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3 results on '"Multi-scale waves"'

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

Author

Stéphane Descombes, Max Duarte, Marc Massot, Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion (EM2C), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec, Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), This research was supported by a fundamental CNRS Ph.D. grant for M. Duarte from the Institute of Mathematics (INSMI) and Engineering Institute (INSIS) of CNRS. This work was granted access to the HPC resources of IDRIS (CNRS Institute of Scientific Computing) under the allocation 2009-i2009066173 made by GENCI (Grand Equipement National de Calcul Intensif)., and ANR-06-CIS6-0007,PITAC,Parallélisation Incluant le Temps pour Accélérer les Calculs(2006)

Subjects
Scale (ratio), Parareal, Context (language use), 010103 numerical & computational mathematics, 01 natural sciences, Parareal algorithm, Multi-scale waves, Convergence analysis, Convergence (routing), Reaction–diffusion system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Operator splitting, 0101 mathematics, Mathematics, Numerical Analysis, Series (mathematics), [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, Applied Mathematics, Numerical analysis, Mathematical analysis, AMS 65Y05, 65M12, 65L04, 35A35, 35K57, 35C07, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Modeling and Simulation, Reaction- diffusion, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

International audience; 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 modeling 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.

Published
2011
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2. New resolution strategy for multi-scale reaction waves using time operator splitting, space adaptive multiresolution and dedicated high order implicit/explicit time integrators

Author

Christian Tenaud, Thierry Dumont, Frédérique Laurent, Max Duarte, Stéphane Descombes, Violaine Louvet, Marc Massot, Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion (EM2C), CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris Saclay (COmUE), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur (LIMSI), Université Paris-Sud - Paris 11 (UP11)-Sorbonne Université - UFR d'Ingénierie (UFR 919), Sorbonne Université (SU)-Sorbonne Université (SU)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Université Paris Saclay (COmUE), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Numerical Medicine (NUMED), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), This research was supported by a fundamental projects grant from ANR (French National Research Agency - ANR Blancs) 'Séchelles', Project leader S. Descombes, by a CNRS Ph.D. grant for M. Duarte from the Institute of Mathematics (INSMI) and Engineering Institute (INSIS) of CNRS, and through a PEPS Maths-ST2I CNRS project (V. Louvet)., ANR-09-BLAN-0075,Séchelles,Simulation et comparaison avec l'expérience pour la validation de modèles de problèmes multi-échelles.(2009), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec, Université Nice Sophia Antipolis (... - 2019) (UNS), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université - UFR d'Ingénierie (UFR 919), Sorbonne Université (SU)-Sorbonne Université (SU)-Université Paris-Saclay-Université Paris-Sud - Paris 11 (UP11), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Inria Grenoble - Rhône-Alpes

Subjects
Spacetime, Computer simulation, Applied Mathematics, Numerical analysis, [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, Mathematical analysis, Context (language use), 010103 numerical & computational mathematics, 01 natural sciences, Term (time), 010101 applied mathematics, Adaptive multiresolution, Computational Mathematics, Nonlinear system, Multi-scale waves, Integrator, Reaction–diffusion system, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Operator splitting, 0101 mathematics, Reaction-diffusion, MCS : 65M50, 35K57, 65M08, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problem 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 steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. The paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two-dimensional spiral waves and three-dimensional scroll waves, as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed.

Published
2012
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3. New Resolution Strategies for Multi-scale Reaction Waves: Optimal Time Operator Splitting and Space Adaptive Multiresolution

Author

Frédérique Laurent, Violaine Louvet, Stéphane Descombes, Marc Massot, Thierry Dumont, Christian Tenaud, Max Duarte, Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion (EM2C), CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris Saclay (COmUE), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur (LIMSI), Université Paris-Sud - Paris 11 (UP11)-Sorbonne Université - UFR d'Ingénierie (UFR 919), Sorbonne Université (SU)-Sorbonne Université (SU)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Université Paris Saclay (COmUE), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Numerical Medicine (NUMED), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Luca Cernuzzi, Ramon Puigjaner, Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec, Université Nice Sophia Antipolis (... - 2019) (UNS), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université - UFR d'Ingénierie (UFR 919), Sorbonne Université (SU)-Sorbonne Université (SU)-Université Paris-Saclay-Université Paris-Sud - Paris 11 (UP11), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Inria Grenoble - Rhône-Alpes

Subjects
adaptive multiresolution, Mathematical optimization, Computer science, [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], Scale (descriptive set theory), Context (language use), 010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, Stability (probability), lcsh:QA75.5-76.95, multiresolución adaptativa, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, Reaction-diffusion, 0101 mathematics, división de operadores, Spacetime, Computer simulation, [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, Numerical analysis, General Medicine, 010101 applied mathematics, Nonlinear system, ondas multi-escala, multi-scale waves, lcsh:Electronic computers. Computer science, difusión-reacción, operator splitting
Abstract

We tackle the numerical simulation of reaction-diffusion equations modeling 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 reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. Based on recent theoretical studies of numerical analysis, such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by restrictive diffusive step stability limits, but only by the physics of the phenomenon. We thus aim at solving accurately complete models including all time and space scales of the phenomenon, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy are finally discussed. Estudiamos la simulación numérica de ecuaciones de reacción-difusión que modelan ondas de reacción multi-escala. Este tipo de problemas induce dificultades peculiares y problemas de condicionamiento potencialmente grandes que surgen del amplio espectro de escalas temporales en el término no-lineal de la fuente química y de la presencia de grandes gradientes espaciales en el frente de reacción, muy localizados espacialmente. En este trabajo, introducimos una nueva estrategia de resolución basada en el particionamiento del operador temporal y en la multiresolución adaptiva espacial. Sobre la base de estudios teóricos recientes en el campo de análisis numérico, esta estrategia conduce a un paso de tiempo que no está restringido por las escalas más veloces en el término fuente ni por límites de estabilidad en la etapa de difusión, sino sólo por la propia física del fenómeno en estudio. De esta forma pretendemos resolver de forma precisa modelos completos que incluyen todas las escalas espaciales y temporales del fenómeno, teniendo en cuenta dominios de simulación amplios utilizando recursos computacionales convencionales. La eficiencia se evalúa a través de la simulación numérica de configuraciones que hasta el momento estaban fuera del alcance de los métodos estándar en el campo de la dinámica química no lineal para ondas espiral 2D y ondas scroll 3D. Extensiones futuras de la estrategia se discuten al final.

Published
2011
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