Analysis of the Computational Singular Perturbation Reduction Method for
Chemical Kinetics

Zagaris, Antonios; Kaper, Hans G.; Kaper, Tasso J.

doi:10.1007/s00332-003-0582-9

Cited by 131 publications

(120 citation statements)

References 33 publications

Supporting

Mentioning

104

Contrasting

Unclassified

Order By: Relevance

“…(4.8)-(4.9), CSP performs the updating in two steps; the first step involves the matrices U , the second the matrices L. In [22] and [23], we showed that the two steps play different roles and that, if one is interested solely in approximating the slow manifold, one may as well skip the second step. The slow manifold constructed with the so-called one-step CSP [22], which involves only the matrices U , has the same asymptotic accuracy as the slow manifold constructed with CSP.…”

Section: The Principle Of Cspmentioning

confidence: 98%

“…Many techniques have been proposed for this purpose, especially in the chemical kinetics literature; see the references in [8,22,23]. Generally, they involve the construction of a coordinate system where the slow manifold has a simple functional representation-the ideal system being one where a number of components (the "fast" components) of the state vector are identically zero, as in the Fenichel normal form [7].…”

Section: Introductionmentioning

confidence: 99%

“…It is being used widely, for example, in combustion modeling [6,15,16,17,20,21] and atmospheric science [18]. It has been analyzed mathematically, and its approximation properties have been established rigorously for fast-slow systems [22,23]. In its original formulation, CSP is essentially an iterative technique to generate a coordinate system for the vector field where the system of differential equations reduces to normal form.…”

Section: Introductionmentioning

confidence: 99%

“…Since the emphasis in the present article is on the approximation of the slow manifold, we restrict the discussion to a one-step CSP, where the second step is omitted. This variant of CSP was introduced in [22].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Two perspectives on reduction of ordinary differential equations

Zagaris

Kaper

2005

Mathematische Nachrichten

View full text Add to dashboard Cite

Key words Nonlinear differential equations, kinetic equations, multiple time scales, dimension reduction, slow manifold, normal form, computational singular perturbation, zero derivative principle MSC (2000) 34C20, 34E13, 34E15, 80A30, 80A25, 92C45 Dedicated to the memory of F. V. AtkinsonThis article is concerned with general nonlinear evolution equations x = g(x) in R N involving multiple time scales, where fast dynamics take the orbits close to an invariant low-dimensional manifold and slow dynamics take over as the state approaches the manifold. Reduction techniques offer a systematic way to identify the slow manifold and reduce the original equation to an autonomous equation on the slow manifold. The focus in this article is on two particular reduction techniques, namely, computational singular perturbation (CSP) proposed by Lam and Goussis [Twenty-Second Symposium (International) on Combustion, The University of Washington, Seattle, Washington, August 14-19, 1988 (The Combustion Institute, Pittsburgh, 1988, pp. 931-941] and the zero-derivative principle (ZDP) proposed recently by Gear and Kevrekidis [Constraint-defined manifolds: A legacy-code approach to low-dimensional computation, SIAM J. Sci. Comput., to appear]. It is shown that the tangent bundle to the state space offers a unifying framework for CSP and ZDP. Both techniques generate coordinate systems in the tangent bundle that are natural for the approximation of the slow manifold. Viewed from this more general perspective, both CSP and ZDP generate, at each iteration, approximate normal forms for the system under examination.

show abstract

Section: The Principle Of Cspmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two perspectives on reduction of ordinary differential equations

Zagaris

Kaper

2005

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…The procedure for generating and numerically solving these DAE models is known as computational singular perturbation (CSP). A discussion of the CSP method is provided in [16,26,27]. Whether one is numerically solving the full model or the reduced, slow-timescale DAE model, it is generally assumed that all kinetic parameters are known.…”

Section: Solving the Full Or Reduced Model Numericallymentioning

confidence: 99%

The QSSA in Chemical Kinetics: As Taught and as Practiced

Pantea¹,

Gupta²,

Rawlings³

et al. 2013

Discrete and Topological Models in Molecular Biology

View full text Add to dashboard Cite

Chemical mechanisms for even simple reaction networks involve many highly reactive and short-lived species (intermediates), present in small concentrations, in addition to the main reactants and products, present in larger concentrations. The chemical mechanism also often contains many rate constants whose values are unknown a priori and must be determined from experimental measurements of the large species concentrations. A classic model reduction method known as the quasi-steady-state assumption (QSSA) is often used to eliminate the highly reactive intermediate species and remove the large rate constants that cannot be determined from concentration measurements of the reactants and products. Mathematical analysis based on the QSSA is ubiquitous in modeling enzymatic reactions. In this chapter, we focus attention on the QSSA, how it is "taught" to students of chemistry, biology, and chemical and biological engineering, and how it is "practiced" when researchers confront realistic and complex examples. We describe the main types of difficulties that appear when trying to apply the standard ideas of the QSSA, and propose a new strategy for overcoming them, based on rescaling the reactive intermediate species.First, we prove mathematically that the program taught to beginning students for applying the 100-year-old approach of classic QSSA model reduction cannot be carried out for many of the relevant kinetics problems, and perhaps even most of them. By using Galois theory, we prove that the required algebraic equations cannot be solved for as few as five bimolecular reactions between five species (with three intermediates). We expect that many practitioners have suspected this situation regarding nonsolvability to exist, but we have seen no statement or proof of this fact, especially when the kinetics are restricted to unimolecular and bimolecular reactions. We describe algorithms that can test any mechanism for solvability. We also show that an alternative to solving the QSSA equations, the Horiuti-Temkin theory, also does not work for many examples.Of course, the reduced model (and the full model, for that matter) can be solved numerically, which is the standard approach in practice. The remaining difficulty, however, is how to obtain the values of the large kinetic parameters appearing in the model. These parameters cannot be estimated from measurements of the largeconcentration reactants and products. We show here how the concept of rescaling the reactive intermediate species allows the large kinetic parameters to be removed from the parameter estimation problem. In general, the number of parameters that can be removed from the full model is less than or equal to the number of intermediate species. The outcome is a reduced model with a set of rescaled parameters that is often identifiable from routinely available measurements. New and freely available computational software (parest_dae) for estimating the reduced model's kinetic parameters and confidence intervals is briefly described.

show abstract

The curse of instability

Kuehn

2015

Complexity

View full text Add to dashboard Cite

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

Cited by 131 publications

References 33 publications

Two perspectives on reduction of ordinary differential equations

Two perspectives on reduction of ordinary differential equations

The QSSA in Chemical Kinetics: As Taught and as Practiced

The curse of instability

Contact Info

Product

Resources

About