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Inspiral of binary black holes occurs over a time-scale of many orbits, farlonger than the dynamical time-scale of the individual black holes. Explicitevolutions of a binary system therefore require excessively many time steps tocapture interesting dynamics. We present a strategy to overcome theCourant-Friedrichs-Lewy condition in such evolutions, one relying on modernimplicit-explicit ODE solvers and multidomain spectral methods for ellipticequations. Our analysis considers the model problem of a forced scalar fieldpropagating on a generic curved background. Nevertheless, we encounter andaddress a number of issues pertinent to the binary black hole problem in fullgeneral relativity. Specializing to the Schwarzschild geometry in Kerr-Schildcoordinates, we document the results of several numerical experiments testingour strategy.

IMEX evolution of scalar fields on curved backgrounds