  The completion of a network of advanced laser-interferometric
gravitational-wave observatories around 2001 will make possible the study of
the inspiral and coalescence of binary systems of compact objects (neutron
stars and black holes), using gravitational radiation. To extract useful
information from the waves, such as the masses and spins of the bodies,
theoretical general relativistic gravitational waveform templates of extremely
high accuracy will be needed for filtering the data, probably as accurate as
$O[(v/c)^6]$ beyond the predictions of the quadrupole formula. We summarize a
method, called DIRE, for Direct Integration of the Relaxed Einstein Equations,
which extends and improves an earlier framework due to Epstein and Wagoner, in
which Einstein's equations are recast as a flat spacetime wave equation with
source composed of matter confined to compact regions and gravitational
non-linearities extending to infinity. The new method is free of divergences or
undefined integrals, correctly predicts all gravitational wave ``tail'' effects
caused by backscatter of the outgoing radiation off the background curved
spacetime, and yields radiation that propagates asymptotically along true null
cones of the curved spacetime. The method also yields equations of motion
through $O[(v/c)^4]$, radiation-reaction terms at $O[(v/c)^5]$ and
$O[(v/c)^7]$, and gravitational waveforms and energy flux through $O[(v/c)^4]$,
in agreement with other approaches. We report on progress in evaluating the
$O[(v/c)^6]$ contributions.
