On the application of explicit spatial filtering to the variables or fluxes of linear equations

A new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed. The method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching. The approach is computationally very simple as it implies the spectral analysis of a symmetric matrix. Notwithstanding its simplicity, stretching-based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory. As a byproduct, a simple reduction method is derived, equivalent to a “local embedding algorithm”, which is based on the local projection of the dynamics onto the “most unstable and/or slow modes” compared to the time scale dictated by the local tangential dynamics. This method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point, limit cycles and chaotic attractors. Several numerical examples deriving from typical models of reaction/diffusion kinetics exhibiting complex dynamics are thoroughly addressed. The application to typical combustion models is also analyzed.