An alternative approach for order conditions of Runge-Kutta-Nyström methods

We present an alternative approach proposed by Albrecht to derive general order conditions for Runge-Kutta-Nyström methods and relate it to the classical RKN-theory. The RKN-methods are treated as composite linear methods to yield the general order conditions as orthogonal relations. We then exploit the orthogonal structure of the order conditions and obtain a simple recursion to generate the order conditions. Implications of this approach on the classical RKN-theory are discussed and it may be worthwhile to generalize the approach to other discretizations.