  The Parareal algorithm, which is related to multiple shooting, was introduced
for solving evolution problems in a time-parallel manner. The algorithm was
then extended to solve time-periodic problems. We are interested here in
time-periodic systems which are forced with quickly-switching discontinuous
inputs. Such situations occur, e.g., in power engineering when electric devices
are excited with a pulse-width-modulated signal. In order to solve those
problems numerically we consider a recently introduced modified Parareal method
with reduced coarse dynamics. Its main idea is to use a low-frequency smooth
input for the coarse problem, which can be obtained, e.g., from Fourier
analysis. Based on this approach, we present and analyze a new Parareal
algorithm for time-periodic problems with highly-oscillatory discontinuous
sources. We illustrate the performance of the method via its application to the
simulation of an induction machine.
