Constructive Methods of Invariant Manifolds for Kinetic Problems
Alexander N. Gorban1,2,3, Iliya V. Karlin1,2, and Andrei Yu. Zinovyev2,3
1ETH-Zentrum, Institute of Polymers, CH-8092 Zurich, Switzerland;
2Institute of Computational Modeling SB RAS, Akademgorodok, Krasnoyarsk 660036, Russia;
3Institut des Hautes Etudes Scientifiques, F-91440, Bures-sur-Yvette, France
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 as a problem of constructing the slow invariant manifold. 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.
A collection of methods for construction of slow invariant manifolds is presented, in particular, the Newton method subject to incomplete linearization is the analogue of KAM methods for dissipative systems. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions. We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. Dynamic and static postprocessing procedures give us the opportunity to estimate the accuracy of obtained approximations, and to improve this accuracy significantly.
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 and dynamical correction of the moment equations for nonequilibrium media ; universal continuous media description of dilute polymeric solution, invariant grids for a two-dimensional catalytic reaction and a four-dimensional oxidation reaction, etc.