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A consequent approach is proposed to construct symplectic force-gradientalgorithms of arbitrarily high orders in the time step for precise integrationof motion in classical and quantum mechanics simulations. Within this approachthe basic algorithms are first derived up to the eighth order by directdecompositions of exponential propagators and further collected using anadvanced composition scheme to obtain the algorithms of higher orders. Contraryto the scheme by Chin and Kidwell [Phys. Rev. E 62, 8746 (2000)], wherehigh-order algorithms are introduced by standard iterations of a force-gradientintegrator of order four, the present method allows to reduce the total numberof expensive force and its gradient evaluations to a minimum. At the same time,the precision of the integration increases significantly, especially withincreasing the order of the generated schemes. The algorithms are tested inmolecular dynamics and celestial mechanics simulations. It is shown, inparticular, that the efficiency of the new fourth-order-based algorithms isbetter approximately in factors 5 to 1000 for orders 4 to 12, respectively. Theresults corresponding to sixth- and eighth-order-based composition schemes arealso presented up to the sixteenth order. For orders 14 and 16, such highlyprecise schemes, at considerably smaller computational costs, allow to reduceunphysical deviations in the total energy up in 100 000 times with respect tothose of the standard fourth-order-based iteration approach.Comment: 23 pages, 2 figures; submitted to Phys. Rev.

Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems