# 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

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topDuarte, 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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