The steady state and equilibrium approximations: A geometrical picture

Abstract

We present a geometrical picture of the steady state (SSA) and equilibrium (EA) approximations for simple chemical reactions. Geometrically, SSA and EA correspond to surfaces in the phase space of species concentrations. In this space, chemical reaction is represented by trajectory motion governed by first order differential equations. Initially, during transient decay, trajectories move quickly into the (narrow) region ℛ between the equilibrium (ℰ) and steady state (𝒮) surfaces, puncturing ℰ or 𝒮 orthogonal to a variable axis. This is rigorously what the differential equations say. Once in ℛ, trajectories approach another surface, the slow manifold ℳ, sandwiched between ℰ and 𝒮. SSA or EA is the restriction of the system motion to 𝒮 or ℰ as a ‘‘shadow’’ of its true, nearby motion along ℳ. We illustrate these ideas in relation to the Lindemann mechanism, showing how ℳ is related to 𝒮 and ℰ, and how it may be obtained by iteration, yielding higher order SSA and EA formulas. This procedure is asymptotic with an interesting error analysis. We briefly discuss the generalization of these ideas.

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

“…The focus of [12] was on the Intrinsic Low-Dimensional Manifold (ILDM) method due to Maas and Pope [19] and an iterative method proposed by Fraser [6] and further developed by Roussel and Fraser [31]. In this article, the focus is on the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [8], [9], [13], [15], [16], [17], [18], [20], [21], [32].…”

Entropy-Related Extremum Principles for Model Reduction of Dissipative Dynamical Systems

“…Prominent numerical techniques are the ILDM-method [4,5], computational singular perturbation (CSP) [6,7], Fraser's algorithm [8][9][10], the method of invariant grids [11,35], the zero derivative principle (ZDP) [13,14], the rate-controlled constrained equilibrium (RCCE) method [15], the invariant constrained equilibrium edge pre-image curve (ICE-PIC) method [16,17], flamelet-generated manifolds [18,19] and finite time Lyapunov exponents [20].…”

Extending the zero-derivative principle for slow–fast dynamical systems

“…Systems without an explicit timescale splitting are much more delicate to deal with at a general level; in a number of cases, computational approaches have been proposed in order to approximate slow manifolds, amongst which are the Intrinsic Low-Dimensional Manifold (ILDM) method [15], the Computational Singular Perturbation (CSP) technique [14], the iterative method of Fraser and Roussel [9,19] and the zero-derivative principle [23]. These approaches supply algorithms which are intended to approximate slow manifolds for systems in the form of (1).…”