Algorithms of singular perturbation type

It has been seen that singular perturbation theory constitutes the natural framework justifying the use of QSSA. Classically, the QSSA allows kineticists to obtain explicit mathematical relationships. 

The first papers dealing with the QSSA as a tool for the numerical integration of kinetic models are those by Snow et al. [170—171]. Programs using QSSA may be found, for example, in Detar [172]. It is worth noting that, when one has recourse to QSSA, only standard numerical procedures are to be used since there is no more stiffness. 

A general numerical algorithm of the boundary layer type for stiff systems of differential equations has been proposed by Miranker [173] and applied to a few kinetic problems by Aiken and Lapidus [174,175]. The principle of the method will be briefly described in the case of the following system of differential equations, involving stiff variable x and non-stiff variable y. 

The theory of singular perturbations leads us to seek asymptotic expansions of x(t) and y (f) of the form 

Equations for each of the perturbation functions xu yh Xu Yl are derived by substituting the asymptotic expansions into the initial differential system, by matching terms with the same power in e, and finally by writing the proper initial and boundary layer conditions. The zeroth-order outer approximation is the solution to the system 

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Although no systematic comparisons of singular perturbation type algorithms with other methods have been carried out, the efficiency... 

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