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Simple derivation of Symplectic Integrators with First Order Correctors

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Abstract

In this paper we consider almost integrable systems for which we show that there is a direct connection between symplectic methods and conventional numerical integration schemes. This enables us to construct several symplectic schemes of varying order. We further show that the symplectic correctors, which formally remove all errors of first order in the perturbation, are directly related to the Euler—McLaurin summation formula. Thus we can construct correctors for these higher order symplectic schemes. Using this formalism we derive the Wisdom—Holman midpoint scheme with corrector and correctors for higher order schemes. We then show that for the same amount of computation we can devise a scheme which is of order O(εh 6) + (ε2 h 2), where ε is the order of perturbation and h the stepsize. Inclusion of a modified potential further reduces the error to O(εh 6) + (ε2 h 4).

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Authors and Affiliations

  1. Tuorla Observatory, University of Turku, 21500, Piikkiö, Finland

    Seppo Mikkola

  2. Surrey Space Centre, University of Surrey, Guildford, GU2 7XH, U.K.

    Phil Palmer

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  1. Seppo Mikkola
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  2. Phil Palmer
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Mikkola, S., Palmer, P. Simple derivation of Symplectic Integrators with First Order Correctors. Celestial Mechanics and Dynamical Astronomy 77, 305–317 (2000). https://doi.org/10.1023/A:1011165912381

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  • DOI: https://doi.org/10.1023/A:1011165912381

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