On explicit multi-revolution Runge-Kutta schemes (Q878086)

Let \(\varphi(y) = y+\varepsilon f(y)\) denote a near identity Poincaré map. The computation of \(\varphi^N\), for \(N\) an integer, is considered using a modified Runge-Kutta method with \(\varepsilon^N\) taking the role of stepsize and with the coefficients in the method depending on \(N\). It is assumed that an auxiliary integrator is available to evaluate \(\varphi\) values. An analysis of these ``Multi-revolution Runge-Kutta methods'' was presented by the authors and \textit{L. O. Jay} [Numer. Math. 97, No. 4, 635--666 (2004; Zbl 1067.65067)]. In the present paper, the order is analysed using Butcher series and this leads to a comprehensive study of fourth-order explicit methods with 4 steps. To obtain order 5, six steps are required, as for classical Runge-Kutta methods, and appropriate adaptation of standard simplifying assumptions are used in the derivation. Numerical experiments are presented showing the behaviour of a fifth order multi-revolution Runge-Kutta method derived in the paper, compared with the standard DOPRI5(4) method.