The spectral characterisation of reduced order models in chemical kinetic systems

The size and complexity of multi-scale problems such as those arising in chemical kinetics mechanisms has stimulated the search for methods that reduce the number of species and chemical reactions but retain a desired degree of accuracy. The time-scale characterisation of the multi-scale problem can be carried out on the basis of local information such as the Jacobian matrix of the model problem and its related eigen-system evaluated at one point P of the system trajectory. While the original problem is usually described by ordinary differential equations (ODEs), the reduced order model is described by a reduced number of ODEs and a number of algebraic equations (AEs), that might express one or more physical conservation laws (mass, momentum, energy), or the fact that the long-term dynamics evolves within a so-called Slow Invariant Manifold (SIM). To fully exploit the benefits offered by a reduced order model, it is required that the time scale characterisation of the n-dimensional reduced order model returns an answer consistent and coherent with the time-scale characterisation of the N-dimensional original model. This manuscript discusses a procedure for obtaining the time-scale characterisation of the reduced order model in a manner that is consistent with that of the original problem. While a standard time scale characterisation of the (original) N-dimensional original model can be carried out by evaluating the eigen-system of the (N x N) Jacobian matrix of the vector field that defines the system dynamics, the time-scale characterisation of the n-dimensional reduced order model (with n< N) can be carried out by evaluating the eigen-system of a (n x n) constrained Jacobian matrix, J(C), of the reduced vector field that accounts for the role of the constraints.