Conservative finite difference schemes for dynamical systems

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The article presents the implementation of one of the approaches to the integration of dynamical systems, which preserves algebraic integrals in the original fdm for Sage system. This approach, which goes back to the paper by del Buono and Mastroserio, makes it possible, based on any two explicit difference schemes, including any two explicit Runge-Kutta schemes, to construct a new numerical algorithm for integrating a dynamical system that preserves the given integral. This approach has been implemented and tested in the original fdm for Sage system. Details and implementation difficulties are discussed. For testing, two Runge-Kutta schemes were taken having the same order, but different Butcher tables, which does not complicate the method due to paralleling. Two examples are considered - a linear oscillator and a Jacobi oscillator with two quadratic integrals. The second example shows that the preservation of one integral of motion does not lead to the conservation of the other. Moreover, this method allows us to propose a practical application of the well-known ambiguity in the definition of Butcher tables.

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1. Introduction Many dynamical systems have algebraic integrals of motion [1], but standard numerical methods do not allow preserving these integrals exactly on the approximate solution [2]. This means that the approximate solution satisfies such fundamental laws of nature as the law of conservation of energy also approximately, and, in view of the importance of this law itself, this circumstance is always striking. Consider the dynamical system

About the authors

Yu Ying

Kaili University

Author for correspondence.
ORCID iD: 0000-0002-4105-2566

Assistant Professor of Department of Algebra and Geometry

Kaiyuan Road 3, Kaili, 556011, China

Zhen Lu

Kaili University

ORCID iD: 0000-0002-7526-9026

Associate Professor, Department of Fine art

Kaiyuan Road 3, Kaili, 556011, China


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Copyright (c) 2022 Ying Y., Lu Z.

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