Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge-Kutta methods of high order (Q928845)

This paper is concerned with the explicit construction of symplectic and/or symmetric multirevolution Runge-Kutta (RK) type algorithms. For a near identity map \( \varphi\), \((\varphi (y) = y + \varepsilon f(y) , 0 < \varepsilon <<1)\) typically associated to the one period Poincaré map of a dynamical system, multirevolution RK algorithms attempt to approximate \( \varphi^N \) (with \(N\) large) by using the \( \varphi\)-map at a few selected points. A study of the preservation of symplecticness of \( \varphi \) by these methods has been developed by \textit{M. Calvo, L. O. Jay, J. I. Montijano} and \textit{L. Rández} [Numer. Math. 97, No. 4, 635--666 (2004; Zbl 1067.65067)] together with some families of symplectic multirevolution RK methods. In the paper under consideration the authors give a step further by using the W-transformation to construct several families Poisson/symplectic and symmetric methods. They generalize the standard W-transformation to construct families of high order RK methods by using the \( B(p)\), \(C(q)\) and \(D(r)\) simplifying assumptions and select the available parameters so that satisfy the symplecticness and symmetry requirements. Several Gauss, Radau and Lobatto type methods are derived and the results of some numerical experiments are presented to confirm the order of some methods.