Collocation Methods and Inertial Manifold Theory for the Modal Reduction of Dissipative Distributed Systems

  • A. Adrover
  • G. Continillo
  • S. Crescitelli
  • M. Giona
  • L. Russo
Conference paper


A collocation method is proposed to approach the reduction of dissipative distributed systems through application of the methods of Inertial Manifold theory. The collocation method proposed provides a numerical framework to develop approximate inertial manifolds (AIMs) in the case of partial differential problems (e.g. reaction/diffusion models) containing non-polynomial nonlinearities. The collocation method is also the starting point for the alternative construction of AIMs by means of a renormalization approach naturally derived from the incremental unknown method developed by Temam in a finite difference framework.


Hopf Bifurcation Collocation Method Trial Function Collocation Point Modal Reduction 
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  1. 1.
    Hyman J.M., Nicolaenko B. (1986), The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems, Physica D 18, 113–126CrossRefGoogle Scholar
  2. 2.
    Sirovich L. (Ed.), (1994), Trends and Perspectives in Applied Mathematics, Springer Verlag, Berlin Heidelberg New YorkGoogle Scholar
  3. 3.
    Robinson J.C., (1995b), Finite-dimensional behavior in dissipati ve partial differential equations, Chaos 5, 330–345CrossRefGoogle Scholar
  4. 4.
    Temam R., (1997), Infinite-Dimensional Dynamical Systems in Mechanics and Physics 2nd ed., Springer Verlag, Berlin Heidelberg New YorkGoogle Scholar
  5. 5.
    Temam R. (1990), Inertial manifolds and multigrid methods, SIAM J. Math. Anal. 21, 154–178Google Scholar
  6. 6.
    Temam R. (1994), Applications of inertial manifolds to scientific computing: a new insight in multilevel methods, in Trends and Perspectives in Applied Mathematics, L. Sirovich (Ed.), Springer Verlag, Berlin Heidelberg New York, pp. 293–336CrossRefGoogle Scholar
  7. 7.
    Villadsen J., Michelsen M.L., (1978), Solution of Differential Equation Models by Polynomial Approximation, Prentice-Hall, Englewood Cliffs (NJ )Google Scholar
  8. 8.
    Continillo G., Maffettone P.L., Crescitelli, S., (1995), On the numerical simulation of chaotic behavior of some distributed-parameter systems, in Chaos and Fractals in Chemical Engineering, G. Biardi et al. (eds.), World Scientific, SingaporeGoogle Scholar
  9. 9.
    Continillo G., Galiero G., Maffettone P.L., Crescitelli S. (1996), Characterization of strange attractors in the autoignition of coal stockpiles, in Chaos and Fractals in Chemical Engineering, M. Giona and G. Biardi (eds.), World Scientific, SingaporeGoogle Scholar
  10. 10.
    Continillo G., Faraoni V., Maffettone P.L. Crescitelli S., (2000), Non-linear dynamics of a self-ignited reaction-diffusion system, Chem. Engng. Sci. 55, 303–309Google Scholar
  11. 11.
    Adrover A., Continillo G., Crescitelli S., Giona M., Russo L., (2000), Waveletlike collocation method for finite-dimensional reduction of distributed systems, Comp, h Chem. Eng., in pressGoogle Scholar
  12. 12.
    Elezgaray J. and Arneodo A. (1992), Crisis-induced intermittent bursting in reaction-diffusion chemical systems, Phys. Rev. Lett. 68, 714–717CrossRefGoogle Scholar
  13. 13.
    Elezgaray J., (1989), Structures spatio-temporelles dans les systèmes chimiques hors d’équilibré: fronts de reaction-diffusion et croissances fractales, Ph. Thesis in Physics, Universite Paris V IGoogle Scholar
  14. 14.
    Graham M.D., Kevrekidis I.G., (1996), Alternative approaches to the Karhunen-Loeve decomposition for modal reduction and data analysis, Comp. Chem. Engn. 20, 495–506CrossRefGoogle Scholar
  15. 15.
    Dödel E. J., Kernevez J.P., (1986), AUTO: A Software for Continuation and Bifurcation problems in Ordinary Differential Equations, California Institute of TechnologyGoogle Scholar

Copyright information

© Springer-Verlag Italia, Milan 2002

Authors and Affiliations

  • A. Adrover
    • 1
  • G. Continillo
    • 2
  • S. Crescitelli
    • 3
  • M. Giona
    • 1
  • L. Russo
    • 3
  1. 1.Dipartimento di Ingegneria ChimicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Facoltà di IngegneriaUniversità del SannioBeneventoItaly
  3. 3.Dipartimento di Ingegneria ChimicaUniversità Federico IINapoliItaly

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