  This paper proposes a frequency-time hybrid solver for the time-dependent
wave equation in two-dimensional interior spatial domains. The approach relies
on four main elements, namely, 1) A multiple scattering strategy that
decomposes a given interior time-domain problem into a sequence of
limited-duration time-domain problems of scattering by overlapping open arcs,
each one of which is reduced (by means of the Fourier transform) to a sequence
of Helmholtz frequency-domain problems; 2) Boundary integral equations on
overlapping boundary patches for the solution of the frequency-domain problems
in point 1); 3) A smooth "Time-windowing and recentering" methodology that
enables both treatment of incident signals of long duration and long time
simulation; and, 4) A Fourier transform algorithm that delivers numerically
dispersionless, spectrally-accurate time evolution for given incident fields.
By recasting the interior time-domain problem in terms of a sequence of
open-arc multiple scattering events, the proposed approach regularizes the full
interior frequency domain problem-which, if obtained by either Fourier or
Laplace transformation of the corresponding interior time-domain problem, must
encapsulate infinitely many scattering events, giving rise to non-uniqueness
and eigenfunctions in the Fourier case, and ill conditioning in the Laplace
case. Numerical examples are included which demonstrate the accuracy and
efficiency of the proposed methodology.
