Open AccessOn conjugate difference schemes: the midpoint scheme and the trapezoidal scheme
Author(s) -
Yu Ying,
М. Д. Малых
Publication year - 2021
Publication title -
discrete and continuous models and applied computational science
Language(s) - English
Resource type - Journals
eISSN - 2658-7149
pISSN - 2658-4670
DOI - 10.22363/2658-4670-2021-29-1-63-72
Subject(s) - mathematics , midpoint , trapezoidal rule , scheme (mathematics) , quadratic equation , midpoint method , mathematical analysis , duality (order theory) , pure mathematics , geometry , numerical integration
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 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 t0.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.