Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2753110.22363/2658-4670-2021-29-3-271-284Research ArticleRichardson-Kalitkin method in abstract descriptionBaddourAli<p>PhD student of Department of Applied Probability and Informatics</p>alibddour@gmail.comhttps://orcid.org/0000-0001-8950-1781MalykhMikhail D.<p>Doctor of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Researcher in Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research</p>malykh_md@pfur.ruhttps://orcid.org/0000-0001-6541-6603Peoples’ Friendship University of Russia (RUDN University)Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research3009202129327128430092021Copyright © 2021, Baddour A., Malykh M.D.2021<p style="text-align: justify;">An abstract description of the RichardsonKalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>Ρ</mi><annotation encoding="LaTeX">{{\Rho}}</annotation></semantics></math> is considered, the solution of which results in a real number <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>u</mi><annotation encoding="LaTeX">u</annotation></semantics></math>. To solve this problem, a numerical method is used, that is, the set <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo></mo><mi mathvariant="normal">ℝ</mi></mrow><annotation encoding="LaTeX">{H\subset \mathbb{R}}</annotation></semantics></math> and the mapping <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>h</mi></msub><mo>:</mo><mi>H</mi><mo></mo><mi mathvariant="normal">ℝ</mi></mrow><annotation encoding="LaTeX">{u_h:H\to\mathbb{R}}</annotation></semantics></math> are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>H</mi><annotation encoding="LaTeX">H</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>u</mi><mi>h</mi></msub><annotation encoding="LaTeX">{u_h}</annotation></semantics></math> can be expanded in a convergent series in powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>u</mi><mi>h</mi></msub><mo>=</mo><mi>u</mi><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msup><mi>h</mi><mi>k</mi></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow><annotation encoding="LaTeX">{h:u_h=u+c_1h^k+...}</annotation></semantics></math>. In this very general situation, the RichardsonKalitkin method is formulated for obtaining estimates for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>u</mi><annotation encoding="LaTeX">u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>c</mi><annotation encoding="LaTeX">c</annotation></semantics></math> from two values of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>u</mi><mi>h</mi></msub><annotation encoding="LaTeX">{u_h}</annotation></semantics></math>. The question of using a larger number of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>u</mi><mi>h</mi></msub><annotation encoding="LaTeX">{u_h}</annotation></semantics></math> values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the RichardsonKalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.</p>finite difference methodordinary differential equationsa posteriori errorsметод конечных разностейобыкновенные дифференциальные уравненияапостериорные ошибки<p>1. Introduction A priori estimates for finding solutions to dynamical systems using the finite difference method predict an exponential growth of the error with increasing time [1]. Therefore, long-term computation requires such a small sampling step that cannot be accepted in practice. Nevertheless, calculations for long times are carried out and it is generally accepted that they reproduce not the coordinates themselves, but some average characteristics of the trajectories. In this case, a posteriori error estimates are used instead of huge a priori ones. As early as in the works of Richardson [2], for estimating the errors arising in the calculation of definite integrals by the method of finite differences, it was proposed to refine the grid, and in the works of Runge a similar technique BaddourA., Malykh M.D., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ was applied to the study of ordinary differential equations. This approach was systematically developed in the works of N.N. Kalitkin and his disciples [3]-[7] as the Richardson method, although, given the role of Kalitkin in its development, it would be more correct to call it the Richardson-Kalitkin method. The method itself is very general and universal, so we set out to present it in general form, divorcing it from the concrete implementation of the finite difference method. However, it soon became clear that this method could be extended to methods that are not finite difference methods, for example, the method of successive approximations, and even problems that are not related to differential equations. In our opinion, this method is especially simply described for a class of problems in mechanics and mathematical physics, when it is necessary to calculate a significant number of auxiliary quantities, although only one value of some combination of them is interesting. Example 1. On the segment [0,</p>[E. Hairer, G. Wanner, and S. P. Nørsett, Solving Ordinary Differential Equations, 3rd ed. New York: Springer, 2008, vol. 1.][L. F. Richardson and J. A. Gaunt, “The deferred approach to the limit,” Phil. Trans. A, vol. 226, pp. 299-349, 1927. DOI: 10.1098/rsta.1927.0008.][N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Calculations on quasi-uniform grids. Moscow: Fizmatlit, 2005, In Russian.][N. N. Kalitkin, Numerical methods [Chislennyye metody]. Moscow: Nauka, 1979, In Russian.][A. A. Belov, N. N. Kalitkin, and I. P. Poshivaylo, “Geometrically adaptive grids for stiff Cauchy problems,” Doklady Mathematics, vol. 93, no. 1, pp. 112-116, 2016. DOI: 10.1134/S1064562416010129.][A. A. Belov and N. N. Kalitkin, “Nonlinearity problem in the numerical solution of superstiff Cauchy problems,” Mathematical Models and Computer Simulations, vol. 8, no. 6, pp. 638-650, 2016. DOI: 10.1134/S2070048216060065.][A. A. Belov, N. N. Kalitkin, P. E. Bulatov, and E. K. Zholkovskii, “Explicit methods for integrating stiff Cauchy problems,” Doklady Mathematics, vol. 99, no. 2, pp. 230-234, 2019. DOI: 10.1134/S1064562419020273.][L. N. Trefethen and J. A. C. Weideman, “The exponentially convergent trapezoidal rule,” SIAM Review, vol. 56, pp. 385-458, 3 2014. DOI: 10.1137/130932132.][A. A. Belov and V. S. Khokhlachev, “Asymptotically accurate error estimates of exponential convergence for the trapezoid rule,” Discrete and Continuous Models and Applied Computational Science, vol. 3, pp. 251- 259, 2021. DOI: 10.22363/2658-4670-2021-29-3-251-259.][A. Baddour, M. D. Malykh, A. A. Panin, and L. A. Sevastianov, “Numerical determination of the singularity order of a system of differential equations,” Discrete and Continuous Models and Applied Computational Science, vol. 28, no. 5, pp. 17-34, 2020. DOI: 10.22363/2658-46702020-28-1-17-34.][The Sage Developers. “SageMath, the Sage Mathematics Software System (Version 7.4).” (2016), [Online]. Available: https://www.sagemath.org.][O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The finite element method: its basis and fundamentals, 7th ed. Elsiver, 2013.][F. Hecht, “New development in FreeFem++,” Journal of Numerical Mathematics, vol. 20, no. 3-4, pp. 251-265, 2012. DOI: 10.1515/jnum2012-0013.][A. A. Panin, “Estimates of the accuracy of approximate solutions and their application in the problems of mathematical theory of waveguides [Otsenki tochnosti priblizhonnykh resheniy i ikh primeneniye v zadachakh matematicheskoy teorii volnovodov],” in Russian, Ph.D. dissertation, MSU, Moscow, 2009.]