  We present a new time-stepping algorithm for nonlinear PDEs that exhibit
scale separation in time. Our scheme combines asymptotic techniques (which are
inexpensive but can have insufficient accuracy) with parallel-in-time methods
(which, alone, can be inefficient for equations that exhibit rapid temporal
oscillations). In particular, we use an asymptotic numerical method for
computing, in serial, a solution with low accuracy, and a more expensive fine
solver for iteratively refining the solutions in parallel. We present examples
on the rotating shallow water equations that demonstrate that significant
parallel speedup and high accuracy are achievable.
