Free Access
Issue
ESAIM: M2AN
Volume 45, Number 5, September-October 2011
Page(s) 825 - 852
DOI https://doi.org/10.1051/m2an/2010104
Published online 23 February 2011
  1. A. Abdulle, Fourth order Chebyshev methods with recurrence relation. J. Sci. Comput. 23 (2002) 2041–2054. [Google Scholar]
  2. G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comp. 67 (1998) 457–477. [Google Scholar]
  3. L. Baffico, S. Bernard, Y. Maday, G. Turinici and G. Zérah, Parallel-in-time molecular-dynamics simulations. Phys. Rev. E 66 (2002) 1–4. [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  6. D. Barkley, A model for fast computer simulation of waves in excitable media. Physica D 49 (1991) 61–70. [CrossRef] [Google Scholar]
  7. P. Chartier and B. Philippe, A parallel shooting technique for solving dissipative ODEs. Computing 51 (1993) 209–236. [CrossRef] [MathSciNet] [Google Scholar]
  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). [Google Scholar]
  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. [MathSciNet] [Google Scholar]
  10. M.S. Day and J.B. Bell, Numerical simulation of laminar reacting flows with complex chemistry. Combust. Theory Modelling 4 (2000) 535–556. [CrossRef] [Google Scholar]
  11. S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems. Math. Comp. 70 (2001) 1481–1501. [CrossRef] [MathSciNet] [Google Scholar]
  12. S. Descombes and T. Dumont, Numerical simulation of a stroke: Computational problems and methodology. Prog. Biophys. Mol. Biol. 97 (2008) 40–53. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  14. S. Descombes and M. Schatzman, Strang's formula for holomorphic semi-groups. J. Math. Pures Appl. 81 (2002) 93–114. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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 Mathematics 84 (2007) 749–765. [CrossRef] [Google Scholar]
  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). [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  19. P. Deuflhard, Newton Methods for Nonlinear Problems – Affine invariance and adaptive algorithms. Springer-Verlag (2004). [Google Scholar]
  20. M. Dowle, R.M. Mantel and D. Barkley, Fast simulations of waves in three-dimensional excitable media. Int. J. Bif. Chaos 7 (1997) 2529–2545. [CrossRef] [Google Scholar]
  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 (http://hal.archives-ouvertes.fr/hal-00457731). [Google Scholar]
  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 (http://hal.archives-ouvertes.fr/hal-00546223). [Google Scholar]
  23. T. Echekki, Multiscale methods in turbulent combustion: Strategies and computational challenges. Computational Science & Discovery 2 (2009) 013001. [CrossRef] [Google Scholar]
  24. I.R. Epstein and J.A. Pojman, An Introduction to Nonlinear Chemical Dynamics – Oscillations, Waves, Patterns and Chaos. Oxford University Press (1998). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  28. M. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method. J. Sci. Comput. 29 (2007) 556–578. [Google Scholar]
  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. [Google Scholar]
  30. I. Garrido, B. Lee, G.E. Fladmark and M.S. Espedal, Convergent iterative schemes for time parallelization. Math. Comput. 75 (2006) 1403–1428. [CrossRef] [Google Scholar]
  31. V. Giovangigli, Multicomponent flow modeling. Birkhäuser Boston Inc., Boston, MA (1999). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  33. P. Gray and S.K. Scott, Chemical oscillations and instabilites. Oxford University Press (1994). [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  35. E. Hairer and G. Wanner, Solving ordinary differential equations II – Stiff and differential-algebraic problems. Second edition, Springer-Verlag, Berlin (1996). [Google Scholar]
  36. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration – Structure-Preserving Algorithms for Odinary Differential Equations. Second edition, Springer-Verlag, Berlin (2006). [Google Scholar]
  37. W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag, Berlin (2003). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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 Mécanique 1 (1937) 1–25. [Google Scholar]
  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. [Google Scholar]
  43. C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141–2153. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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. [Google Scholar]
  46. G.I. Marchuk, Splitting and alternating direction methods, in Handbook of numerical analysis I, North-Holland, Amsterdam (1990) 197–462. [Google Scholar]
  47. M. Massot, 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] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  51. M. Schatzman, Toward non commutative numerical analysis: High order integration in time. J. Sci. Comput. 17 (2002) 107–125. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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). [Google Scholar]
  56. B. Sportisse, An analysis of operator splitting techniques in the stiff case. J. Comput. Phys. 161 (2000) 140–168. [CrossRef] [MathSciNet] [Google Scholar]
  57. B. Sportisse and R. Djouad, Reduction of chemical kinetics in air pollution modeling. J. Comput. Phys. 164 (2000) 354–376. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  59. G. Strang, Accurate partial difference methods. I. Linear Cauchy problems. Arch. Ration. Mech. Anal. 12 (1963) 392–402. [Google Scholar]
  60. G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  61. P. Sun, A pseudo non-time splitting method in air quality modeling. J. Comp. Phys. 127 (1996) 152–157. [CrossRef] [Google Scholar]
  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. [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  65. J.G. Verwer and B. Sportisse, Note on operator splitting in a stiff linear case. Rep. MAS-R9830 (1998). [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  67. A.I. Volpert, V.A. Volpert and V.A. Volpert, Traveling wave solutions of parabolic systems. American Mathematical Society, Providence, RI (1994). [Google Scholar]
  68. N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Springer-Verlag, New York (1971). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you