  We introduce a new strategy for coupling the parallel in time (parareal)
iterative methodology with multiscale integrators. Following the parareal
framework, the algorithm computes a low-cost approximation of all slow
variables in the system using an appropriate multiscale integrator, which is
refined using parallel fine scale integrations. Convergence is obtained using
an alignment algorithm for fast phase-like variables. The method may be used
either to enhance the accuracy and range of applicability of the multiscale
method in approximating only the slow variables, or to resolve all the state
variables. The numerical scheme does not require that the system is split into
slow and fast coordinates. Moreover, the dynamics may involve hidden slow
variables, for example, due to resonances. We propose an alignment algorithm
for almost-periodic solution, in which case convergence of the parareal
iterations is proved. The applicability of the method is demonstrated in
numerical examples.

Numerical Analysis