The periodic standing wave (PSW) method for the binary inspiral of black
holes and neutron stars computes exact numerical solutions for periodic
standing wave spacetimes and then extracts approximate solutions of the
physical problem, with outgoing waves. The method requires solution of a
boundary value problem with a mixed (hyperbolic and elliptic) character.
We present here a new numerical method for such problems, based on three
innovations: (i) a coordinate system adapted to the geometry of the problem,
(ii) an expansion in multipole moments of these coordinates and a filtering out
of higher moments, and (iii) the replacement of the continuum multipole moments
with their analogs for a discrete grid. We illustrate the efficiency and
accuracy of this method with nonlinear scalar model problems. Finally, we take
advantage of the ability of this method to handle highly nonlinear models to
demonstrate that the outgoing approximations extracted from the standing wave
solutions are highly accurate even in the presence of strong nonlinearities.Comment: RevTex, 32 pages, 13 figures, 6 table
