Numerical integrators based on modified differential equations

Inspired by the theory of modified equations (backward error analysis), a new approach to high-order, structure-preserving numerical integrators for ordinary differential equations is developed. This approach is illustrated with the implicit midpoint rule applied to the full dynamics of the free rigid body. Special attention is paid to methods represented as B-series, for which explicit formulae for the modified differential equation are given. A new composition law on B-series, called substitution law, is presented.