  Inspiral of binary black holes occurs over a time-scale of many orbits, far
longer than the dynamical time-scale of the individual black holes. Explicit
evolutions of a binary system therefore require excessively many time steps to
capture interesting dynamics. We present a strategy to overcome the
Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern
implicit-explicit ODE solvers and multidomain spectral methods for elliptic
equations. Our analysis considers the model problem of a forced scalar field
propagating on a generic curved background. Nevertheless, we encounter and
address a number of issues pertinent to the binary black hole problem in full
general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild
coordinates, we document the results of several numerical experiments testing
our strategy.
