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

By Frédéric Legoll, Tony Lelièvre, Giovanni Samaey. 2012

51 citations recorded by the source corpus.

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 converges to 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 ...

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Ansel L. Blumers
2. G. Em Karniadakis
3. Zhen Li
4. Beth A. Wingate
5. Terry S. Haut
6. Maizar Raissi
7. Paris Perdikaris
8. Suchuan Dong

In Computational Theory and Mathematics

1. G. Em Karniadakis
2. Ansel L. Blumers
3. Zhen Li
4. Beth A. Wingate
5. Terry S. Haut
6. Paris Perdikaris
7. Dongbin Xiu
8. Maizar Raissi

In Numerical Analysis

1. Ansel L. Blumers
2. G. Em Karniadakis
3. Zhen Li
4. Beth A. Wingate
5. Terry S. Haut

Citations in the corpus

• Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics (2019, methodology): “While most applications employ continuum solvers, a few utilize other methods such as particle solvers [21–26].”
• An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs (2013, methodology): “In contrast to [25], [12], and [22] , here we investigate this procedure for a model nonlinear PDE whose scale separation is of a highly oscillatory nature, and where methods that work well for stiff dissipative problems (e.”