Résumé en anglais

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the significant improvement of the parallel efficiency of the parareal in time method, an iterative predictor-corrector algorithm. 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 rigourous 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. In addition to this, we discuss the potential benefits of reusing information from previous parareal iterations to enhance efficiency even more.