Numerical error analysis in solving ordinary differential equations with emphasis on the accuracy and stability of classical methods

Abstract

Ordinary differential equations (ODEs) play a crucial role in modeling various natural, physical, engineering, and biological phenomena. In most cases, analytical solutions to these equations are not possible or practical; therefore, the use of numerical methods to find approximate solutions is essential. The aim of this paper is to carefully review classical numerical methods, including Euler, Heuwen, and fourth-order Runge–Kutta (RK4), in solving ordinary differential equations. The primary focus is on analyzing numerical error, the stability of the methods, and the accuracy of the results obtained from each algorithm. First, basic concepts related to types of errors, including local and global errors, are presented. Then, convergence and stability criteria in numerical methods are explained. Each of the methods mentioned is introduced and analyzed separately, and then they are used to solve a simple experimental differential equation numerically. The results obtained from the numerical implementation are reviewed through tables and comparative analyses. Comparing the approximate values with the analytical answer shows that the fourth-order Runge–Kutta method has significantly higher accuracy than the Euler and Heun's methods and is also more suitable for slowly decaying issues. It was also observed that reducing the time interval leads to a reduction in error in all methods, but at the cost of increasing computational cost. Finally, the paper provides guidance on selecting the most suitable method, considering the problem's characteristics, the desired level of accuracy, and computational constraints. This research can be useful for students and researchers in the field of applied mathematics and engineering.