Lack of dissipativity is not symplecticness (Q1899935)

A new fourth-order three stage Runge-Kutta-Nyström (RKN) algorithm is developed for the approximate solution of Hamiltonian systems. The algorithm has very small error constants. It is designed to be nondissipative and it is symplectic when applied to linear Hamiltonian systems. The algorithm is compared with a symplectic RKN algorithm of the same order and number of stages due to \textit{E. Forest} and \textit{R. D. Ruth} [Physica D 43, No. 1, 105-117 (1990; Zbl 0713.65044)] which has larger error constants. Numerical calculations show that for the Kepler problem the nondissipative algorithm is initially more accurate for small time values but later is less accurate for large time values.