Comparative Analysis of Euler and Order Four Runge-Kutta Methods in Adams-Bashforth-Moulton Predictor-Corrector Method

Abstract

This study conducts a comparative analysis of the Euler and Runge-Kutta 4 methods within the Adams Predictor-Corrector framework for solving second-order ordinary differential equations (ODEs) and coupled differential equations. The Euler method, a basic first-order explicit scheme, and the Runge-Kutta 4 method, a higher-order explicit scheme, are widely utilized numerical methods for ODEs. However, their effectiveness varies based on the problem’s characteristics. We specifically investigate these methods integrated into the Adams Predictor-Corrector method, renowned for its stability and efficiency in solving initial value problems. Through both numerical experiments and theoretical analysis, we establish that the Runge-Kutta method demonstrates superior performance over the Euler method within this framework. Additionally, we observe that the 4th order of the Adams Predictor-Corrector method yields more accurate results than the 5th order for second-order ODEs, while the 5th order performs better for coupled ODEs. Overall, the Runge-Kutta method exhibits enhanced accuracy and stability compared to the Euler method in the context of the Adams Predictor-Corrector method.