Quasi‐Spectral Method for the Solution of the Master Equation for Unimolecular Reaction Systems

Abstract

Rate constants of elementary reactions involving unimolecular steps can be calculated from molecular data in a most general way by solving appropriate master equations. The conventional numerical solution requires rather a fine discretization applied over a sufficiently large energy range to achieve a reasonable accuracy. This leads to linear but very high‐dimensional systems of differential equations. We propose a quasi‐spectral method that uses Gaussian radial basis functions to establish a low‐dimensional linear model to speed up the numerical integration. The combination with an iterative adaptation provides a further improvement of computational efficiency. The suggested approach is illustrated and exemplified by means of the unimolecular decomposition of 2,3‐dihydro‐2,5‐dimethylfuran‐3‐yl, an intermediate radical occurring in the pyrolysis and oxidation of 2,5‐dimethylfuran. A comparison of the conventional and the proposed method is presented to validate the novel approach and to demonstrate its performance.

Study of internal flame front structure of accelerating hydrogen/oxygen flames with detailed chemical kinetics and diffusion models

“…when using power series of fixed order within one element (e.g. second order) and using interval indicators as test functions, leads to the classical finite volume method (see [1,25] for more details).…”

Some Aspects of Time-Reversal in Chemical Kinetics

“…There are a variety of different formulations of the master equation [ 7 ], and, although, in many cases, the master equation is cast in a continuum form, we shall investigate the discrete form because it allows an analysis similar to the previous sections. We adopt a simple formulation according to [ 56 ], which is based on a discretization into energy parcels. Then, we obtain a finite dimensional ordinary differential equation system for the vector of average number densities in energy interval i out of a total of energy intervals [ 56 ].…”

“…We adopt a simple formulation according to [ 56 ], which is based on a discretization into energy parcels. Then, we obtain a finite dimensional ordinary differential equation system for the vector of average number densities in energy interval i out of a total of energy intervals [ 56 ]. The rate of change of each energy parcel is given by or in vector notation where denotes the rate of reaction that produces the species (input flux), f is the corresponding distribution, the transition probability form energy interval j to energy interval i , I the identity matrix, and K the diagonal matrix of averaged unimolecular rate coefficients , and denotes the collision frequency with bath gas molecules.…”

“…If (chemical reaction of the species to products), then the eigenvalues shift, but they are still negative (can be verified using the arguments used in [ 57 ] for determining the Gershgorin circles). Even in the case of chemical reaction, the eigenvalues are typically well separated from the largest one (the one governing the “macroscopic” behavior), unless there are very special conditions such as very high temperatures [ 56 ].…”