  The computational cost of inspiral and merger simulations for black-hole
binaries increases in inverse proportion to the square of the mass ratio
$q:=m_2/m_1\leq 1$. One factor of $q$ comes from the number of orbital cycles,
which is proportional to $1/q$, and another is associated with the required
number of time steps per orbit, constrained (via the Courant-Friedrich-Lewy
condition) by the need to resolve the two disparate length scales. This
problematic scaling makes simulations progressively less tractable at smaller
$q$. Here we propose and explore a method for alleviating the scale disparity
in simulations with mass ratios in the intermediate astrophysical range
($10^{-4} \lesssim q\lesssim 10^{-2}$), where purely perturbative methods may
not be adequate. A region of radius much larger than $m_2$ around the smaller
object is excised from the numerical domain, and replaced with an analytical
model approximating a tidally deformed black hole. The analytical model
involves certain a priori unknown parameters, associated with unknown bits of
physics together with gauge-adjustment terms; these are dynamically determined
by matching to the numerical solution outside the excision region. In this
paper we develop the basic idea and apply it to a toy model of a scalar charge
in a circular geodesic orbit around a Schwarzschild black hole, solving for the
massless Klein-Gordon field in a 1+1D framework. Our main goal here is to
explore the utility and properties of different matching strategies, and to
this end we develop two independent implementations, a finite-difference one
and a spectral one. We discuss the extension of our method to a full 3D
numerical evolution and to gravity.
