A micro/macro parareal algorithm for singularly perturbed ODEs

Giovanni Samaey
Computer Science

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate macroscopic model), which is iteratively corrected by means of a set of more accurate simulations using a fine-scale propagator (simulating the full microscopic dynamics). This correction is done in paral- lel, reducing wall-clock time compared to the integration of the full microscopic model. We provide a numerical convergence analysis for a prototypical example of a micro- macro model, namely singularly perturbed ordinary differential equations. We show that the method only converges to the full microscopic solution (as a function of the number of parareal iterations) if special care is taken during the coupling of the mi- croscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate this result with numerical experiments.

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