Constructive Methods of Invariant Manifolds for Kinetic Problems

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.

Significantly extended and corrected review, 245 pages, 26 GIF figures, 279 references, accepted to Physics Reports

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Cited by in corpus (30)

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• Field Theory of Reaction-Diffusion: Mass Action with an Energetic Variational Approach
• Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift
• Essential equivalence of the GENERIC and Steepest Entropy Ascent models of dissipation for non-equilibrium thermodynamics
• Uniqueness of thermodynamic projector and kinetic basis of molecular individualism
• Order--disorder separation: Geometric revision
• Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation
• Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics
• On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: toward the derivation of constitutive laws of the rate type
• Slow manifold reduction for plasma science
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• Normal stability of slow manifolds in nearly-periodic Hamiltonian systems
• A continuation method for the efficient solution of parametric optimization problems in kinetic model reduction
• Coarse Variables of Autonomous ODE Systems and Their Evolution
• A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs
• Collisional Gyrokinetics Teases the Existence of Metriplectic Reduction
• Reduction of chemical systems by delayed quasi-steady state assumptions
• Methods of nonlinear kinetics
• Efficient Spectral Methods for Quasi-Equilibrium Closure Approximations of Symmetric Problems on Unit Circle and Sphere
• Steepest-Entropy-Ascent and Maximal-Entropy-Production Dynamical Models of Irreversible Relaxation to Stable Equilibrium from Any Non-Equilibrium State. Unified Treatment for Six Non-Equilibrium Frameworks
• Dealing with complexity of biological systems: from data to models
• On the quasi-steady-state approximation in an open Michaelis--Menten reaction mechanism
• Exploring Hydrodynamic Closures for the Lid-driven Micro-cavity
• Invariance correction to Grad's equations: Where to go beyond approximations?
• Combined micro--macro integration scheme from an invariance principle: application to ferrofluid dynamics
• Attractiveness of Invariant Manifolds