On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

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We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Gröbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

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1. Quadratic invariant and conservative RK scheme One of most widespread mathematical models is an autonomous system of ordinary differential equations, i.e., the system of the form ⎧{


About the authors

Yu Ying

Kaili University

Author for correspondence.
Email: yingy6165@gmail.com

Assistant Professor of Department of Algebra and Geometry

3, Kaiyuan Road, Kaili, 556011, China

Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University)

Email: malykh_md@pfur.ru

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

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation


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Copyright (c) 2020 Ying Y., Malykh M.D.

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