Constructive methods of invariant manifolds for kinetic problems

By Alexander N Gorban, I. Karlin, A. Zinovyev. 2003

169 citations recorded by the source corpus.

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 di6erential equation for a manifold immersed inthe phase space ( the invariance equation). The equationof motionfor immersed man ifolds is obtained (the 1lm extension of the dynamics). Invariant manifolds are 8xed points for this equation, and slow invariant manifolds are Lyapunov stable 8xed 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 ∼ 1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of exten sionof list of variables) to gainmore accuracy indescriptionof highly n equilibrium =ows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data;

Subfields: Applied Mathematics, Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f6a708ff9236261143ad9728507e2dc24de03260

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. I. Karlin
2. Alexander N Gorban
3. David R. Emerson
4. Xiao-Jun Gu

In Computational Mechanics

1. Alexander N Gorban
2. I. Karlin
3. David R. Emerson
4. Xiao-Jun Gu
5. Dieter Bothe
6. Michel Pierre

In Statistical and Nonlinear Physics

1. I. Karlin
2. Alexander N Gorban
3. David R. Emerson
4. Hod Lipson
5. Xiao-Jun Gu

Citations in the corpus

• Comparison of Invariant Manifolds for Model Reduction in Chemical Kinetics (2007, background): “[3, 4], when one reduces the grid spacing in order to refine the grid, then, once the grid spacing becomes small enough, one can face the problem of the Courant instability [7].”
• Automated refinement and inference of analytical models for metabolic networks (2011, background): “Genetic programming techniques, previously demonstrated for the reverse engineering of chemical reaction networks from observed time series data, can be used to infer both the topology and numerical parameters of metabolic networks [30, 41].”
• Invariant Manifolds for Physical and Chemical Kinetics (2005, methodology): “– Derivation of the post–Navier–Stokes hydrodynamics from the Boltzmann equation [11,12,14,17].”
• A high-order moment approach for capturing non-equilibrium phenomena in the transition regime (2009, background): “In this sense, the moments outside the GMM are referred to as ‘fast’ moments, and those in the GMM are ‘slow’ moments (Gorban et al. 2004).”