  Numerical approximation of the long time behavior of a stochastic
differential equation (SDE) is considered. Error estimates for time-averaging
estimators are obtained and then used to show that the stationary behavior of
the numerical method converges to that of the SDE. The error analysis is based
on using an associated Poisson equation for the underlying SDE. The main
advantage of this approach is its simplicity and universality. It works equally
well for a range of explicit and implicit schemes including those with simple
simulation of random variables, and for hypoelliptic SDEs. To simplify the
exposition, we consider only the case where the state space of the SDE is a
torus and we study only smooth test functions. However we anticipate that the
approach can be applied more widely. An analogy between our approach and
Stein's method is indicated. Some practical implications of the results are
discussed.
