Constructive Methods of Invariant Manifolds for Kinetic Problems

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 a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the di erential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the lm extension of the dynamics). Invariant manifolds are xed points for this equation, and slow invariant manifolds are Lyapunov stable xed points, thus slowness is presented as stability. 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 signi cantly. The following examples of applications are presented: Nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from [email protected], [email protected], [email protected]