness of these methods for solving IVPs in ODEs.Early comparisons of the Euler's and Runge-Kutta methods, such as those by Butcher (2016), Atkinson et al. (2019) and Boyce and DiPrima (2001) emphasized the relative simplicity of Euler's method but also pointed out its limitations in terms of accuracy and stability, particularly for stiff equations.On the contrary, the RK4 method was shown to provide Volume 5,