  A variant of the Parareal method for highly oscillatory systems of PDEs was
proposed by Haut and Wingate (2014). In that work they proved superlinear
conver- gence of the method in the limit of infinite time scale separation.
Their coarse solver features a coordinate transformation and a fast-wave
averag- ing method inspired by analysis of multiple scales PDEs and is
integrated using an HMM-type method. However, for many physical applications
the timescale separation is finite, not infinite. In this paper we prove con-
vergence for finite timescale separaration by extending the error bound on the
coarse propagator to this case. We show that convergence requires the solution
of an optimization problem that involves the averaging win- dow interval, the
time step, and the parameters in the problem. We also propose a method for
choosing the averaging window relative to the time step based as a function of
the finite frequencies inherent in the problem.
