  A method is introduced for solving Einstein's equations using two distinct
coordinate systems. The coordinate basis vectors associated with one system are
used to project out components of the metric and other fields, in analogy with
the way fields are projected onto an orthonormal tetrad basis. These field
components are then determined as functions of a second independent coordinate
system. The transformation to the second coordinate system can be thought of as
a mapping from the original ``inertial'' coordinate system to the computational
domain. This dual-coordinate method is used to perform stable numerical
evolutions of a black-hole spacetime using the generalized harmonic form of
Einstein's equations in coordinates that rotate with respect to the inertial
frame at infinity; such evolutions are found to be generically unstable using a
single rotating coordinate frame. The dual-coordinate method is also used here
to evolve binary black-hole spacetimes for several orbits. The great
flexibility of this method allows comoving coordinates to be adjusted with a
feedback control system that keeps the excision boundaries of the holes within
their respective apparent horizons.
