A simplified Nyström-tree theory for extended Runge-Kutta-Nyström integrators solving multi-frequency oscillatory systems

In the study of extended Runge-Kutta-Nyström (abbr. ERKN) methods for the integration of multi-frequency oscillatory systems, a quite complicated set of algebraic conditions arises which must be satisfied for a method to achieve some specified order. A theory of tri-colored tree was proposed by Yang et al. (2009), for achieving the order conditions of ERKN methods which are designed specially for multi-frequency and multidimensional perturbed oscillators. The tri-colored tree theory for the order conditions in that paper is useful, but not completely satisfactory due to the existence of redundant trees. In this paper, a simplified tri-colored theory and the order conditions for ERKN integrators are developed by constructing a set of simplified special extended Nyström trees (abbr. SSENT) and defining some real-valued mappings on it. In order to simplify the tri-colored tree theory, two special mappings, the extended elementary differential and the sign mapping for a tree are investigated in detail. This leads to a novel Nyström-tree theory for the order conditions for ERKN methods without any redundant trees, which simplifies the tri-colored theory.