Approximation of near identity Poincaré maps by multi-revolution Runge-Kutta methods

Laurent O. Jay, Manuel Calvo, Juan I. Montijano, Luis Rández

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

na.ljay@na-net.ornl.gov

http://www.math.uiowa.edu/ljay/

Contributed talk

The so-called multi-revolution methods were introduced in celestial mechanics as an efficient tool for the long-term numerical integration of orbits of artificial satellites. Multi-revolution methods attempt to track the solution after a large number $N$ of periods by using the one period map $\varphi_T$ at only a few suitable points. In this talk we will provide a theoretical background of such methods used in some problems of celestial mechanics and in highly oscillatory problems of other fields. For those problems it is assumed in advance that the solutions of the differential equations for the initial conditions under consideration are quasi periodic. We will give a general presentation and analysis of multi-revolution Runge-Kutta (MRRK) methods similar to the one developed by John Butcher for standard RK methods applied to ordinary differential equations. An $s$-stage multi-revolution Runge-Kutta method is an algorithm that approximates the map $\varphi_T^N$ of N quasi-periods in terms of the one quasi-period Poincaré map $\varphi_T$ evaluated at $s\ll N$ suitable selected points. We assume that that the one quasi-period map is a near identity map in some neighborhood of an initial value and that the quasi-period T is known in advance. Some order conditions, simplifying assumptions, and order estimates will be presented. The construction of high order MRIRK methods will be described based on some families of orthogonal polynomials. Some references are given by [1,2,3,4,5].