American Institute of Mathematical Sciences

Advanced
August  2002, 2(3): 433-456. doi: 10.3934/dcdsb.2002.2.433

Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure

Marc Massot 1,

1. 

Laboratoire de Mathématiques Appliquées de Lyon, Université Claude Bernard, Lyon 1, 69622 Villeurbanne Cedex, France

Received  December 2001 Published  May 2002

We investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions. We assume that their rates of progress are given by the law of mass action and that their equilibrium constants are compatible with thermodynamics; it thus provides an entropic structure [14] [23]. We study a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. The entropic structure brings in a global convex Lyapounov function and the well-posedness of the associated finite dimensional dynamical system. We then assume that a subset of the reactions is constituted of "Fast" reactions. The partial equilibrium constraint is linear in the entropic variable and thus identifies the "Slow" and "Fast" variables uniquely in the concentration space through constant orthogonal projections. It is proved that there exists a convex compact polyhedron invariant by the dynamical system which contains an affine foliation associated with a Tikhonov normal form. The reduction step is then identified using the orthogonal projection onto the partial equilibrium manifold and proved to be compatible with the entropy production. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. Asymptotic expansions are obtained.
Keywords: entropic structure., Singular perturbation, reduced complex chemistry.
Mathematics Subject Classification: Primary: 34E15, 34D15, 34D23, 80A25, 80A3.
Citation: Marc Massot. Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 433-456. doi: 10.3934/dcdsb.2002.2.433
[1]

Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019
[2]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279
[3]

Manfred Deistler. Singular arma systems: A structure theory. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 383-391. doi: 10.3934/naco.2019025
[4]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783
[5]

Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2349-2364. doi: 10.3934/dcdss.2019147
[6]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[7]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks and Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[8]

Chris Guiver. The generalised singular perturbation approximation for bounded real and positive real control systems. Mathematical Control and Related Fields, 2019, 9 (2) : 313-350. doi: 10.3934/mcrf.2019016
[9]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305
[10]

Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175
[11]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241
[12]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks and Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
[13]

Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137
[14]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
[15]

Zheng-Jian Bai, Xiao-Qing Jin, Seak-Weng Vong. On some inverse singular value problems with Toeplitz-related structure. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 187-192. doi: 10.3934/naco.2012.2.187
[16]

Hiroshi Morishita, Eiji Yanagida, Shoji Yotsutani. Structure of positive radial solutions including singular solutions to Matukuma's equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 871-888. doi: 10.3934/cpaa.2005.4.871
[17]

Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics and Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015
[18]

Robert H. Dillon, Jingxuan Zhuo. Using the immersed boundary method to model complex fluids-structure interaction in sperm motility. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 343-355. doi: 10.3934/dcdsb.2011.15.343
[19]

Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313
[20]

Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy