  In this paper, a time-periodic MGRIT algorithm is proposed as a means to
reduce the time-to-solution of numerical algorithms by exploiting the time
periodicity inherent to many applications in science and engineering. The
time-periodic MGRIT algorithm is applied to a variety of linear and nonlinear
single- and multiphysics problems that are periodic-in-time. It is demonstrated
that the proposed parallel-in-time algorithm can obtain the same time-periodic
steady-state solution as sequential time-stepping. It is shown that the
required number of MGRIT iterations can be estimated a priori and that the new
MGRIT variant can significantly and consistently reduce the time-to-solution
compared to sequential time-stepping, irrespective of the number of dimensions,
linear or nonlinear PDE models, single-physics or coupled problems and the
employed computing resources. The numerical experiments demonstrate that the
time-periodic MGRIT algorithm enables a greater level of parallelism yielding
faster turnaround, and thus, facilitating more complex and more realistic
problems to be solved.
