Novel numerical decomposition approaches for multiscale combustion and kinetic models

Abstract

A general theoretical procedure of decomposition of original multiscale system of ODEs into fast and slow subsystems is presented. Multiscale systems arise in modelling of chemical, biochemical, mechanical systems. The performed analysis shows that existing algorithms can be interpreted as possible realizations of the general framework for identification of slow invariant manifolds (slow motions). The aim of this framework is a decomposition of the original system of equations into two separate sets - fast and slow sub-systems. The analysis is based on a new concept of a singularly perturbed vector field. General procedures for decomposition of singularly perturbed fields onto fast and slow parts are presented and this permits us to develop the quasi-linearization method for identification of the fast and the slow subprocesses for complicated kinetics and combustion problems. Application of the suggested numerical technique (the quasi-linearization method) demonstrates that a number of the uncovered restrictions on existing numerical procedures are successfully overcome. The proposed numerical procedure is applied to the highly non-linear problems of mathematical theory of combustion and demonstrates an essentially better performance with respect to existing ones.