In the search of new approaches for the efficient exploitation of large scale compu- tational platforms, the parallelization in the time direction for time dependent problems is a very promising approach. Among the existing methods in this frame, the parallel in time method, since its introduction in [10], has been developed in many ways that, altogether, allow to identify its pros and cons. Among the cons, the current approaches present in practice some efficiency limitations as regards correct scalings that “spoil” the huge potential of the idea. This article is a contribution towards overcoming this major obstruction by exploiting the idea that the numerical schemes to parallelize time could be coupled to other iterative numerical algorithms that are needed to solve the PDE. We present a parareal scheme in which these alternative iterations are truncated (i.e. not converged) during each parareal iteration but in which convergence is nevertheless achieved across the parareal iterations. In order to limit the use of too much memory necessitated by the recovery of these alternative iterations over the parareal iterations, we propose also a compression procedure via proper orthogonal decomposition. After a mathematical analysis of the convergence properties of this new approach, we present some numerical results dealing with the application of the scheme to ac- celerate the time-dependent neutron diffusion equation in a reactor core. The numerical results show a significant improvement of the performances with respect to the plain parareal algorithm, which is an important step towards making the parallelization in the time direction be a fully competitive option for the exploitation of massively parallel computers
