Cory Frontin published the first journal paper from his doctoral work in Physics of Fluids titled "Output error behavior for discretizations of ergodic, chaotic systems of ordinary differential equations".  Cory shows how statistical error due to sampling of long-time averages significant decreases the impact of higher-order discretizations.  Being proponents and developers of higher-order methods, this was not a result we were hoping to show.  However, it is quite thought provoking.  Above is one of my favorite (or least favorite depending on your perspective) results from the paper showing how this trade between statistical and discretization error as a function of the timestep (for fixed number of timesteps). Cory also includes a start-up error model as well as the impact of parallelization.