Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I Theory

A method has been developed to investigate the sensitivity of the solutions of large sets of coupled nonlinear rate equations to uncertainties in the rate coefficients. This method is based on varying all the rate coefficients simultaneously through the introduction of a parameter in such a way that the output concentrations become periodic functions of this parameter at any given time t. The concentrations of the chemical species are then Fourier analyzed at time t. We show via an application of Weyl's ergodic theorem that a subset of the Fourier coefficients is related to 〈∂ci/∂kl〉, the rate of change of the concentration of species i with respect to the rate constant for reaction l averaged over the uncertainties of all the other rate coefficients. Thus a large Fourier coefficient corresponds to a large sensitivity, and a small Fourier coefficient corresponds to a small sensitivity. The amount of numerical integration required to calculate these Fourier coefficients is considerably less than that required in tests of sensitivity where one varies one rate coefficient at a time, while holding all others fixed. The Fourier method developed in this paper is not limited to chemical rate equations, but can be applied to the study of the sensitivity of any large system of coupled, nonlinear differential equations with respect to the uncertainties in the modeling parameters.