Forward symplectic integrators and the long-time phase error in periodic motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth order accuracy in the periodic energy. We study the one-dimensional harmonic oscillator and the two-dimensional Kepler problem in great detail, and compare the effectiveness of some recent fourth order algorithms.