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

top
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

top

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 -

References

top
  1. A. Abdulle, Fourth order Chebyshev methods with recurrence relation. J. Sci. Comput.23 (2002) 2041–2054.  Zbl1009.65048
  2. G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comp.67 (1998) 457–477.  Zbl0896.65066
  3. L. Baffico, S. Bernard, Y. Maday, G. Turinici and G. Zérah, Parallel-in-time molecular-dynamics simulations. Phys. Rev. E66 (2002) 1–4.  
  4. 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.  
  5. 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.  Zbl1022.65096
  6. D. Barkley, A model for fast computer simulation of waves in excitable media. Physica D49 (1991) 61–70.  
  7. P. Chartier and B. Philippe, A parallel shooting technique for solving dissipative ODEs. Computing51 (1993) 209–236.  Zbl0788.65079
  8. 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).  
  9. Y. D'Angelo and B. Larrouturou, Comparison and analysis of some numerical schemes for stiff complex chemistry problems. RAIRO Modél. Math. Anal. Numér.29 (1995) 259–301.  Zbl0829.76062
  10. M.S. Day and J.B. Bell, Numerical simulation of laminar reacting flows with complex chemistry. Combust. Theory Modelling4 (2000) 535–556.  Zbl0970.76065
  11. S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp.70 (2001) 1481–1501.  Zbl0981.65107
  12. S. Descombes and T. Dumont, Numerical simulation of a stroke: Computational problems and methodology. Prog. Biophys. Mol. Biol.97 (2008) 40–53.  
  13. S. Descombes and M. Massot, Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction. Numer. Math.97 (2004) 667–698.  Zbl1060.65105
  14. S. Descombes and M. Schatzman, Strang's formula for holomorphic semi-groups. J. Math. Pures Appl.81 (2002) 93–114.  Zbl1030.35095
  15. 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.  
  16. S. Descombes, T. Dumont, V. Louvet and M. Massot, On the local and global errors of splitting approximations of reaction-diffusion equations with high spatial gradients. Int. J. Computer Mathematics84 (2007) 749–765.  Zbl1122.65061
  17. 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).  
  18. P. Deuflhard, 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.  Zbl0313.65070
  19. P. Deuflhard, Newton Methods for Nonlinear Problems – Affine invariance and adaptive algorithms. Springer-Verlag (2004).  Zbl1056.65051
  20. M. Dowle, R.M. Mantel and D. Barkley, Fast simulations of waves in three-dimensional excitable media. Int. J. Bif. Chaos7 (1997) 2529–2545.  Zbl0899.92002
  21. 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 ().  Zbl1243.65107URIhttp://hal.archives-ouvertes.fr/hal-00457731
  22. 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 ().  Zbl1322.92019URIhttp://hal.archives-ouvertes.fr/hal-00546223
  23. T. Echekki, Multiscale methods in turbulent combustion: Strategies and computational challenges. Computational Science & Discovery2 (2009) 013001.  
  24. I.R. Epstein and J.A. Pojman, An Introduction to Nonlinear Chemical Dynamics – Oscillations, Waves, Patterns and Chaos. Oxford University Press (1998).  
  25. C. Farhat and M. Chandesris, 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.  Zbl1032.74701
  26. 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.  Zbl1309.76060
  27. 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.  Zbl1140.65336
  28. M. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method. J. Sci. Comput.29 (2007) 556–578.  Zbl1141.65064
  29. 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.  Zbl1228.76107
  30. I. Garrido, B. Lee, G.E. Fladmark and M.S. Espedal, Convergent iterative schemes for time parallelization. Math. Comput.75 (2006) 1403–1428.  Zbl1089.76038
  31. V. Giovangigli, Multicomponent flow modeling. Birkhäuser Boston Inc., Boston, MA (1999).  Zbl0956.76003
  32. S.A. Gokoglu, 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. P. Gray and S.K. Scott, Chemical oscillations and instabilites. Oxford University Press (1994).  
  34. E. Grenier, M.A. Dronne, S. Descombes, H. Gilquin, A. Jaillard, M. Hommel and J.P. Boissel, A numerical study of the blocking of migraine by Rolando sulcus. Prog. Biophys. Mol. Biol.97 (2008) 54–59.  
  35. E. Hairer and G. Wanner, Solving ordinary differential equations II – Stiff and differential-algebraic problems. Second edition, Springer-Verlag, Berlin (1996).  Zbl0859.65067
  36. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration – Structure-Preserving Algorithms for Odinary Differential Equations. Second edition, Springer-Verlag, Berlin (2006).  Zbl1094.65125
  37. W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag, Berlin (2003).  Zbl1030.65100
  38. W. Jahnke, W.E. Skaggs and A.T. Winfree, Chemical vortex dynamics in the Belousov-Zhabotinsky reaction and in the two-variable Oregonator model. J. Phys. Chem.93 (1989) 740–749.  
  39. J. Kim and S.Y. Cho, Computation accuracy and efficiency of the time-splitting method in solving atmosperic transport-chemistry equations. Atmos. Environ.31 (1997) 2215–2224.  
  40. O.M. Knio, H.N. Najm and P.S. Wyckoff, A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation. J. Comput. Phys.154 (1999) 467–482.  Zbl0958.76061
  41. 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 Mécanique1 (1937) 1–25.  Zbl0018.32106
  42. J.L. Lions, Y. Maday and G. Turinici, Résolution d'EDP par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris Sér. I Math.332 (2001) 661–668.  
  43. C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp.77 (2008) 2141–2153.  Zbl1198.65186
  44. Y. Maday and G. Turinici, A parareal in time procedure for the control of partial differential equations. C. R., Math.335 (2002) 387–391.  Zbl1006.65071
  45. 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.  Zbl1067.65102
  46. G.I. Marchuk, Splitting and alternating direction methods, in Handbook of numerical analysis I, North-Holland, Amsterdam (1990) 197–462.  Zbl0875.65049
  47. M. Massot, Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete Contin. Dyn. Syst. Ser. B2 (2002) 433–456.  Zbl1001.80006
  48. G.J. McRae, W.R. Goodin and J.H. Seinfeld, Numerical solution of the atmospheric diffusion equation for chemically reacting flows. J. Comput. Phys.45 (1982) 1–42.  Zbl0502.76098
  49. H.N. Najm and O.M. Knio, Modeling Low Mach number reacting flow with detailed chemistry and transport. J. Sci. Comput.25 (2005) 263–287.  Zbl1203.80025
  50. H.N. Najm, P.S. Wyckoff and O.M. Knio, A semi-implicit numerical scheme for reacting flow. I. Stiff chemistry. J. Comput. Phys.143 (1998) 381–402.  Zbl0936.76064
  51. M. Schatzman, Toward non commutative numerical analysis: High order integration in time. J. Sci. Comput.17 (2002) 107–125.  Zbl0999.65095
  52. L.F. Shampine, B.P. Sommeijer and J.G. Verwer, IRKC: An IMEX solver for stiff diffusion-reaction PDEs. J. Comput. Appl. Math.196 (2006) 485–497.  Zbl1100.65075
  53. M.D. Smooke, 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.  Zbl0487.65045
  54. B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math.88 (1998) 315–326.  Zbl0910.65067
  55. 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).  
  56. B. Sportisse, An analysis of operator splitting techniques in the stiff case. J. Comput. Phys.161 (2000) 140–168.  Zbl0953.65062
  57. B. Sportisse and R. Djouad, Reduction of chemical kinetics in air pollution modeling. J. Comput. Phys.164 (2000) 354–376.  Zbl0961.92038
  58. 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.  Zbl1066.65079
  59. G. Strang, Accurate partial difference methods. I. Linear Cauchy problems. Arch. Ration. Mech. Anal.12 (1963) 392–402.  Zbl0113.32303
  60. G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal.5 (1968) 506–517.  Zbl0184.38503
  61. P. Sun, A pseudo non-time splitting method in air quality modeling. J. Comp. Phys.127 (1996) 152–157.  Zbl0859.65133
  62. R. Témam, 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.  Zbl0195.46001
  63. R. Témam, 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.  Zbl0207.16904
  64. J.G. Verwer and B.P. Sommeijer, An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. SIAM J. Sci. Comput.25 (2004) 1824–1835.  Zbl1061.65090
  65. J.G. Verwer and B. Sportisse, Note on operator splitting in a stiff linear case. Rep. MAS-R9830 (1998).  
  66. J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKC time-stepping for advection-diffusion-reaction problems. J. Comput. Phys.201 (2004) 61–79.  Zbl1059.65085
  67. A.I. Volpert, V.A. Volpert and V.A. Volpert, Traveling wave solutions of parabolic systems. American Mathematical Society, Providence, RI (1994).  Zbl0805.35143
  68. N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971).  Zbl0209.47103

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.