Электронный каталог

Статья

Lapshenkova, L.O.
On a Finite-Difference Scheme Defining a Birational Non-Quadratic Map between Time Layers / L.O.Lapshenkova, K.S.Mashkovtseva, A.A.Trusova, M.D.Malykh. – Text : electronic // Discrete and Continuous Models and Applied Computational Science. – 2026. – Vol. 34, No. 1. – P. 98–112. – URL: https://doi.org/10.22363/2658-4670-2026-34-1-98-112. – Bibliogr.: 21.

The article considers reversible difference schemes for dynamical systems based on the system doubling method proposed by V.N. Abrashin and S.N. Sytova. The method duplicates the original variables, leading to an extended system whose finite-difference approximation defines a birational map between time layers. The preservation of algebraic integrals in such schemes is investigated. It is proved that if the original system admits a homogeneous quadratic first integral, the corresponding bilinear form is exactly preserved by the discrete scheme. This property is demonstrated on the Jacobi oscillator, where the geometric mean of the duplicated variables ensures exact conservation of the quadratic integral. A more detailed analysis is performed on the non-trivial Vanhaecke system, an integrable Hamiltonian system with two degrees of freedom and higher-degree polynomial integrals. Numerical experiments carried out in the computer algebra system Sage using the package fdm.sage confirm that the two copies oscillate synchronously around the exact values of the first integrals, and averaging reduces the oscillation amplitude. For separable Hamiltonian systems, the scheme is shown to be symplectic. The results obtained allow recommending the doubling method for constructing stable and structure-preserving numerical integrators for a wide class of dynamical systems with polynomial right-hand sides, including high-dimensional systems.



Спец.(статьи,препринты) = С 17 д - Численное решение дифференциальных и интегральных уравнений. Разностные методы
ОИЯИ = ОИЯИ (JINR)2026
Бюллетени = 33/026