Testing of adaptive symplectic conservative numerical methods for solving the Kepler problem

The properties of a family of new adaptive symplectic conservative numerical methods for solving the Kepler problem are examined. It is shown that the methods preserve all first integrals of the problem and the orbit of motion to high accuracy in real arithmetic. The time dependences of the phase variables have the second, fourth, or sixth order of accuracy. The order depends on the chosen values of the free parameters of the family. The step size in the methods is calculated automatically depending on the properties of the solution. The methods are effective as applied to the computation of elongated orbits with an eccentricity close to unity.