Computational singular perturbation method

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

The computer program uses the CSP method (Computational Singular Perturbation). This method allows the quasi-stationary species and the quasi-equilibrated reactions to be identified, and a reduced mechanism to be generated. 

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called time scale separation methods. The most common methods are those of (i) Intrinsic Low Dimensional Manifolds (ILDM), (ii) Computational Singular Perturbation CSF), and (iii) level of importance (LOl) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species. 

Three time-scale separation methods will be described below. These are the Intrinsic Low Dimensional Manifold method (ILDM), Computational Singular Perturbation (CSP), and the lifetime analysis based on tbe Level Of Imjaortance (LOI). 

Lam and Goussis elaborated a detailed theory based on the application of computational perturbation methods for the investigation of reaction mechanisms. This family of methods is called computational singular perturbation theory and is often... 

Gupta, S., Im, H.G., Valorani, M. Classification of ignition regimes in HCCI cranbustion using computational singular perturbation. Proc. Combust. InsL 33, 2991-2999 (2011) Hadjinicolaou, M., Goussis, D.A. Asymptotic solution of stiff PDEs with the CSP method the reaction diffusion equation. SIAM J. Sci. Comput. 20, 781-810 (1998)... 

Singular Perturbation Problems in Chemical Physics Analytic and Computational Methods, Edited by John J. H. Miller, Advances in Chemical Physics Series, Vol. XCVII. 

In the case of singularly perturbed boundary value problems, for which it is required to find diffusion fluxes, computational difficulties arise. These lead to theoretical and applied problems that require special numerical methods allowing us to approximate both the problem solution and the... 

P. W. Hemker and G. I. Shishkin, Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition, in Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods, J. J. H. Miller, Ed., Front Range Press, Copper Mountain, CO, 1992, pp. 3-4. 

SINGULAR PERTURBATION PROBLEMS IN CHEMICAL PHYSICS Analytic and Computational Methods... 

A. G. Xiao and Y. X. Zhao, Convergence of parallel multistep hybrid methods for singular perturbation problems. Applied Mathematics and Computation, 2009, 215(6), 2139-2148. 

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