On the properties of numerical solutions of dynamical systems obtained using the midpoint method

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Abstract

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

About the authors

Vladimir P. Gerdt

Joint Institute for Nuclear Research

Author for correspondence.
Email: gerdt@jinr.ru

Doctor of Physical and Mathematical Sciences, Full Professor at the Joint Institute for Nuclear Research (JINR) where he is the head of the Group of Algebraic and Quantum Computations

Joliot-Curie St. 6, Dubna, Moscow Region 141980, Russian Federation

Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University)

Email: malykh-md@rudn.ru

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia

Miklukho-Maklaya St. 6, Moscow 117198, Russian Federation

Leonid A. Sevastianov

Joint Institute for Nuclear Research; Peoples’ Friendship University of Russia (RUDN University)

Email: sevastianov-la@rudn.ru

professor, Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

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

Yu Ying

Peoples’ Friendship University of Russia (RUDN University); Kaili University

Email: yingy6165@gmail.com

postgraduate student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); assistant professor of Department of Algebra and Geometry, Kaili University

Miklukho-Maklaya St. 6, Moscow 117198, Russian Federation; Kaiyuan Road 3, Kaili 556011, China

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