Approximate compositions of a near identity map by multi-revolution Runge-Kutta methods (Q1889897)

Properties of multi-revolution Runge-Kutta methods applied to ordinary differential equations in a neighbourhood of nearly periodic orbits are analyzed. The basic idea of a multi-revolution method is to approximate the \(N\)th power \(\varphi^N\) of a near identity map \(\varphi\) by \(s\ll N\) evalutions of \(\varphi\) at suitable points. Using this approach multi-revolution methods can be reinterpreted as classical Runge-Kutta methods with a different step size. The authors give order conditions and order estimations. In addition, multi-revolution methods which preserve symplectic structures are described.