Joint work with Andrew Stuart, John Terry (Warwick University), Hersir Sigurgeirsson and James Warren (Stanford University)
We are concerned with the numerical integration of dynamical systems over long time-intervals. Classical numerical analysis only guarantees the accuracy of trajectories on time-scales much shorter than those of interest. However, it is believed that numerically computed trajectories may still be able to accurately reproduce the statistical properites of such systems.
As an example, we consider deterministic systems of interacting particles. As the number of particles goes to infinity, it can be shown that the trajectory of a single tracer particle converges to that of a well-known stochastic process, e.g. Brownian motion. Our aim is to show that numerically computed trajectories will also approximate the same stochastic process. We will present some examples where results have been obtained, and others where work is still in progress.