An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems

Journal of Computational Physics

Volumn 214, Issue 1, 2006, Pages 316-346

  Goussis, Dimitris A ^a   Valorani, Mauro ^b  

^a Dipartimento di Meccanica e Aeronautica ^*  (Greece)

^b Dipartimento di Meccanica e Aeronautica   (Italy)

Author keywords

Asymptotic analysis; Invariant manifolds; Model reduction; Multiple time scales; Singular perturbation analysis

Indexed keywords

APPROXIMATION ALGORITHMS; COMPUTATIONAL EFFICIENCY; DYNAMICS; ITERATIVE METHODS; PHASE SPACE METHODS; VECTOR SPACES;

COMPUTATIONAL SINGULAR PERTURBATION; INVARIANCE EQUATIONS; INVARIANT MANIFOLDS; ITERATIVE ALGORITHM; MODEL REDUCTION; MULTIPLE TIME SCALE; SINGULAR PERTURBATION ANALYSIS; SLOW INVARIANT MANIFOLDS; STATE VECTOR; STIFF SYSTEMS;

ASYMPTOTIC ANALYSIS;

EID: 33644500803     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2005.09.019     Document Type: Article

References (62)

1. N. Fenichel Geometric singular perturbation theory for ordinary differential equations J. Diff. Eqs. 31 1979 53-98
2. P. Constantin C. Foias B. Nicolaenko R. Temam Integral manifolds and inertial manifolds for dissipative partial differential equations Appl. Math. Sci. 70 1989
3. S.J. Fraser The steady state and equilibrium approximations: A geometrical picture J. Chem. Phys. 88 1988 4732-4738
4. A.H. Nguyen S.J. Fraser Geometrical picture of reaction in enzyme kinetics J. Chem. Phys. 91 1989 186-193
5. M.R. Roussel S.J. Fraser Geometry of the steady-state approximation: perturbation and accelerated convergence methods J. Chem. Phys. 93 1990 1072-1081
6. M.R. Roussel S.J. Fraser On the geometry of transient relaxation J. Chem. Phys. 94 1991 7106-7113
7. M.R. Roussel S.J. Fraser Accurate steady-state approximations-implications for kinetics experiments and mechanism J. Phys. Chem. 95 1991 8762-8770
8. M.R. Roussel S.J. Fraser Global analysis of enzyme-inhibition kinetics J. Phys. Chem. 97 1993 8316-8327
9. M.R. Roussel Forced convergence iterative schemes for the approximation of invariant manifolds J. Math. Chem. 21 1997 385-393
10. M.R. Roussel S.J. Fraser Invariant manifold for metabolic reduction Chaos 11 2001 196-206
11. S.J. Fraser Slow manifold for a bimolecular association mechanism J. Chem. Phys. 120 2004 3075-3085
12. M.J. Davis R.T. Skodje Geometric investigation of low-dimensional manifolds in systems approaching equilibrium J. Chem. Phys. 111 1999 859-874
13. M.J. Davis R.T. Skodje Geometric approach to multiple-time-scale kinetics: A nonlinear master equation describing vibration-to-vibration relaxation Z. Phys. Chem. 215 2001 233-252
14. R.T. Skodje M.J. Davis Geometrical simplification of complex kinetic systems J. Phys. Chem. A 105 2001 10356-10365
15. J. Nafe U. Maas A general algorithm for improving ILDMs Combust. Theory Model 6 2002 697-709
16. S.H. Lam D.A. Goussis Understanding complex chemical kinetics with computational singular perturbation Proc. Combust. Inst. 22 1988 931-941
17. S.H. Lam D.A. Goussis Conventional asymptotic and computational singular perturbation for simplified kinetics modelling M.O. Smooke Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames Springer Lecture Notes 1991 Springer Berlin 227-242
18. D.A. Goussis S.H. Lam A study of homogeneous methanol oxidation kinetics using CSP Proc. Combust. Inst. 24 1992 113-120
19. S.H. Lam Using CSP to understand complex chemical kinetics Combust. Sci. Technol. 1993 375-404
20. S.H. Lam D.A. Goussis The CSP method for simplifying kinetics Int. J. Chem. Kinet. 26 1994 461-486
21. D.A. Goussis On the construction and use of reduced chemical kinetics mechanisms produced on the basis of given algebraic relations J. Comput. Phys. 128 1996 261-273
22. M. Hadjinicolaou D.A. Goussis Asymptotic solution of Stiff PDE's - the reaction diffusion equation SIAM J. Sci. Comput. 20 1999 781-810
23. A. Massias D. Diamantis E. Mastorakos D.A. Goussis An algorithm for the construction of global reduced mechanisms with CSP data Combust. Flame 117 1999 685-708
24. 4 -air mechanism at lean conditions Combust. Sci. Tech. 159 2000 281-303
25. T.F. Lu Y.G. Ju C.K. Law Complex CSP for chemistry reduction and analysis Combust. Flame 126 2001 1445-1455
26. M. Valorani D.A. Goussis Explicit time-scale splitting algorithms for Stiff problems: Auto-ignition of gaseous mixtures behind a steady shock J. Comput. Phys. 169 2001 44-79
27. M. Valorani, F. Creta, D.A. Goussis, H.N. Najm, Local and global manifolds in stiff reaction-diffusion systems, in: 2nd MIT Conf. on Comput. Fluid and Solid Mechanics, Boston, June, 2003, pp. 1548-1551.
28. M. Valorani H.N. Najm D.A. Goussis CSP analysis of a transient flame vortex interaction: Time scales and manifolds Combust. Flame 134 2003 35-53
29. U. Maas S.B. Pope Simplifying chemical kinetics - intrinsic low dimensional manifolds in composition space Combust. Flame 88 1992 239-264
30. U. Maas S.B. Pope Implementation of simplified chemical kinetics based on intrinsic low dimensional manifolds Proc. Combust. Inst. 24 1992 103-112
31. D. Schmidt U. Maas Warnatz Simulation of nonequilibrium hypersonic flows Comput. Fluids 22 1993 285-294
32. U. Maas S.B. Pope Laminar flame calculations using simplified chemical kinetics based on intrinsic low dimensional manifolds Proc. Combust. Inst. 25 1994 1349-1356
33. R.L.G.M. Eggels L.P.H. DeGoey Mathematically reduced reaction mechanisms applied to adiabatic flat hydrogen-air flames Combust. Flame 100 1995 559-570
34. R.L.G.M. Eggels L.P.H. DeGoey Modeling of burner-stabilized hydrogen/air flames using mathematically reduced reaction schemes Combust. Sci. Tech. 107 1995 165-180
35. D. Schmidt T. Blasenderey U. Maas Intristic low-dimensional manifold for strained and unstrained flames Combust. Theory Model 2 1998 135-152
36. 2-air flames using automatically simplified kinetics Proc. Combust. Inst. 27 1998 1073-1080
37. T. Blasenderey U. Maas ILDMs of higher hydrocarbons and the hierarchy of chemical kinetics Proc. Combust. Inst. 28 2000 1623-1630
38. 2-Ar using intristic low-dimensionals and wavelet adaptive multilevel representation Combust. Theory Model. 5 2001 163-184
39. S. Singh J.M. Powers S. Paolucci On slow manifolds of chemically reactive systems J. Chem. Phys. 117 2002 1482-1496
40. I. Goldfarb V. Goldshtein U. Maas Comparative analysis of two asymptotic approaches based on integral manifolds IMA J. Appl. Math. 69 2004 353-374
41. H.G. Kaper T.J. Kaper Asymptotic analysis of two reduction methods for systems of chemical reactions Physica D 165 2002 66-93
42. A. Zagaris H.G. Kaper T.J. Kaper Analysis of the computational singular perturbation reduction method for chemical kinetics J. Nonlinear Sci. 14 2004 59-91
43. A. Zagaris H.G. Kaper T.J. Kaper Fast and slow dynamics for the CSP method SIAM J. Multiscale Model. 2 2004 613-638
44. D.A. Goussis, M. Valorani, F. Creta, H.N. Najm, Inertial manifolds with CSP, in: 2nd MIT Conference on Computational Fluid and Solid Mechanics, Boston, June, 2003, pp. 1951-1954.
45. M. Valorani D.A. Goussis F. Creta H.N. Najm Higher order corrections in the approximation of inertial manifolds and the construction of simplified problems with the CSP method J. Comput. Phys. 209 2005 754-786
46. A.N. Gorban I.V. Karlin Method of invariant manifold for chemical kinetics Chem. Eng. Sci. 58 2003 4751-4768
47. A.N. Gorban I.V. Karlin Invariant manifolds for physical and chemical kinetics Lectutre Notes in Physics vol. 660 2004 Springer Berlin
48. O. Axelsson Iterative Solution Methods 1994 Cambridge University Press England
49. G. Strang Linear Algebra and its Applications third ed. 1988 Saunders Philadelphia, PA
50. A. Jennings J.J. McKeown Matrix Computations 1992 Wiley England
51. W. Richardson J. Volk K.H. Lau S.H. Lin H. Eyring Application of the singular perturbation method to reaction kinetics Proc. Natl. Acad. Sci. USA 70 1973 1588-1592
52. K.J. Laidler Chemical Kinetics 1950 McGraw-Hill New York pp. 76-85
53. S.M.T. De La Selva E. Pina Some mathematical properties of the Lindemann mechanism Rev. Mex. Fis. 42 1996 431-448
54. 6 J. Phys. Chem. A 106 2002 5860-5879
55. R.Y. Qian A new formulation of the Lindemann mechanism of unimolecular reactions Chin. J. Chem. 21 2003 1562-1564
56. K.R.A. Kumar A.C. McIntosh J. Brindley et al. Effect of two-step chemistry on the critical extinction-pressure drop for pre-mixed flames Combust. Flame 134 2003 157-167
57. K. Saito Effect of different choices of the Boltzmannized flux operator on rate constants in the Lindemann mechanism Chem. Phys. Lett. 383 2004 282-287
58. A.N. Gorban I.V. Karlin A.Y. Zinovyev Constructive methods of invariant manifolds for kinetic problems Phys. Rep. 396 2004 197-403
59. D. Givon R. Kupferman A. Stuart Extracting macroscopic dynamics: Model problems and algorithms Nonlinearity 17 2004 55-127
60. C.W. Gear I.G. Kevrekidis Telescopic projective methods for parabolic differential equations J. Comput. Phys. 187 1993 95-109
61. E. Pesheck N. Boivin P. Christophe Nonlinear modal analysis of structural systems using multi mode invariant manifolds Nonlinear Dyn. 25 2001 183-205
62. C.W. Gear T.J. Kaper I.G. Kevrekidis A. Zagaris Projecting to a slow manifold: Singularly perturbed systems and legacy codes SIAM J. Appl. Dyn. Syst. 4 2005 711-732