In this contribution we present the parareal algorithm in its plain version. We show how it works and in particular it inherent lack of scalability.

We the consider the problem of accelerating its numerical implementation. This is achieved by first reformulating the algorithm in a rigorous infinite dimensional functional space setting. We then formulate implementable versions where time dependent subproblems are solved at increasing accuracy across the parareal iterations (in opposition to the classical version where the subproblems are solved at a fixed high accuracy). Aside from the important improvement in parallel efficiency and as a natural by product, the new approach provides a rigorous online stopping criterion with a posteriori error estimators and the numerical cost to achieve a certain final accuracy is designed to be near-minimal.

We illustrate the gain in efficiency of the new approach on simple numerical experiments.

We then present application to molecular dynamics.