Method of invariant manifold for chemical kinetics

Abstract

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.Comment: 41 pages, 1 figur

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

“…This approach joins ideas of Ehrenfests on coarse-graining [35], methods of projection operators, and methods of invariant manifold [7,8,15]. The essence of this approach can be formulated very simply: we turn from infinitesimal projection of vector fields to "natural projection" of segments of trajectories.…”

“…It was invented first for Lattice-Boltzmann kinetics [42]. In many cases it is more simple than the BGK model, because the gradient model is local in the sense that it uses only the entropy function and its derivatives at a current state, and it is not necessary to compute the equilibrium (or quasiequilibrium for quasiequilibrium models [15,40]. The entropic gradient model has an one-point relaxation spectrum, because the gradient vector field (19) has near an equilibrium Ψ eq an extremely simple linear approximation:…”

“…The thermodynamic projector developed in this paper and a series of previous works [7,8,15,43] can be applied to any system with entropy, i.e. to a system with a specified function (functional) whose time derivative should be preserved in model reduction.…”

“…There are different physically motivated ways to select the scalar product and create the symmetrization [14,15,16]. But symmetrization does not provide termodinamicity and the entropy for the projected equations can decrease.…”

Method of invariant grid for model reduction of hydrogen combustion

“…In the sequel, the Method of Invariant Grid (MIG) introduced in [3] as a computational realization of the Method of Invariant Manifold (MIM) [4] is used for the first time to study a combustion problem. The Quasi-Equilibrium Grid algorithm introduced in [5] is adopted to obtain a first approximation of the SIM, which is afterwards refined via MIG iterations.…”