On Difference Schemes for the Many-Body Problem Preserving All Algebraic Integrals

Abstract: A new approach to creating difference schemes of any order for the many-body problem preserving all its algebraic integrals is proposed. It is based on a combination of two ideas: the method of energy quadratization and the rejection of inheritance symplectic structure. Results of the tests with simplest scheme of this class are presented. A flat three-body problem with equal masses is selected for testing. The case when bodies pass close to each other is considered, for which the algorithm of time step scaling near numerical singularities is specially developed. A comparison with the explicit Runge–Kutta method of the 4th order and the simplest symplectic method, the midpoint scheme, was made. © 2022, Pleiades Publishing, Ltd.