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Approximate compositions of a near identity map by multi-revolution Runge-Kutta methods

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The so-called multi-revolution methods were introduced in celestial mechanics as an efficient tool for the long-term numerical integration of nearly periodic orbits of artificial satellites around the Earth. A multi-revolution method is an algorithm that approximates the map φ T N of N near-periods T in terms of the one near-period map φ T evaluated at few s << N selected points. More generally, multi-revolution methods aim at approximating the composition φN of a near identity map φ. In this paper we give a general presentation and analysis of multi-revolution Runge-Kutta (MRRK) methods similar to the one developed by Butcher for standard Runge-Kutta methods applied to ordinary differential equations. Order conditions, simplifying assumptions, and order estimates of MRRK methods are given. MRRK methods preserving constant Poisson/symplectic structures and reversibility properties are characterized. The construction of high order MRRK methods is described based on some families of orthogonal polynomials.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Pza. San Francisco s/n, Universidad de Zaragoza, 50009, Zaragoza, Spain

    Manuel Calvo, Juan I. Montijano & Luis Rández

  2. Department of Mathematics, 14 MacLean Hall, The University of Iowa, Iowa City, IA 52242-1419, USA

    Laurent O. Jay

Authors
  1. Manuel Calvo
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  2. Laurent O. Jay
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  3. Juan I. Montijano
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  4. Luis Rández
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Correspondence to Laurent O. Jay.

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Mathematics Subject Classification (1991): 65L05, 65L06

This material is based upon work supported by the National Science Foundation Grant No. 9983708 and by the DGI Grant BFM2001–2562

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Calvo, M., Jay, L., Montijano, J. et al. Approximate compositions of a near identity map by multi-revolution Runge-Kutta methods. Numer. Math. 97, 635–666 (2004). https://doi.org/10.1007/s00211-004-0518-9

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  • DOI: https://doi.org/10.1007/s00211-004-0518-9

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