Discrete and Continuous Models and Applied Computational ScienceDiscrete 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)2270310.22363/2658-4670-2019-27-3-242-262Computational modeling and simulationЧисленное и имитационное моделированиеResearch ArticleOn the properties of numerical solutions of dynamical systems obtained using the midpoint methodО свойствах численных решений динамических систем, полученных по схеме средней точкиGerdtVladimir P.ГердтВ. П.

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

gerdt@jinr.ruMalykhMikhail D.МалыхМ. Д.

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

malykh-md@rudn.ruSevastianovLeonid A.СевастьяновЛ. А.

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

sevastianov-la@rudn.ruYingYuИнЮй

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

yingy6165@gmail.comJoint Institute for Nuclear ResearchОбъединенный институт ядерных исследованийPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народовKaili UniversityУниверситет Каили15122019273VOL 27, NO3 (2019)ТОМ 27, №3 (2019)24226222012020Copyright ©; 2019, Gerdt V.P., Malykh M.D., Sevastianov L.A., Ying Y.Copyright ©; 2019, Гердт В.П., Малых М.Д., Севастьянов Л.А., Ин Ю.2019Gerdt V.P., Malykh M.D., Sevastianov L.A., Ying Y.Гердт В.П., Малых М.Д., Севастьянов Л.А., Ин Ю.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/22703

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 .̂

В статье рассматривается схема средней точки как разностная схема для динамической системы вида ̇ = (). Эта схема замечательна тем, что в силу теоремы Купера сохраняет все квадратичные интегралы движения, более того, это - простейшая схема из числа симплектических схем Рунге-Кутты, обладающих названным свойством. Свойства приближённых решений изучены в рамках численных экспериментов с линейным и нелинейным осцилляторами, а также с системой нескольких связанных осцилляторов. Показано, что помимо сохранения всех интегралов движения, приближённые решения наследуют периодичность движения. При этом уделено внимание обсуждению введения понятие периодичности приближённого решения, найденного по разностной схеме. В случае нелинейного осциллятора выполнение каждого шага требует решения системы нелинейных алгебраических уравнений. Обсуждены вопросы организации вычислений по таким схемам. Дано сравнение с другими схемами, в том числе симметрическими относительно перестановки и .̂

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