Discrete 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)2614110.22363/2658-4670-2021-29-1-63-72Research ArticleOn conjugate difference schemes: the midpoint scheme and the trapezoidal schemeYingYu<p>Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Algebra and Geometry</p>45384377@qq.comMalykhMikhail D.<p>Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics</p>malykhmd@pfur.ruKaili UniversityPeoples’ Friendship University of Russia (RUDN University)30032021291637230032021Copyright © 2021, Ying Y., Malykh M.D.2021<div class="data" style="text-align: justify;">The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo></mo></mover><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, 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 Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers 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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo></mo><mi>t</mi><mo></mo><mn>0</mn></math>.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. 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