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)2614110.22363/2658-4670-2021-29-1-63-72ArticlesСтатьиResearch ArticleOn conjugate difference schemes: the midpoint scheme and the trapezoidal schemeО сопряжённых разностных схемах: схема средней точки и схема трапецийYingYuИнЮй

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Algebra and Geometry

45384377@qq.comMalykhMikhail D.МалыхМ. Д.

Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

malykhmd@pfur.ruKaili UniversityУниверситет КайлиPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народов30032021291VOL 29, NO1 (2021)ТОМ 29, №1 (2021)637230032021Copyright ©; 2021, Ying Y., Malykh M.D.Copyright ©; 2021, Ин Ю., Малых М.Д.2021Ying Y., Malykh M.D.Ин Ю., Малых М.Д.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/26141
The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x˙=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Cooper’s theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Cooper’s theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as ∆t→0. Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

В статье исследован вопрос о сохранении квадратичных интегралов на приближённых решениях автономных систем обыкновенных дифференциальных уравненийx˙=f(x), найденных по схеме трапеций. Установлена связь между схемой трапеции и схемой средней точки, которая сохраняет все квадратичные интегралы движения в силу теоремы Купера. Эта связь позволяет рассматривать схему трапеций как двойственную к схеме средней точки и отыскать двойственный аналог для теоремы Купера. Доказано, что на приближённом решении, найденном по симметрической схеме, сохраняется не сам квадратичный интеграл, а более сложное выражение, которое переходит в интеграл в пределе при ∆t→0. Результаты проиллюстрированы примерами — линейным и эллиптическим осцилляторами. В обоих случаях в явном виде выписаны выражения, которые сохраняет схема трапеций.

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