Gilles Vilmart (University of Geneva)

Modified differential equations in geometric numerical integration.

abstract: Joint work with my PhD advisors Philippe Chartier (INRIA Rennes) and Ernst Hairer (Genève).

The aim of geometric integration in numerical analysis is to find and study numerical integrators for differential equations that preserve geometric properties of the exact solutions (e.g. energy conservation).

First we explain the main ideas of the theory of modified equations (backward error analysis). Then we present a new approach to increase the order of accuracy of a numerical integrator arbitrarily high by applying it to a modified differential equation.

This is illustrated in the case of the motion of a rigid body where new efficient integrators are derived. Also, our approach was recently used by Roman Kozlov (2007) to develop a new efficient integrator for the Kepler problem.