New algorithmic implementation of exact three-point difference schemes for systems of nonlinear ordinary differential equations of the second ord
Abstract
UDC 519.62
In this paper, we propose and justify three-point difference schemes of higher order of accuracy on a non-uniform grid for systems of nonlinear ordinary differential equations of the second order with a derivative on the right-hand side and boundary conditions of the first kind. We construct a new approximation of the derivative of the solution to the boundary value problem at grid nodes, prove the existence and uniqueness of the solution, and establish the order of accuracy of the difference schemes. Additionally, an iterative Newton-type method for determining the solution of these schemes is developed and an algorithm for the automatic selection of grid points, which guarantees the achievement of the specified accuracy, is suggested. We also present numerical solutions of some examples, which confirm the efficiency and reliability of our algorithm.
References
V. L. Makarov, A. A. Samarskii, Exact three-point difference schemes for second-order nonlinear ordinary differential equations and their implementation, Soviet Math. Dokl., 41, № 3, 495 – 500 (1991).
M. V. Kutniv, V. L. Makarov, A. A. Samarskii, Accurate three-point difference schemes for second-order nonlinear ordinary differential equations and their implementation, Comput. Math. and Math. Phys., 39, № 1, 45 – 60 (1999).
M. V. Kutniv, Accurate three-point difference schemes for second-order monotone ordinary differential equations and their implementation, Comput. Math. and Math. Phys., 40, № 3, 368 – 382 (2000).
M. V. Kutniv, Modified three-point difference schemes of high-accuracy order for second order nonlinear ordinary differential equations, Comput. Methods Appl. Math., 3, № 2, 287 – 312 (2003). DOI: https://doi.org/10.2478/cmam-2003-0019
L. B. Gnativ, M. V. Kutniv, A. I. Chukhrai, Generalized three-point difference schemes of high order of accuracy for nonlinear ordinary differential equations of second order, J. Math. Sci., 167, № 1, 62 – 75 (2010), https://doi.org/10.1007/s10958-010-9902-4 DOI: https://doi.org/10.1007/s10958-010-9902-4
M. Krol, M. V. Kutniv, Three-point difference schemes of high-order accuracy for second-order nonlinear differential equations with boundary conditions of the third kind, J. Comput. and Appl. Math., № 2(116), 43 – 62 (2014).
I. P. Gavrilyuk, M. Hermann, V. L. Makarov, M. V. Kutniv, Exact and truncated difference schemes for boundary value ODEs, Springer AG, Basel (2011), https://doi.org/10.1007/978-3-0348-0107-2 DOI: https://doi.org/10.1007/978-3-0348-0107-2
M. V. Kutniv, Three-point difference schemes of high accuracy order for systems of nonlinear ordinary differential equations of the second order, Comput. Math. and Math. Phys., 41, № 6, 860 – 873 (2001).
M. V. Kutniv, High-order accurate three-point difference schemes for systems of second-order ordinary differential equations with a monotone operator, Comput. Math. and Math. Phys., 42, № 5, 724 – 738 (2002).
L. B. Gnativ, M. V. Kutniv, V. L. Makarov, Generalized three-point difference schemes of high-order accuracy for systems of second order nonlinear ordinary differential equations, Different. Equat., 45, № 7, 998 – 1019 (2009), https://doi.org/10.1134/S0012266109070088 DOI: https://doi.org/10.1134/S0012266109070088
E. Hairer, S. P. N rsett, G. Wanner, Solving ordinary differential equations I. Nonstiff problems, Springer-Verlag, Berlin etc. (1987), https://doi.org/10.1007/978-3-662-12607-3 DOI: https://doi.org/10.1007/978-3-662-12607-3
L. B. Gnativ, M. Krol, M. V. Kutniv, Exact three-point difference schemes for second order nonlinear differential equations with boundary conditions of the third kind, J. Numer. Appl. Math., № 3(109), 34 – 52 (2012), https://doi.org/10.1016/j.cam.2015.12.003 DOI: https://doi.org/10.1016/j.cam.2015.12.003
A. A. Samarskii, E. S. Nikolaev, Numerical methods for grid equations, vol. 2, Iterative methods, Birkh¨auser, Basel etc. (1989). DOI: https://doi.org/10.1007/978-3-0348-9272-8
V. A. Trenohyn, Funktsyonalnыi analyz, Nauka, Moskva (1980).
J. R. Cash, F. Mazzia, Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems, J. Numer. Anal., Industrial and Appl. Math., 1, № 6, 81 – 90 (2006).
Ch. Tsitouras, S. N. Papakostas, Cheap error estimation for Runge-Kutta methods, SIAM J. Sci. Comput., 20, № 6, 2067 – 2088 (1999), https://doi.org/10.1137/S1064827596302230 DOI: https://doi.org/10.1137/S1064827596302230
Copyright (c) 2022 Мирослав Володимирович Кутнів
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.