Formal series and numerical integrators, part I: Systems of ODEs and symplectic integrators Original Research Article

The study of the order conditions of numerical integrators for systems of differential equations and differential-algebraic equations often leads to different kinds of series expansions. We present a unified approach for the derivation and handling of those series. In this first part of our work we restrict ourselves to the study of one-step methods for ODEs: We apply our general approach to the study of NB-series methods (a general class of methods including all Runge-Kutta type methods). Special attention is paid to the integration of Hamiltonian systems by symplectic methods.