On the Accuracy of a Family of Adaptive Symplectic Conservative Methods for the Kepler Problem

Abstract: The results of the analysis of the accuracy of a new one-parameter family of adaptive symplectic conservative numerical methods for solving the Kepler problem are presented. The methods implement a symplectic mapping of the initial state into the current state and, as a consequence, preserve the phase volume. In contrast to the existing symplectic methods, for example, the Verlet method, they preserve, within the exact arithmetic, all the first integrals inherent in the problem, namely, the angular momentum, total energy, and the Laplace–Runge–Lenz vector. In addition, the orbit and velocity hodograph are preserved. The variable step of integration is selected automatically based on the local properties of the solution to the problem. The step decreases where the phase variables change most rapidly. The methods approximate the dependence of phase variables on time with either the second or fourth order, depending on the value of the parameter. The limits of the number of calculated points for the solution period are established, which provide a certain order of accuracy. When the number of calculated points exceeds the upper limit, it is impractical to carry out calculations due to the determining influence of roundoff errors. With an increase in the eccentricity of the orbit, the upper limit of the number of calculated points decreases. It is shown that there is a relationship between the value of the parameter and the number of calculated points, at which the approximate solution is exact within the framework of exact arithmetic. One of the problems of computational mathematics is as follows: to date, there is no numerical method that preserves all the global properties of exact solutions to the Cauchy problem for Hamiltonian systems in the general case. The investigated methods for the Kepler problem are an example of a positive solution to the indicated problem. © 2021, Pleiades Publishing, Ltd.