  We introduce numerical methods for simulating the diffusive motion of rigid
bodies of arbitrary shape immersed in a viscous fluid. We parameterize the
orientation of the bodies using normalized quaternions, which are numerically
robust, space efficient, and easy to accumulate. We construct a system of
overdamped Langevin equations in the quaternion representation that accounts
for hydrodynamic effects, preserves the unit-norm constraint on the quaternion,
and is time reversible with respect to the Gibbs-Boltzmann distribution at
equilibrium. We introduce two schemes for temporal integration of the
overdamped Langevin equations of motion, one based on the Fixman midpoint
method and the other based on a random finite difference approach, both of
which ensure the correct stochastic drift term is captured in a computationally
efficient way. We study several examples of rigid colloidal particles diffusing
near a no-slip boundary, and demonstrate the importance of the choice of
tracking point on the measured translational mean square displacement (MSD). We
examine the average short-time as well as the long-time quasi-two-dimensional
diffusion coefficient of a rigid particle sedimented near a bottom wall due to
gravity. For several particle shapes we find a choice of tracking point that
makes the MSD essentially linear with time, allowing us to estimate the
long-time diffusion coefficient efficiently using a Monte Carlo method.
However, in general such a special choice of tracking point does not exist, and
numerical techniques for simulating long trajectories, such as the ones we
introduce here, are necessary to study diffusion on long timescales.
