Constructive methods of invariant manifolds for kinetic problems
Constructive methods of invariant manifolds for kinetic problems
The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.
A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn similar to 1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium, flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, etc.
197-403
Gorban, A.N.
3b4ae629-6486-47c8-9d6c-6d89a9985dd5
Karlin, I.V.
3f0e01a2-c4d9-4210-9ef3-47e6c426cc8a
Zinovyev, A.Y.
ee4e8073-51de-4a57-a22a-8803c041c07c
June 2004
Gorban, A.N.
3b4ae629-6486-47c8-9d6c-6d89a9985dd5
Karlin, I.V.
3f0e01a2-c4d9-4210-9ef3-47e6c426cc8a
Zinovyev, A.Y.
ee4e8073-51de-4a57-a22a-8803c041c07c
Gorban, A.N., Karlin, I.V. and Zinovyev, A.Y.
(2004)
Constructive methods of invariant manifolds for kinetic problems.
Physics Reports, 396 (4-6), .
(doi:10.1016/j.physrep.2004.03.006).
Abstract
The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.
A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn similar to 1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium, flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, etc.
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Published date: June 2004
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Local EPrints ID: 49203
URI: http://eprints.soton.ac.uk/id/eprint/49203
ISSN: 0370-1573
PURE UUID: fc2169ba-e19e-4fe2-bad6-7e0465a96749
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Date deposited: 24 Oct 2007
Last modified: 15 Mar 2024 09:54
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Author:
A.N. Gorban
Author:
I.V. Karlin
Author:
A.Y. Zinovyev
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