Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-24T11:16:16.755Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  23 February 2011

Max Duarte ,
Marc Massot  and
Stéphane Descombes
Max Duarte
Affiliation:
Laboratoire EM2C – UPR CNRS 288, École Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France. max.duarte@em2c.ecp.fr; marc.massot@em2c.ecp.fr
Marc Massot
Affiliation:
Laboratoire EM2C – UPR CNRS 288, École Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France. max.duarte@em2c.ecp.fr; marc.massot@em2c.ecp.fr
Stéphane Descombes
Affiliation:
Laboratoire J.A. Dieudonné – UMR CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France. sdescomb@unice.fr
Get access

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.

Keywords

Parareal algorithmoperator splittingconvergence analysisreaction-diffusionmulti-scale waves
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 825 - 852
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdulle, A., Fourth order Chebyshev methods with recurrence relation. J. Sci. Comput. 23 (2002) 2041–2054.
Akrivis, G., Crouzeix, M. and Makridakis, C., Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comp. 67 (1998) 457–477. CrossRef
Baffico, L., Bernard, S., Maday, Y., Turinici, G. and Zérah, G., Parallel-in-time molecular-dynamics simulations. Phys. Rev. E 66 (2002) 1–4. CrossRef
G. Bal, 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.
G. Bal and Y. Maday, 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.
Barkley, D., A model for fast computer simulation of waves in excitable media. Physica D 49 (1991) 61–70. CrossRef
Chartier, P. and Philippe, B., A parallel shooting technique for solving dissipative ODEs. Computing 51 (1993) 209–236. CrossRef
Y. D'Angelo, Analyse et Simulation Numérique de Phénomènes liés à la Combustion Supersonique. Ph.D. thesis, École Nationale des Ponts et Chaussées, France (1994).
D'Angelo, Y. and Larrouturou, B., Comparison and analysis of some numerical schemes for stiff complex chemistry problems. RAIRO Modél. Math. Anal. Numér. 29 (1995) 259–301. CrossRef
Day, M.S. and Bell, J.B., Numerical simulation of laminar reacting flows with complex chemistry. Combust. Theory Modelling 4 (2000) 535–556. CrossRef
Descombes, S., Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70 (2001) 1481–1501. CrossRef
Descombes, S. and Dumont, T., Numerical simulation of a stroke: Computational problems and methodology. Prog. Biophys. Mol. Biol. 97 (2008) 40–53. CrossRef
Descombes, S. and Massot, M., Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction. Numer. Math. 97 (2004) 667–698. CrossRef
Descombes, S. and Schatzman, M., Strang's formula for holomorphic semi-groups. J. Math. Pures Appl. 81 (2002) 93–114. CrossRef
S. Descombes, T. Dumont and M. Massot, 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.
Descombes, S., Dumont, T., Louvet, V. and Massot, M., 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. CrossRef
S. Descombes, T. Dumont, V. Louvet, M. Massot, F. Laurent and J. Beaulaurier, 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).
Deuflhard, P., 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. CrossRef
P. Deuflhard, Newton Methods for Nonlinear Problems – Affine invariance and adaptive algorithms. Springer-Verlag (2004).
Dowle, M., Mantel, R.M. and Barkley, D., Fast simulations of waves in three-dimensional excitable media. Int. J. Bif. Chaos 7 (1997) 2529–2545. CrossRef
M. Duarte, M. Massot, S. Descombes, C. Tenaud, T. Dumont, V. Louvet and F. Laurent, 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).
T. Dumont, M. Duarte, S. Descombes, M.A. Dronne, M. Massot and V. Louvet, 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).
Echekki, T., Multiscale methods in turbulent combustion: Strategies and computational challenges. Computational Science & Discovery 2 (2009) 013001. CrossRef
I.R. Epstein and J.A. Pojman, An Introduction to Nonlinear Chemical Dynamics – Oscillations, Waves, Patterns and Chaos. Oxford University Press (1998).
Farhat, C. and Chandesris, M., 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. CrossRef
F. Fischer, F. Hecht and Y. Maday, 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.
M. Gander and E. Hairer, Nonlinear convergence analysis for the parareal algorithm, in Domain Decomposition Methods in Science and Engineering XVII, Springer, Berlin (2008) 45–56.
Gander, M. and Vandewalle, S., Analysis of the parareal time-parallel time-integration method. J. Sci. Comput. 29 (2007) 556–578.
I. Garrido, M.S. Espedal and G.E. Fladmark, 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.
Garrido, I., Lee, B., Fladmark, G.E. and Espedal, M.S., Convergent iterative schemes for time parallelization. Math. Comput. 75 (2006) 1403–1428. CrossRef
V. Giovangigli, Multicomponent flow modeling. Birkhäuser Boston Inc., Boston, MA (1999).
Gokoglu, S.A., 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. CrossRef
P. Gray and S.K. Scott, Chemical oscillations and instabilites. Oxford University Press (1994).
Grenier, E., Dronne, M.A., Descombes, S., Gilquin, H., Jaillard, A., Hommel, M. and Boissel, J.P., A numerical study of the blocking of migraine by Rolando sulcus. Prog. Biophys. Mol. Biol. 97 (2008) 54–59. CrossRef
E. Hairer and G. Wanner, Solving ordinary differential equations II – Stiff and differential-algebraic problems. Second edition, Springer-Verlag, Berlin (1996).
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration – Structure-Preserving Algorithms for Odinary Differential Equations. Second edition, Springer-Verlag, Berlin (2006).
W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag, Berlin (2003).
Jahnke, W., Skaggs, W.E. and Winfree, A.T., Chemical vortex dynamics in the Belousov-Zhabotinsky reaction and in the two-variable Oregonator model. J. Phys. Chem. 93 (1989) 740–749. CrossRef
Kim, J. and Cho, S.Y., Computation accuracy and efficiency of the time-splitting method in solving atmosperic transport-chemistry equations. Atmos. Environ. 31 (1997) 2215–2224. CrossRef
Knio, O.M., Najm, H.N. and Wyckoff, P.S., A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation. J. Comput. Phys. 154 (1999) 467–482. CrossRef
A.N. Kolmogoroff, I.G. Petrovsky and N.S. Piscounoff, 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.
Lions, J.L., Maday, Y. and Turinici, G., Résolution d'EDP par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 661–668. CrossRef
Lubich, C., On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141–2153. CrossRef
Y. Maday and G. Turinici, A parareal in time procedure for the control of partial differential equations. C. R., Math. 335 (2002) 387–391.
Y. Maday and G. Turinici, 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.
G.I. Marchuk, Splitting and alternating direction methods, in Handbook of numerical analysis I , North-Holland, Amsterdam (1990) 197–462.
Massot, M., 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. CrossRef
McRae, G.J., Goodin, W.R. and Seinfeld, J.H., Numerical solution of the atmospheric diffusion equation for chemically reacting flows. J. Comput. Phys. 45 (1982) 1–42. CrossRef
Najm, H.N. and Knio, O.M., Modeling Low Mach number reacting flow with detailed chemistry and transport. J. Sci. Comput. 25 (2005) 263–287. CrossRef
Najm, H.N., Wyckoff, P.S. and Knio, O.M., A semi-implicit numerical scheme for reacting flow. I. Stiff chemistry. J. Comput. Phys. 143 (1998) 381–402. CrossRef
Schatzman, M., Toward non commutative numerical analysis: High order integration in time. J. Sci. Comput. 17 (2002) 107–125. CrossRef
Shampine, L.F., Sommeijer, B.P. and Verwer, J.G., IRKC: An IMEX solver for stiff diffusion-reaction PDEs. J. Comput. Appl. Math. 196 (2006) 485–497. CrossRef
Smooke, M.D., 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. CrossRef
Sommeijer, B.P., Shampine, L.F. and Verwer, J.G., RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88 (1998) 315–326. CrossRef
B. Sportisse, 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).
Sportisse, B., An analysis of operator splitting techniques in the stiff case. J. Comput. Phys. 161 (2000) 140–168. CrossRef
Sportisse, B. and Djouad, R., Reduction of chemical kinetics in air pollution modeling. J. Comput. Phys. 164 (2000) 354–376. CrossRef
G.A. Staff and E.M. Rønquist, 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.
Strang, G., Accurate partial difference methods. I. Linear Cauchy problems. Arch. Ration. Mech. Anal. 12 (1963) 392–402. CrossRef
Strang, G., On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. CrossRef
Sun, P., A pseudo non-time splitting method in air quality modeling. J. Comp. Phys. 127 (1996) 152–157. CrossRef
Témam, R., 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. CrossRef
Témam, R., 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. CrossRef
Verwer, J.G. and Sommeijer, B.P., An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. SIAM J. Sci. Comput. 25 (2004) 1824–1835. CrossRef
J.G. Verwer and B. Sportisse, Note on operator splitting in a stiff linear case. Rep. MAS-R9830 (1998).
Verwer, J.G., Sommeijer, B.P. and Hundsdorfer, W., RKC time-stepping for advection-diffusion-reaction problems. J. Comput. Phys. 201 (2004) 61–79. CrossRef
A.I. Volpert, V.A. Volpert and V.A. Volpert, Traveling wave solutions of parabolic systems. American Mathematical Society, Providence, RI (1994).
N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971).