  We investigate the influence of numerical discretization errors on computed
averages in a molecular dynamics simulation of TIP4P liquid water at 300 K
coupled to different deterministic (Nos\'e-Hoover and Nos\'e-Poincar\'e) and
stochastic (Langevin) thermostats. We propose a couple of simple practical
approaches to estimating such errors and taking them into account when
computing the averages. We show that it is possible to obtain accurate
measurements of various system quantities using step sizes of up to 70% of the
stability threshold of the integrator, which for the system of TIP4P liquid
water at 300 K corresponds to the step size of about 7 fs.
