A Micro-Macro Parareal Algorithm: Application to Singularly Perturbed Ordinary Differential Equations

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model over the complete time interval. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution are better and better approximations of the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The error bound depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments. ∗Laboratoire Navier, École Nationale des Ponts et Chaussées, Université Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France; INRIA Rocquencourt, MICMAC teamproject, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France †CERMICS, École Nationale des Ponts et Chaussées, Université Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France; INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France ‡Scientific Computing, Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium Scientific Computing, Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium