  Numerical simulations of binary black holes---an important predictive tool
for the detection of gravitational waves---are computationally expensive,
especially for binaries with high mass ratios or with rapidly spinning
constituent holes. Existing codes for evolving binary black holes rely on
explicit timestepping methods, for which the timestep size is limited by the
smallest spatial scale through the Courant-Friedrichs-Lewy condition. Binary
inspiral typically involves spatial scales (the spatial resolution required by
a small or rapidly spinning hole) which are orders of magnitude smaller than
the relevant (orbital, precession, and radiation-reaction) timescales
characterizing the inspiral. Therefore, in explicit evolutions of binary black
holes, the timestep size is typically orders of magnitude smaller than the
relevant physical timescales. Implicit timestepping methods allow for larger
timesteps, and they often reduce the total computational cost (without
significant loss of accuracy) for problems dominated by spatial rather than
temporal error, such as for binary-black-hole inspiral in corotating
coordinates. However, fully implicit methods can be difficult to implement for
nonlinear evolution systems like the Einstein equations. Therefore, in this
paper we explore implicit-explicit (IMEX) methods and use them for the first
time to evolve black-hole spacetimes. Specifically, as a first step toward IMEX
evolution of a full binary-black-hole spacetime, we develop an IMEX algorithm
for the generalized harmonic formulation of the Einstein equations and use this
algorithm to evolve stationary and perturbed single-black-hole spacetimes.
Numerical experiments explore the stability and computational efficiency of our
method.
