# Richardson-Kalitkin method in abstract description

**Authors:**Baddour A.^{1}, Malykh M.D.^{1}^{,2}-
**Affiliations:**- Peoples’ Friendship University of Russia (RUDN University)
- Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research

**Issue:**Vol 29, No 3 (2021)**Pages:**271-284**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/27531**DOI:**https://doi.org/10.22363/2658-4670-2021-29-3-271-284

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## Abstract

An abstract description of the Richardson–Kalitkin 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 ${\rm P}$ is considered, the solution of which results in a real number $u$. To solve this problem, a numerical method is used, that is, the set $H\subset \mathrm{\mathbb{R}}$ and the mapping ${u}_{h}:H\to \mathrm{\mathbb{R}}$ are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set $H$ and ${u}_{h}$ can be expanded in a convergent series in powers of $h:{u}_{h}=u+{c}_{1}{h}^{k}+...$. In this very general situation, the Richardson–Kalitkin method is formulated for obtaining estimates for $u$ and $c$ from two values of ${u}_{h}$. The question of using a larger number of ${u}_{h}$ values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the Richardson–Kalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.

## Full Text

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,

## About the authors

### Ali Baddour

Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: alibddour@gmail.com

ORCID iD: 0000-0001-8950-1781

PhD student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation### Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University); Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research
Email: malykh_md@pfur.ru

ORCID iD: 0000-0001-6541-6603

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

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation## References

- 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.