Hans G. Kaper

Asymptotic analysis of two reduction methods for systems of chemical reactions

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.

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

Abstract This article is concerned with the asymptotic accuracy of the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [The CSP method for simplifying kinetics, Int. J. Chem. Kin. 26 (1994) 461–486] to reduce the dimensionality of a system of chemical kinetics equations. The method, which is generally applicable to multiple-time scale problems arising in a broad array of scientific disciplines, exploits the presence of disparate time scales to model the dynamics by an evolution equation on a lower-dimensional slow manifold. In this article it is shown that the successive applications of the CSP algorithm generate, order by order, the asymptotic expansion of a slow manifold. The results are illustrated on the Michaelis–Menten–Henri equations of enzyme kinetics.

On the bounded solutions of a nonlinear convolution equation

This investigation is concerned with the nonlinear convolution equation u(x) − (gou) * k(x) = 0 on the real line IR. The kernel k is nonnegative and integrable on IR , with ∫ IR k(x)dx = 1; the function g is real-valued and continuous on IR, g(0) = 0 , and there exists a p > 0 such that g(x) > x for x ∈ (0,p) and g(p) = p. Sufficient conditions are given for the non-existence of bounded nontrivial solutions. Implications for the solution of the inhomogeneous equation u(x) − (g o u) * k(x) = f(x), x ∈ IR , are discussed. Finally, uniqueness (modulo translation) is shown to hold. The results are applied to a problem of mathematical epidemiology.

Bifurcation of Pulsating and Spinning Reaction Fronts in Condensed Two-Phase Combustion

Abstract We employ a nonlinear stability analysis to describe the bifurcation of pulsating and spinning modes of combustion in condensed media. We adopt the two-phase model of Margolis (1983) in which the modified nondimensional activation energy Δ of the reaction is large, but finite, and in which the limiting component of the mixture melts during the reaction process, as characterized by a nondimensional melting parameter M. We identify several types of non-steady solution branches which bifurcate from the steady palanar solution and show that they are supercritical and stable only for certain realistic ranges of M. For example, the spinning modes, though supercritical and stable for a range of M > 0, are subcritical and unstable for M = 0.

Fast and Slow Dynamics for the Computational Singular Perturbation Method

The computational singular perturbation (CSP) method ofLam and Goussis is an iterative method to reduce the dimensionality of systems of ordinary differential equations with multiple time scales. In [J. Nonlinear Sci., 14 (2004), pp. 59--91], the authors of this paper showed that each iteration of the CSP algorithm improves the approximation of the slow manifold by one order. In this paper, it is shown that the CSP method simultaneously approximates the tangent spaces to the fast fibers along which solutions relax to the slow manifold. Again, each iteration adds one order of accuracy. In some studies, the output of the CSP algorithm is postprocessed by linearly projecting initial data onto the slow manifold along these approximate tangent spaces. These projections, in turn, also become successively more accurate.

Data sonification and sound visualization

Sound can help us explore and analyze complex data sets in scientific computing. The authors describe a digital instrument for additive sound synthesis (Diass) and a program to visualize sounds in a virtual reality environment (M4Cave). Both are part of a comprehensive music composition environment that includes additional software for computer-assisted composition and automatic music notation.

Quenching for semilinear singular parabolic problems

Let f be a real-valued function that is nondecreasing and continuously differentiable on

On Nonadiabatic Condensed Phase Combustion

[0,c)

Heterogeneous Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems

for some finite

Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights *

c(c > 0)