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.
Mots-clés : parareal algorithm, operator splitting, convergence analysis, reaction-diffusion, multi-scale waves
@article{M2AN_2011__45_5_825_0,
author = {Duarte, Max and Massot, Marc and Descombes, St\'ephane},
title = {Parareal operator splitting techniques for multi-scale reaction waves: {Numerical} analysis and strategies},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {825--852},
publisher = {EDP-Sciences},
volume = {45},
number = {5},
year = {2011},
doi = {10.1051/m2an/2010104},
mrnumber = {2817546},
zbl = {1269.65089},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2010104/}
}
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 PY - 2011 SP - 825 EP - 852 VL - 45 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2010104/ DO - 10.1051/m2an/2010104 LA - en ID - M2AN_2011__45_5_825_0 ER -
%0 Journal Article %A Duarte, Max %A Massot, Marc %A Descombes, Stéphane %T Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 825-852 %V 45 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2010104/ %R 10.1051/m2an/2010104 %G en %F M2AN_2011__45_5_825_0
Duarte, Max; Massot, Marc; Descombes, Stéphane. Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 825-852. doi : 10.1051/m2an/2010104. http://archive.numdam.org/articles/10.1051/m2an/2010104/
[1] , Fourth order Chebyshev methods with recurrence relation. J. Sci. Comput. 23 (2002) 2041-2054. | MR | Zbl
[2] , and , Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comp. 67 (1998) 457-477. | MR | Zbl
[3] , , , and , Parallel-in-time molecular-dynamics simulations. Phys. Rev. E 66 (2002) 1-4.
[4] , On the convergence and the stability of the parareal algorithm to solve partial differential equations, in Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng. 40, Springer, Berlin (2003) 426-432. | MR | Zbl
[5] and , A “parareal” time discretization for non-linear PDE's with application to the pricing of an American put, in Recent Developments in Domain Decomposition Methods, Lect. Notes Comput. Sci. Eng. 23, Springer, Berlin (2003) 189-202. | MR | Zbl
[6] , A model for fast computer simulation of waves in excitable media. Physica D 49 (1991) 61-70.
[7] and , A parallel shooting technique for solving dissipative ODEs. Computing 51 (1993) 209-236. | MR | Zbl
[8] , Analyse et Simulation Numérique de Phénomènes liés à la Combustion Supersonique. Ph.D. thesis, École Nationale des Ponts et Chaussées, France (1994).
[9] and , Comparison and analysis of some numerical schemes for stiff complex chemistry problems. RAIRO Modél. Math. Anal. Numér. 29 (1995) 259-301. | EuDML | Numdam | MR | Zbl
[10] and , Numerical simulation of laminar reacting flows with complex chemistry. Combust. Theory Modelling 4 (2000) 535-556. | Zbl
[11] , Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70 (2001) 1481-1501. | MR | Zbl
[12] and , Numerical simulation of a stroke: Computational problems and methodology. Prog. Biophys. Mol. Biol. 97 (2008) 40-53.
[13] and , Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction. Numer. Math. 97 (2004) 667-698. | MR | Zbl
[14] and , Strang's formula for holomorphic semi-groups. J. Math. Pures Appl. 81 (2002) 93-114. | MR | Zbl
[15] , and , Operator splitting for stiff nonlinear reaction-diffusion systems: Order reduction and application to spiral waves, in Patterns and waves (Saint Petersburg, 2002), AkademPrint, St. Petersburg (2003) 386-482. | MR
[16] , , and , On the local and global errors of splitting approximations of reaction-diffusion equations with high spatial gradients. Int. J. Computer Mathematics 84 (2007) 749-765. | MR | Zbl
[17] , , , , and , Operator splitting techniques for multi-scale reacting waves and application to low mach number flames with complex chemistry: Theoretical and numerical aspects. In preparation (2011).
[18] , A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math. 22 (1974) 289-315. | MR | Zbl
[19] , Newton Methods for Nonlinear Problems - Affine invariance and adaptive algorithms. Springer-Verlag (2004). | MR | Zbl
[20] , and , Fast simulations of waves in three-dimensional excitable media. Int. J. Bif. Chaos 7 (1997) 2529-2545. | MR | Zbl
[21] , , , , , and , New resolution strategy for multi-scale reaction waves using time operator splitting, space adaptive multiresolution and dedicated high order implicit/explicit time integrators. J. Sci. Comput. (to appear) available on HAL (http://hal.archives-ouvertes.fr/hal-00457731). | MR | Zbl
[22] , , , , and , Simulation of human ischemic stroke in realistic 3D geometry: A numerical strategy. Bull. Math. Biol. (to appear) available on HAL (http://hal.archives-ouvertes.fr/hal-00546223). | MR
[23] , Multiscale methods in turbulent combustion: Strategies and computational challenges. Computational Science & Discovery 2 (2009) 013001.
[24] and , An Introduction to Nonlinear Chemical Dynamics - Oscillations, Waves, Patterns and Chaos. Oxford University Press (1998). | Zbl
[25] and , Time-decomposed parallel time-integrators: Theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Methods Eng. 58 (2003) 1397-1434. | MR | Zbl
[26] , and , A parareal in time semi-implicit approximation of the Navier-Stokes equations, in Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng. 40, Springer, Berlin (2003) 433-440. | MR
[27] and , Nonlinear convergence analysis for the parareal algorithm, in Domain Decomposition Methods in Science and Engineering XVII, Springer, Berlin (2008) 45-56. | MR | Zbl
[28] and , Analysis of the parareal time-parallel time-integration method. J. Sci. Comput. 29 (2007) 556-578. | MR | Zbl
[29] , and , A convergence algorithm for time parallelization applied to reservoir simulation, in Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng. 40, Springer, Berlin (2003) 469-476. | MR | Zbl
[30] , , and , Convergent iterative schemes for time parallelization. Math. Comput. 75 (2006) 1403-1428. | MR | Zbl
[31] , Multicomponent flow modeling. Birkhäuser Boston Inc., Boston, MA (1999). | MR | Zbl
[32] , Significance of vapor phase chemical reactions on cvd rates predicted by chemically frozen and local thermochemical equilibrium boundary layer theories. J. Electrochem. Soc. 135 (1988) 1562-1570.
[33] and , Chemical oscillations and instabilites. Oxford University Press (1994).
[34] , , , , , and , A numerical study of the blocking of migraine by Rolando sulcus. Prog. Biophys. Mol. Biol. 97 (2008) 54-59.
[35] and , Solving ordinary differential equations II - Stiff and differential-algebraic problems. Second edition, Springer-Verlag, Berlin (1996). | MR | Zbl
[36] , and , Geometric Numerical Integration - Structure-Preserving Algorithms for Odinary Differential Equations. Second edition, Springer-Verlag, Berlin (2006). | MR | Zbl
[37] and , Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag, Berlin (2003). | MR | Zbl
[38] , and , Chemical vortex dynamics in the Belousov-Zhabotinsky reaction and in the two-variable Oregonator model. J. Phys. Chem. 93 (1989) 740-749.
[39] and , Computation accuracy and efficiency of the time-splitting method in solving atmosperic transport-chemistry equations. Atmos. Environ. 31 (1997) 2215-2224.
[40] , and , A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation. J. Comput. Phys. 154 (1999) 467-482. | MR | Zbl
[41] , and , Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. Bulletin de l'Université d'état Moscou, Série Internationale Section A Mathématiques et Mécanique 1 (1937) 1-25. | Zbl
[42] , and , Résolution d'EDP par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 661-668. | MR | Zbl
[43] , On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141-2153. | MR | Zbl
[44] and , A parareal in time procedure for the control of partial differential equations. C. R., Math. 335 (2002) 387-391. | MR | Zbl
[45] and , The parareal in time iterative solver: A further direction to parallel implementation, in Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng. 40, Springer, Berlin (2003) 441-448. | MR | Zbl
[46] , Splitting and alternating direction methods, in Handbook of numerical analysis I, North-Holland, Amsterdam (1990) 197-462. | MR | Zbl
[47] , Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete Contin. Dyn. Syst. Ser. B 2 (2002) 433-456. | MR | Zbl
[48] , and , Numerical solution of the atmospheric diffusion equation for chemically reacting flows. J. Comput. Phys. 45 (1982) 1-42. | MR | Zbl
[49] and , Modeling Low Mach number reacting flow with detailed chemistry and transport. J. Sci. Comput. 25 (2005) 263-287. | MR | Zbl
[50] , and , A semi-implicit numerical scheme for reacting flow. I. Stiff chemistry. J. Comput. Phys. 143 (1998) 381-402. | MR | Zbl
[51] , Toward non commutative numerical analysis: High order integration in time. J. Sci. Comput. 17 (2002) 107-125. | MR | Zbl
[52] , and , IRKC: An IMEX solver for stiff diffusion-reaction PDEs. J. Comput. Appl. Math. 196 (2006) 485-497. | MR | Zbl
[53] , Error estimate for the modified Newton method with applications to the solution of nonlinear, two-point boundary value problems. J. Optim. Theory Appl. 39 (1983) 489-511. | MR | Zbl
[54] , and , RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88 (1998) 315-326. | MR | Zbl
[55] , Contribution à la modélisation des écoulements réactifs : Réduction des modèles de cinétique chimique et simulation de la pollution atmosphérique. Ph.D. thesis, École Polytechnique, France (1999).
[56] , An analysis of operator splitting techniques in the stiff case. J. Comput. Phys. 161 (2000) 140-168. | MR | Zbl
[57] and , Reduction of chemical kinetics in air pollution modeling. J. Comput. Phys. 164 (2000) 354-376. | MR | Zbl
[58] and , Stability of the parareal algorithm, in Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng. 40, Springer, Berlin (2003) 449-456. | MR | Zbl
[59] , Accurate partial difference methods. I. Linear Cauchy problems. Arch. Ration. Mech. Anal. 12 (1963) 392-402. | MR | Zbl
[60] , On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506-517. | MR | Zbl
[61] , A pseudo non-time splitting method in air quality modeling. J. Comp. Phys. 127 (1996) 152-157. | Zbl
[62] , Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I. Arch. Rational Mech. Anal. 32 (1969) 135-153. | MR | Zbl
[63] , Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Arch. Rational Mech. Anal. 33 (1969) 377-385. | MR | Zbl
[64] and , An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. SIAM J. Sci. Comput. 25 (2004) 1824-1835. | MR | Zbl
[65] and , Note on operator splitting in a stiff linear case. Rep. MAS-R9830 (1998).
[66] , and , RKC time-stepping for advection-diffusion-reaction problems. J. Comput. Phys. 201 (2004) 61-79. | MR | Zbl
[67] , and , Traveling wave solutions of parabolic systems. American Mathematical Society, Providence, RI (1994). | MR | Zbl
[68] , The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971). | MR | Zbl
Cité par Sources :
