Asymptotic analysis of two reduction methods for systems of chemical reactions

By H. Kaper, T. L. A. N. Laboratory, Boston University. 2001

171 citations recorded by the source corpus.

Abstract This paper concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope [Combust. Flame 88 (1992) 239] and an iterative method due to Fraser [J. Chem. Phys. 88 (1988) 4732] and further developed by Roussel and Fraser [J. Chem. Phys. 93 (1990) 1072]. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The asymptotic expansions of these manifolds (e↓0, where e measures the ratio of the reaction time scales) are compared with the asymptotic expansion of M e , the slow manifold given by geometric singular perturbation theory. It is shown that the expansions of the ILDM and M e agree up to and including terms of O (e) ; the former has an error at O (e 2 ) that is proportional to the local curvature of M 0 . The error vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of M e . Starting from M 0 , the ith application of the algorithm yields the correct expansion coefficient at O (e i ) , while leaving the lower-order coefficients invariant. Thus, after l applications, the expansion is accurate up to and including the terms of O (e l ) . The analytical results are illustrated on a planar system from enzyme kinetics (Michaelis–Menten–Henri) and a model planar system due to Davis and Skodje.

Subfields: Computational Mechanics, Fluid Flow and Transfer Processes, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4f6f193195b0b31df97799d57a44af42e5962d31

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Fluid Flow and Transfer Processes

In Statistical and Nonlinear Physics

Citations in the corpus

Invariant Manifolds for Physical and Chemical Kinetics (2005, methodology): “These methods were thoroughly analyzed in two papers [94, 95].”
The extension of the ILDM concept to reaction–diffusion manifolds (2007, methodology): “This improves the convergence, because problems with existence of the ILDM are overcome due to the linear prolongation, and in the domain of existence of the ILDM the slow system manifold is already approximated with second-order accuracy [37, 38, 46].”
Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: a chemical explosive mode analysis (2010, methodology): “…to treat both linear and nonlinear systems with time scale separation, CSP provides a refinement procedure to decouple fast and slow processes iteratively until an arbitrarily high order of accuracy is achieved (Lam & Goussis 1994; Kaper & Kaper 2002; Zagaris, Kaper & Kaper 2004a, 2004b, 2005).”
Tikhonov's Theorem and Quasi-Steady State (2011, unknown): “For instance, Kaper and Kaper [12] discuss various reduction methods and asymptotic expansions in the light of Fenichel’s invariant manifold theory.”
A general algorithm for improving ILDMs (2002, methodology): “Furthermore, some nice theoretical investigations of the method have been undertaken [17, 18].”
A constructive approach to quasi-steady state reductions (2014, background): “By now it is customary to interpret QSS as a singular perturbation phenomenon ([16,17,21,28,29] and others), and in the present paper we will adhere to this interpretation.”
Constructive methods of invariant manifolds for kinetic problems (2003, methodology): “These methods were thoroughly analyzed in two papers [63, 64].”
Advanced models of fuel droplet heating and evaporation (2006, unknown): “[376–378]) and Computational Singular Perturbation (CSP) (e.”