Multiscale Adaptive Model Reduction in Reactive Flows*

The numerical solution of mathematical models for reacting flows is a challenging task because of the simultaneous contribution of a wide range of time scales present in the system. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active scales. The complexity of the problem can be reduced when fast/active and slow/active time scales gaps becomes large. We propose a numerical technique named the \textsl{G-Scheme}, to achieve multiscale adaptive model reduction. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We introduce a locally curvilinear frame of reference, defined by a set of orthonormal basis vectors, with corresponding coordinates, attached to this decomposition. The evolution of the coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the \textsl{G-Scheme}, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem. The effectiveness of the \textsl{G-Scheme}, is demonstrate by solving a number of relevant problems.