Two approaches to the construction of optimal second-order numerical methods and their application to the analysis of oscillatory nonlinear systems

Authors

  • V.M. Zayats

DOI:

https://doi.org/10.15407/dopovidi2014.01.037

Keywords:

numerical methods, oscillatory nonlinear systems

Abstract

Iterative and direct approaches to the minimization of errors at a discretization of second-order numerical methods are proposed. The iterative approach is based on a modification of the method of trapezoids and setting the time when the explicit and implicit Euler methods give the same contribution to the amendment to the next discretization point of a dynamical system. Combining the derived formula with the method of trapezoids, the possibility of constructing the optimal precision numerical method is shown. The direct approach is based on determining a time when the tangents drawn to the nearby points of discretization of the continuous system intersect, which provides the zero error of a discretization. The expediency of their application to the analysis of nonlinear dynamical oscillatory systems with a low coefficient of attenuation, long transients, and high power is confirmed.

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References

Bondarenko V. M., Gerasymiv I. I., Mandzy B. A., Maranov A. V. Analysis of the accuracy and qualitative correspondence of discrete models of electrical circuits. Kyiv, 1983 (in Russian).

Butenin N. V., Neimark Yu. I., Fufaev N. A. Introduction to the theory of nonlinear oscillations. Moscow: Nauka, 1987 (in Russian).

Vasilev V. I., Shevchenko A. I. Combined algorithm of optimal complexity. Proceedings of the International Conference “Artificial Intelligence”. Vol. 1. Crimea, 2002: 308–310 (in Russian).

Zayats V. M. Kibernetika i sistemnyi analiz, 2000, No. 4: 161–165 (in Russian).

Zayats V. M. Visn. NU “Lvivska politekhnika”. Informatsiini systemy ta merezhi, 2004, No. 519: 132–142 (in Ukrainian).

Zayats V. M. Izv. vuzov. Radioelektronika, 1993, No. 3: 26–32 (in Russian).

Zayats V. M. Building composite second order difference methods. Proceedings of the Scientific Conference “Computational methods and systems transformation information”, Lviv: FMI NANU, 2011: 34–36 (in Ukrainian).

Petrenko A. I. Numerical methods in computer science. Kyiv: V-vo BHV, 1999 (in Ukrainian).

Samoilenko A. M., Ronto N. I. Numerical-analytical methods for investigating periodic solutions. Kyiv: Vyshcha shkola, 1976 (in Russian).

Chua L. O., Lin P.-M. Computer analysis of electronic circuits (algorithms and computational methods). Moscow: Energiia, 1980 (in Russian).

Published

24.03.2025

How to Cite

Zayats, V. (2025). Two approaches to the construction of optimal second-order numerical methods and their application to the analysis of oscillatory nonlinear systems . Reports of the National Academy of Sciences of Ukraine, (1), 37–42. https://doi.org/10.15407/dopovidi2014.01.037

Issue

Section

Information Science and Cybernetics