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ART--2009-003

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

Calvo, M. (calvo@unizar.es) ; Jay, J.O. (ljay@math.uiowa.edu) ; Montijano, Juan I. (monti@unizar.es) ; Rández, Luis (randez@unizar.es)


Published in:   Numerische Mathematik
  Numer. Math. Vol. 97, Num. 4 (junio 2004), pp. 635-666


Abstract: 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 phgrTN of N near-periods T in terms of the one near-period map phgrT evaluated at few s << N selected points. More generally, multi-revolution methods aim at approximating the composition phgrN of a near identity map phgr. 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. 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

Language: eng
Department: Matemática Aplicada

Keyword(s): multi-revolution methods ; Runge-Kutta ; Butcher ; compositions of a near identity map

 

 Record created 2009-05-07, last modified 2011-08-09



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