On the accuracy of a family of adaptive symplectic conservative methods for the Kepler problem

In this work we describe results of the analysis of accuracy for the new single-parameter family of adaptive symplectic conservative numerical methods for the Kepler problem. The methods perform symplectic mapping from the initial to the current state and, therefore, they preserve phase volume. In contrast to the existing symplectic methods, e.g., Verlet integrator, they preserve all first integrals of the Kepler problem i.e., angular momentum, full energy and Laplace–Runge–Lenz vector in the frame of the exact arithmetic. The orbit and the velocity hodograph are preserved as well. The numerical integration adaptive step is chosen automatically based on the local features of the solution. The step decreases where phase variables change most rapidly. The methods approximate dependence of phase variables on time with either 2-nd or 4-th order depending on parameter value. The limits of computational points per orbital period are identified to guaranty the prescribed order of accuracy. When number of points is exceeding the upper limit, round-off errors dominate and further increase in the number of points is not reasonable. The upper limit of computational points decreases as an eccentricity of the trajectory increases. It is shown that there is a relationship between the value of the parameter and the number of computational points, at which the approximate solution is exact within the framework of exact arithmetic. One of the computational mathematics problems is the following: by now there is no numerical algorithm which preserve all global characteristics of the exact solution of the Cauchy problem for Hamiltonian systems in general case. The numerial methods for the Kepler problem discussed in this paper provide an example of the positive solution of this problem.