The extension of the ILDM concept to reaction–diffusion manifolds

REFERENCES

• Pope , S. B. 1990 . Computations of turbulent combustion: Progress and challenges . Proc. Comb. Inst. , 23 : 591 – 612 .
   Google Scholar
• Mitropolskiy , Yu. A. and Lykova , O. B. 1968 . Lectures on the Methods of Integral Manifolds , Kiev : Institute of Mathematics, Ukrainian Academy of Science . in Russian
   Google Scholar
• Fenichel , N. 1979 . Geometric singular perturbation theory for ordinary differential equations . J. Differential Equations , 31 : 53 – 98 .
   Web of Science ®Google Scholar
• Gol'dshtein , V. and Sobolev , V. 1988 . Qualitative Analysis of Singularly Perturbed Systems , Novosibirsk : Institute of Mathematics, Siberian Branch of USSR Academy of Science .
   Google Scholar
• Gol'dshtein , V. and Sobolev , V. 1992 . “ Integral manifolds in chemical kinetics and combustion ” . In Singularity Theory and Some Problems of Functional Analysis , Edited by: Gindikin , S. G. 73 – 92 . Providence, RI : American Mathematical Society .
   Google Scholar
• Bykov , V. , Goldfarb , I. and Gol'dshtein , V. 2004 . On the integral manifold approach in flame propagation problem: pressure driven flame in porous media . IMA J. of Applied Mathematics , 69 : 335 – 352 .
   Google Scholar
• Bykov , V. , Goldfarb , I. and Gol'dshtein , V. 2005 . Novel numerical decomposition approaches for multiscale combustion and kinetic models . Journal of Physics: Conference Series , 22 : 1 – 29 .
   Google Scholar
• Roussel , M. R. and Fraser , S. J. 1993 . Global analysis of enzyme inhibition kinetics . J. Phys. Chem. , 97 : 8316 – 8327 . errata, ibid. 98, 5174
   Web of Science ®Google Scholar
• Roussel , M. R. and Fraser , S. J. 2001 . Invariant manifold methods for metabolic model reduction . Chaos , 11 : 196 – 206 .
   Google Scholar
• Singh , S. , Rastigejev , Y. , Paolucci , S. and Powers , J. M. 2001 . Viscous detonation in H2/O2/Ar using intrinsic low dimensional manifolds and wavelet adaptive multilevel representation . Combustion Theory and Modelling , 5 : 163 – 184 .
   Web of Science ®Google Scholar
• Singh , S. , Powers , J. M. and Paolucci , S. 2002 . On slow manifolds of chemical reaction flows . J. Phys. Chem. , 117 : 1482 – 1496 .
   Google Scholar
• Lam , S. H. and Goussis , D. M. 1994 . The GSP method for simplifying kinetics . International Journal of Chemical Kinetics , 26 : 461 – 486 .
   Web of Science ®Google Scholar
• Hadjinicolaou , M. and Goussis , D. A. 1999 . Asymptotic solutions of stiff PDEs with the CSP method: The reaction diffusion equation . SIAM journal of Scientific Computing , 20 : 781 – 910 .
   Web of Science ®Google Scholar
• Valorani , M. and Goussis , D. A. 2001 . Explicit time-scale splitting algorithm for stiff problems: Auto-ignition of gaseous mixtures behind a steady shock . Journal of Computational Physics , 169 : 44 – 79 .
   Web of Science ®Google Scholar
• Peters , N. 2000 . Turbulent Combustion , Cambridge : Cambridge University Press .
   Google Scholar
• Gicquel , O. , Darabiha , N. and Thevenin , D. 2000 . Laminar pre- mixed hydrogen/air counterflow flame simulations using flame prolongation of ILDM with differential diffusion . Proc. Combust. Inst. , 28 : 1901 – 1908 .
   Web of Science ®Google Scholar
• Maas , U. and Pope , S. B. 1992 . Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space . Combustion and Flame , 88 : 239 – 264 .
   Web of Science ®Google Scholar
• Maas , U. and Pope , S. B. 1992 . Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds . Proc. Comb. Inst. , 88 : 103 – 112 .
   Google Scholar
• Ren , Z. , Pope , S. B. , Vladimirsky , A. and Guckenheimer , J. M. 2006 . The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics . Journal of Chemical Physics , 124 : 114111
   PubMed Web of Science ®Google Scholar
• Ren , Z. and Pope , S. B. 2006 . The geometry of reaction trajectories and attracting manifolds in composition space . Combustion Theory and Modelling , 10 : 361 – 388 .
   Web of Science ®Google Scholar
• Davis , M. J. 2006 . Low-dimensional manifolds in reaction–diffusion equations – 1. Fundamental aspects . Journal of Physical Chemistry A , 110 : 5235 – 5256 .
   PubMed Web of Science ®Google Scholar
• Davis , M. J. 2006 . Low-dimensional manifolds in reaction–diffusion equations – 2. Numerical analysis and method development . Journal of Physical Chemistry A , 110 : 5257 – 5272 .
   Google Scholar
• Zagaris , A. , Kaper , H. G. and Kaper , T. J. 2004 . Fast and slow dynamics for the computational singular perturbation method . Multiscale Modelling and Simulation , 2 : 613 – 638 .
   Web of Science ®Google Scholar
• Valorani , M. , Goussis , D. A. , Creta , F. and Najm , H. N. 2005 . Higher order corrections in the approximation of lowdimensional manifolds and the construction of simplified problems with the CSP method . Journal of Computational Physics , 209 : 754 – 786 .
   Web of Science ®Google Scholar
• Goussis , D. A. , Valorani , M. , Creta , F. and Najm , H. N. 2005 . Reactive and reactive–diffusive time scales in stiff reaction–diffusion systems . Progress in Computational Fluid Dynamics , 5 : 316 – 326 .
   Web of Science ®Google Scholar
• Valorani , M. , Creta , F. , Goussis , D. A. , Lee , J. C. and Najm , H. N. 2006 . An automatic procedure for the simplification of chemical kinetic mechanisms based on CSP . Combustion and Flame , 146 : 29 – 51 .
   Web of Science ®Google Scholar
• Giovangigli , V. 1991 . Convergent iterative methods for multikomponent diffusion . Impact Comput. Sci. Eng. , 3 : 244 – 276 .
   Google Scholar
• Warnatz , J. , Maas , U. and Dibble , R. W. 2000 . Combustion , Berlin : Springer-Verlag .
   Google Scholar
• Golub , G. H. and van Loan , C. F. 1989 . Matrix Computation , Baltimore : The Hopkins University Press .
   Google Scholar
• Maas , U. 1998 . Efficient calculation of intrinsic low-dimensional manifolds for the simplification of chemical kinetics . Computing and Visualization in Science , 1 : 69 – 82 .
   Google Scholar
• Maas , U. and Pope , S. B. 1994 . Laminar flame calculations using simplified chemical kinetics based on intrinsic low-dimensional manifolds . Proc. Comb. Inst. , 25 : 1349 – 1356 .
   Google Scholar
• Blasenbrey , T. , Schmidt , D. and Maas , U. 1998 . Automatically simplified chemical kinetics and molecular transport and its application in premixed and non-premixed laminar flame calculations . Proc. Comb. Inst. , 27 : 505 – 511 .
   Google Scholar
• Blasenbrey , T. and Maas , U. 2000 . ILDMs of higher hydrocarbons and the hierarchy of chemical kinetics . Proc. Comb. Inst. , 28 : 1623 – 1630 .
   Google Scholar
• Nafe , J. and Maas , U. 2002 . Modeling of NO-formation based on ILDM reduced chemistry . Proc. Comb. Inst. , 29 : 1379 – 1385 .
   Google Scholar
• Maas , U. 1993 . Automatische Reduktion von Reaktionsmechanismen zur Simulation reaktiver Strömungen , Habilitation thesis Institut für Technische Verbrennung, Universität Stuttgart .
   Google Scholar
• Brakke , K. 1978 . The Motion of a Surface by its Mean Curvature, Mathematical Notes, vol. 20 , Princeton, NJ : Princeton University Press .
   Google Scholar
• Rhodes , C. , Morari , M. and Wiggins , S. 1999 . Identification of the low order manifolds: Validating the algorithm of Maas and Pope . Chaos , 9 : 108 – 123 .
   PubMed Web of Science ®Google Scholar
• Kaper , H. G. and Kaper , T. J. 2001 . Asymptotic analysis of two reduction methods for systems of chemical reactions , Argonne National Laboratory . ANL/MCS-P912-1001
   Google Scholar
• van Oijen , J. A. and de Goey , L. P.H. 2002 . Modelling of premixed counterflow flames using the flamelet generated manifold method . Combustion Theory and Modelling , 6 : 463 – 478 .
   Web of Science ®Google Scholar
• Bykov , V. and Maas , U. . Generation of reduced models by decoupling of chemical kinetics and convection/diffusion processes . Proceedings of the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems . pp. 244 Montreal, , Canada : McGill University .
   Google Scholar
• Bykov , V. and Maas , U. 2007 . Extension of the ILDM method to the domain of slow chemistry . Proc. Comb. Inst. , 31 : 465 – 472 .
   Web of Science ®Google Scholar
• Gorban , A. and Karlin , I. 2002 . Methods of invarinat manifolds for kinetic problems . Journal of Chemical kinetics , 396 : 197 – 403 .
   Google Scholar
• Gorban , A. , Karlin , I. and Zinoviev , A. 2004 . Invariant Manifolds for Physical and Chemical Kinetics, Lecture Notes in Physics, vol. 660 , Berlin : Springer-Verlag .
   Google Scholar
• Ren , Z. , Pope , S. B. , Vladimirsky , A. and Guckenheimer , J. M. 2007 . Application of the ICE-PIC method for the dimension reduction of chemical kinetics coupled with transport . Proc. Comb. Inst. , 31 : 473 – 481 .
   Web of Science ®Google Scholar
• Nafe , J. and Maas , U. 2002 . A general algorithm for improving ILDMs . Combustion Theory and Modelling , 6 : 697 – 709 .
   Web of Science ®Google Scholar
• Goldfarb , I. , Gol'dshtein , V. and Maas , U. 2004 . Comparative analysis of two assymptotic approaches based on integral manifolds . IMA J. of Applied Mathematics , 69 : 353 – 374 .
   Web of Science ®Google Scholar
• Wolfram , S. 2003 . The Mathematica Book, , 5th edn , Wolfram Media .
   Google Scholar
• John , F. 1991 . Partial Differential Equations, , 4th edn , New York : Springer-Verlag .
   Google Scholar
• Deuflhardt , P. , Hairer , E. and Zugk , J. 1987 . One-step and extrapolation methods for differential-algebraic systems . J. Num. Math. , 51 : 501 – 506 .
   Google Scholar
• Maas , U. 1995 . Coupling of chemical reaction with flow and molecular transport . Applications of Mathematics , 3 : 249 – 266 .
   Google Scholar
• Bauer , J. , Bykov , V. and Maas , U. . Implementation of ILDMs based on a representation in generalized coordinates . Paper presented at the European Conference on Computational Fluid Dynamics . ECCOMAS CFD
   Google Scholar
• Davis , M. J. and Skodje , R. T. 1999 . Geometric investigation of low-dimensional manifolds in systems approaching equilibrium . J. Chem. Phys. , 111 : 859 – 874 .
   Web of Science ®Google Scholar
• Stahl , G. and Warnatz , J. 1991 . Numerical investigation of time-dependent properties and extinction of strained methane- and propane-air flamelets . Combustion and Flame , 85 : 285 – 299 .
   Web of Science ®Google Scholar
• Blasenbrey , T. , Schmidt , D. and Maas , U. . Intrinsic low-dimensional manifolds of strained und unstrained flames . Proceedings of the 16th International Colloquium on the Dynamics of Explosions and Reactive Systems . Vol. 16 , pp. 329 – 332 . Cracow, , Poland : University of Mining and Metallurgy, AGH .
   Google Scholar
• Maas , U. and Warnatz , J. 1988 . Ignition processes in hydrogen–oxygen mixtures . Combustion and Flame , 74 : 53 – 69 .
   Web of Science ®Google Scholar