  A projection-based formulation is presented for non-linear model reduction of
problems with extreme scale disparity. The approach allows for the selection of
an arbitrary, but complete, set of solution variables while preserving the
structure of the governing equations. Least-squares-based minimization is
leveraged to guarantee symmetrization and discrete consistency with the
full-order model (FOM). Two levels of scaling are used to achieve the
conditioning required to effectively handle problems with extremely disparate
physical phenomena, characterized by extreme stiffness in the system of
equations. The formulation -- referred to as model-form preserving
least-squares with variable transformation (MP-LSVT) -- provides global
stabilization for both implicit and explicit time integration schemes. To
achieve computational efficiency, a pivoted QR decomposition is used with
oversampling, and adapted to the MP-LSVT method. The framework is demonstrated
in representative two- and three-dimensional reacting flow problems, and the
MP-LSVT is shown to exhibit improved stability and accuracy over standard
projection-based ROM techniques. Physical realizability and local stability are
promoted by enforcing limiters in both temperature and species mass fractions.
These limiters are demonstrated to be important in eliminating regions of
spurious burning, thus enabling the ROMs to provide accurate representations of
the heat release rate and flame propagation speed. In the 3D application, it is
shown that more than two orders of magnitude acceleration in computational
efficiency can be achieved, while also providing reasonable future-state
predictions. A key contribution of this work is the development and
demonstration of a comprehensive ROM formulation that targets highly
challenging multi-scale transport-dominated problems.
