A Competitive Study on the Euler and Different Order Runge-Kutta Methods with Accuracy and Stability

H. Rahman¹ , A. Khair² , N. Sultana³

Section:Research Paper, Product Type: Journal-Paper
Vol.9 , Issue.1 , pp.14-18, Feb-2022

Online published on Feb 28, 2022

Copyright © H. Rahman, A. Khair, N. Sultana . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
Numerical methods in solving ordinary differential equations (ODEs) play an essential role in dealing with different mathematical, physical, and engineering problems. The accuracy and stability of the numerical methods are the critical factors for examining their performances in solving related problems. In this study, a comparison is made between some well-known methods, namely the Euler and different order Runge-Kutta (RK) methods, which are RK(2,2), RK(3,3), and RK(4,4) schemes, in terms of accuracy and stability. We have tested the numerical accuracy of the mentioned methods through the root mean square error (RMSE) value. Furthermore, the stability polynomials of the methods are determined, and their contour plot is made to analyze the methods’ stability. We have got the ordered relation “Euler RK(2,2) RK(3,3) RK(4,4)” both for the accuracy and stability tests. According to the ordered relation, it is clear that the RK(4,4) method is the most accurate and stable scheme among the test methods.

Key-Words / Index Term :
Euler method, Different order Runge-Kutta methods, Accuracy, Stability, Root mean square error, Stability polynomial

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