In this article, we present some properties on Runge-Kutta methods. The first is to show that the equidistant subdivision of the interval  into s parts and that the use of Lagrange interpolation do not make it possible to find the coefficients resulting from the equations of Runge-Kutta rooted tree of order p. The second consists in showing that we cannot find a Runge-Kutta method for a certain subdivision resulting from the description of the method.

subdivision, Lagrange interpolation, Runge-Kutta methods, rooted tree.

Received: November 14, 2022; Accepted: December 27, 2022; Published: January 5, 2023

How to cite this article: SÉKA Hippolyte, Some properties on Runge-Kutta methods, Universal Journal of Mathematics and Mathematical Sciences 18(1) (2023), 53-65. http://dx.doi.org/10.17654/2277141723004

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] S. Brian, Méthodes numériques de résolution d’équations différentielles. http://www.fresnel.fr/perso/stout/Anal_numer/Cours4.pdf.
[2] J.-C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2nd ed., 2008, p. 175.
[3] Christophe Besse, Résolution numérique des Équations Différentielles Ordinaires, L3 Mapi, 2016.
[4] D. Francine and D. Marc, Systèmes Dynamiques, Lecture Notes, September 3, 2017. https://math.unice.fr/diener/SysDyn/AllSysDyn.pdf.
[5] C. François, E. D. O.: Méthodes numériques (cours 3), 2015, Lecture Notes, Institut Galilée, Université Paris XIII, January 13, 2015.
https://www.math.univ-paris13.fr/~cuvelier/docs/Enseignements/Energetique/ AnaNumII/12-13/ PresentationEDO_PartII.pdf.
[6] K. Guido, Numerical Analysis of Ordinary Differential Equations, Notes on Applied Mathematics, 2018.
http://simweb.iwr.uni-heidelberg.de/gkanscha/notes/ode.pdf.
[7] L. Guy and P. Christian, Cours d’analyse 1 Licence 1er semester, Lecture notes, 2006. https://math.unice.fr/pauly/ca1.pdf.
[8] Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons, Inc., 2009.
[9] C. Jérémy, Méthodes numériques ROCK4 pour la résolution d’équations différentielles et étude de l’algèbre pré-Lie sous-jacente, Lecture Notes, 2011. http://math.univ-lyon1.fr/caldero/Cochoy-TIPE.pdf.
[10] R. Julien, Méthodes d’Intégration Numérique, Université Paul Sabatier, 2014-2015.
[11] H. Séka and K. R. Assui, Order of the Runge-Kutta method and evolution of stability method, Ural Mathematical Journal 5(2) (2019), 64-71. https://doi.10.15826/umj.2019.2.006.