Constructive Methods of Invariant Manifolds for Kinetic Problems

TL;DR

This paper reviews and develops constructive methods for invariant manifolds to achieve model reduction in kinetic problems, focusing on slow invariant manifolds and their applications in physical and chemical kinetics.

Contribution

It introduces new methods for constructing slow invariant manifolds, including analogues of KAM methods for dissipative systems and invariant grids, with diverse applications.

Findings

Nonperturbative deviations of hydrodynamics from Boltzmann equation

Construction of moment equations with dynamical correction

Invariant grids for chemical reactions and polymer solutions

Abstract

We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The problem of reduced description is studied 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 for construction of slow invariant manifolds is presented, in particular the analogue of KAM methods for dissipative systems. We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. 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 ~ 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; invariant grids for chemical reactions; universal continuous media description of dilute polymeric solution, etc.