Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies
Laboratoire EM2C – UPR CNRS 288, École Centrale
Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex,
2 Laboratoire J.A. Dieudonné – UMR CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France. firstname.lastname@example.org
In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy 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 the operator splitting as well as the parareal algorithm has 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 characterization of such techniques for these stiff configurations. Therefore, we conduct in this work a precise numerical analysis that considers the combination of time operator splitting and the parareal algorithm in the context of stiff reaction fronts. The impact of the stiffness featured by these fronts on the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution of such problems. We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimates and examine the performance of such strategies in the context of academical one-dimensional test cases as well as multi-dimensional configurations simulated on parallel architecture.
Mathematics Subject Classification: 65Y05 / 65M12 / 65L04 / 35A35 / 35K57 / 35C07
Key words: Parareal algorithm / operator splitting / convergence analysis / reaction-diffusion / multi-scale waves
© EDP Sciences, SMAI, 2011