International Journal of Numerical Methods and Applications

The International Journal of Numerical Methods and Applications publishes research articles on numerical methods and their applications in various fields, including differential equations, fluid dynamics, and bioinformatics. It also welcomes survey articles on new methods in numerical analysis.

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A TWO-STEP FIFTH ORDER RUNGE-KUTTA METHOD

Authors

  • Séka Hippolyte
  • N’gohissé Konan Firmin
  • Kouadio Amos Michaêl Akpini

Keywords:

Runge-Kutta method, ordinary differential equation, one-step, error, stability

DOI:

https://doi.org/10.17654/0975045226002

Abstract

In this article, we develop a two-step Runge-Kutta method of order 5 that depends on a free parameter $a_{6,3}$ to solve ordinary differential equation. For $a_{6,3}=6 / 7$, the two-step and one-step methods of order 5 of Butcher are found. An estimation of the error of this two-step method is proposed, and the comparison of the stability region with that of the one-step method is made.

Received: July 2, 2025
Revised: July 23, 2025
Accepted: August 4, 2025

References

[1] K. A. M. Akpini et al., Optimal method of Runge-Kutta of order 5, Journal of Mathematics Research 11(1) (2019), 92-101.

[2] J.-C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., Cambridge University Press, 2008.

[3] T. Feagin, High-order Explicit Runge-Kutta Methods, 2013.

http://sce.uhcl.edu/rungekutta.

[4] H. Séka and K. R. Assui, A new eighth order Runge-Kutta family method, Journal of Mathematics Research 11(2) (2019), 190-199.

[5] H. Séka and K. R. Assui, Order of the Runge-Kutta method and evolution of the stability region, Ural Mathematical Journal 25 (2019), 64-71.

doi: 10.15826/umj.2019.2.006.

[6] H. Séka and K. R. Assui, A new seventh order Runge-Kutta family: comparison with the method of Butcher and presentation of a calculation software, Mathematics and Computer Science 34 (2019), 68-75.

https://10.11648/j.mcs.20190403.12.

[7] Sergey Khashin, Estimating the error in the classical Runge-Kutta methods, Zh. Vychisl. Mat. Mat. Fiz. 54(5) (2014), 746-754.

https://doi.org/10.7868/S0044466914050172.

[8] Sergey Khashin, List of some known Runge-Kutta methods family (Preliminary version). http://math.ivanovo.ac.ru/dalgebra/Khashin/rk/sh_rk.html.

[9] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71 (2009), 1311-1328. http://dx.doi.org/10.1016/j.na.2008.12.005.

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Published

2025-09-11

Issue

Vol. 26 No. 1 (2026)

Section

Articles

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Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA

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How to Cite

A TWO-STEP FIFTH ORDER RUNGE-KUTTA METHOD. (2025). International Journal of Numerical Methods and Applications, 26(1), 39-54. https://doi.org/10.17654/0975045226002

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