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

Max Duarte; Marc Massot; Stéphane Descombes

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 5, page 825-852
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Duarte, Max, Massot, Marc, and Descombes, Stéphane. "Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 825-852. <http://eudml.org/doc/197611>.

@article{Duarte2011,
abstract = { 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. },
author = {Duarte, Max, Massot, Marc, Descombes, Stéphane},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Parareal algorithm; operator splitting; convergence analysis; reaction-diffusion; multi-scale waves; parareal algorithm; reaction-diffusion equation; parallel computation; numerical examples; semidiscretization; algorithm; stiff reaction fronts; convergence; nonlinear chemical dynamics},
language = {eng},
month = {2},
number = {5},
pages = {825-852},
publisher = {EDP Sciences},
title = {Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies},
url = {http://eudml.org/doc/197611},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Duarte, Max
AU - Massot, Marc
AU - Descombes, Stéphane
TI - Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/2//
PB - EDP Sciences
VL - 45
IS - 5
SP - 825
EP - 852
AB - 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.
LA - eng
KW - Parareal algorithm; operator splitting; convergence analysis; reaction-diffusion; multi-scale waves; parareal algorithm; reaction-diffusion equation; parallel computation; numerical examples; semidiscretization; algorithm; stiff reaction fronts; convergence; nonlinear chemical dynamics
UR - http://eudml.org/doc/197611
ER -

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