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

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 φNT of N near-periods T in terms of the one near-period map φT evaluated at few s ≪ N selected points. More generally, multi-revolution methods aim at approximating the composition φ 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.