The performance of phase-lag enhanced explicit Runge–Kutta Nyström pairs on [formula omitted]-body problems

Abstract

Explicit Runge–Kutta Nyström methods with enhanced phase-lag order are intended for long integrations of initial value ordinary differential equations describing free oscillations or free oscillations of high frequency and forced oscillations of low frequency. Numerical comparisons by others of RKN4(3), RKN6(4) and RKN8(6) pairs has established that the pairs with enhanced phase-lag order are more efficient on the intended problems than general purpose pairs.We investigate if these gains in efficiency extend to N-body problems used to model the orbital dynamics of the Solar System. The emphasis in our comparisons is on the RKN8(6) pairs because we are interested in long, accurate integrations. We have included the RKN4(3) and RKN6(4) pairs principally to gain insight about how the gains in efficiency depend on the order.Our main finding is that the gains for the RKN8(6) pair extend to the system of major planets except at severe accuracy requirements, and to the system of regular satellites of these planets. In addition, we found for Kepler’s two-body problem that the gains can be sensitive to small changes in eccentricity.