New families of symplectic Runge-Kutta-Nyström integration methods
by
Fernando Casas
Universitat Jaume I, 12071-Castellon, Spain
Coauthors: Sergio Blanes (University of Cambridge), José Ros (Universitat de Valencia)

In this paper we present new families of sixth and eighth order Runge-Kutta-Nyström symplectic methods for integrating numerically the equations of motion arising in Hamiltonian dynamics. These methods use the processing technique to reduce the otherwise prohibitively large number of order conditions and consequently the number of evaluations involved. Both the processor and the kernel are written as a composition of explicitly computable flows. By combining the processing technique with non-trivial flows associated with different elements of the Lie algebra involved, we obtain symplectic integration schemes with superior efficiency to other previously known algorithms of equivalent order. This improvement (in some cases up to four orders of magnitude) is mainly due to the reduction in the number of evaluations. The performance of the new methods is illustrated on a perturbed Kepler Hamiltonian which describes the motion of a satellite in Dynamical Astronomy. The treatment can also be easily extended to more general second order systems of ordinary differential equations with geometric properties which is convenient to preserve by the numercal scheme.

Date received: January 26, 2000