  Binary black-hole systems are expected to be important sources of
gravitational waves for upcoming gravitational-wave detectors. If the spins are
not colinear with each other or with the orbital angular momentum, these
systems exhibit complicated precession dynamics that are imprinted on the
gravitational waveform. We develop a new procedure to match the precession
dynamics computed by post-Newtonian (PN) theory to those of numerical binary
black-hole simulations in full general relativity. For numerical relativity NR)
simulations lasting approximately two precession cycles, we find that the PN
and NR predictions for the directions of the orbital angular momentum and the
spins agree to better than $\sim 1^{\circ}$ with NR during the inspiral,
increasing to $5^{\circ}$ near merger. Nutation of the orbital plane on the
orbital time-scale agrees well between NR and PN, whereas nutation of the spin
direction shows qualitatively different behavior in PN and NR. We also examine
how the PN equations for precession and orbital-phase evolution converge with
PN order, and we quantify the impact of various choices for handling partially
known PN terms.
