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We show that when time-reversible symplectic algorithms are used to solveperiodic motions, the energy error after one period is generally two ordershigher than that of the algorithm. By use of correctable algorithms, we showthat the phase error can also be eliminated two orders higher than that of theintegrator. The use of fourth order forward time step integrators can result insixth order accuracy for the phase error and eighth accuracy in the periodicenergy. We study the 1-D harmonic oscillator and the 2-D Kepler problem ingreat details, and compare the effectiveness of some recent fourth orderalgorithms.Comment: Submitted to Phys. Rev. E, 29 Page

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