  Simulations of binary black hole systems using the Spectral Einstein Code
(SpEC) are done on a computational domain that excises the regions inside the
black holes. It is imperative that the excision boundaries are outflow
boundaries with respect to the hyperbolic evolution equations used in the
simulation. We employ a time-dependent mapping between the fixed computational
frame and the inertial frame through which the black holes move. The
time-dependent parameters of the mapping are adjusted throughout the simulation
by a feedback control system in order to follow the motion of the black holes,
to adjust the shape and size of the excision surfaces so that they remain
outflow boundaries, and to prevent large distortions of the grid. We describe
in detail the mappings and control systems that we use. We show how these
techniques have been essential in the evolution of binary black hole systems
with extreme configurations, such as large spin magnitudes and high mass
ratios, especially during the merger, when apparent horizons are highly
distorted and the computational domain becomes compressed. The techniques
introduced here may be useful in other applications of partial differential
equations that involve time-dependent mappings.
