Constructive Methods of Invariant Manifolds for Kinetic

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 con-structing the slow invariant manifold. The invariance conditions are formulated as the dierential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the lm ex-tension 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 ther-modynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions. We systemati-cally consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. Dynamic and static postprocessing proce-dures give us the opportunity to estimate the accuracy of obtained approximations, and to improve this accuracy signicantly. The following examples of applications are presented: Nonperturbative devia-tion of physically consistent hydrodynamics from the Boltzmann equation and fro