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Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies

Published online by Cambridge University Press:  23 February 2011

Max Duarte ,
Marc Massot  and
Stéphane Descombes
Max Duarte
Affiliation:
Laboratoire EM2C – UPR CNRS 288, École Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France. max.duarte@em2c.ecp.fr; marc.massot@em2c.ecp.fr
Marc Massot
Affiliation:
Laboratoire EM2C – UPR CNRS 288, École Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France. max.duarte@em2c.ecp.fr; marc.massot@em2c.ecp.fr
Stéphane Descombes
Affiliation:
Laboratoire J.A. Dieudonné – UMR CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France. sdescomb@unice.fr
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Abstract

In this paper, we investigate the coupling between operator splitting techniques and a timeparallelization scheme, the parareal algorithm,as a numericalstrategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts,spatially very localized.In a series of previous studies, the numerical analysis of theoperator splitting as well as the parareal algorithmhas been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterizationof such techniques for these stiff configurations.Therefore,we conduct in this work a precise numerical analysis that considers thecombination of time operator splitting and the parareal algorithmin the context of stiff reaction fronts. The impact of the stiffness featured by these frontson the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution ofsuch problems.We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimatesand examine the performance of such strategiesin the context of academical one-dimensional test casesas well as multi-dimensional configurationssimulated on parallel architecture.

Keywords

Parareal algorithmoperator splittingconvergence analysisreaction-diffusionmulti-scale waves
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 825 - 852
Copyright
© EDP Sciences, SMAI, 2011

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