Order conditions and symmetry for two-step hybrid methods

Abstract

In this study of two-step hybrid methods for second-order equations of the type y″ = f(y), we apply P-series [Hairer, E., Lubich, C. and Wanner, G. (2002). Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics.] to formalise the approach of Chan [Chan, R. P. K. (2002). Two-step hybrid methods. Internal Publication.] to the order conditions, and present two characterizations of symmetry. Although order conditions can be obtained through the classical theory for the Nyström methods, it is of interest to derive particular simpler formulas for the class of two-step hybrid methods in order to facilitate the search for high-order methods. Moreover, the approach proves useful in analysing the symmetry of the hybrid methods.

Keywords:

• Two-step hybrid methods
• Second-order differential equation
• Order conditions
• Symmetry

Acknowledgements

The authors would like to thank Ernst Hairer for his constructive suggestions. The authors are also indebted to John Butcher and the participants of the Numerical Analysis workshops at the University of Auckland for helpful suggestions during the preparation of the paper. Pierre Leone was supported by the Fonds National Suisse pour la Recherche Scientifique during the preparation of the paper.