  A quasi-equilibrium (QE) computational scheme was recently developed in
general relativity to calculate the complete gravitational wavetrain emitted
during the inspiral phase of compact binaries. The QE method exploits the fact
that the the gravitational radiation inspiral timescale is much longer than the
orbital period everywhere outside the ISCO. Here we demonstrate the validity
and advantages of the QE scheme by solving a model problem in relativistic
scalar gravitation theory. By adopting scalar gravitation, we are able to
numerically track without approximation the damping of a simple, quasi-periodic
radiating system (an oscillating spherical matter shell) to final equilibrium,
and then use the exact numerical results to calibrate the QE approximation
method. In particular, we calculate the emitted gravitational wavetrain three
different ways: by integrating the exact coupled dynamical field and matter
equations, by using the scalar-wave monopole approximation formula
(corresponding to the quadrupole formula in general relativity), and by
adopting the QE scheme. We find that the monopole formula works well for weak
field cases, but fails when the fields become even moderately strong. By
contrast, the QE scheme remains quite reliable for moderately strong fields,
and begins to breakdown only for ultra-strong fields. The QE scheme thus
provides a promising technique to construct the complete wavetrain from binary
inspiral outside the ISCO, where the gravitational fields are strong, but where
the computational resources required to follow the system for more than a few
orbits by direct numerical integration of the exact equations are prohibitive.
