Approximation of Slow Attracting Manifolds in Chemical Kinetics by Trajectory-Based Optimization Approaches

Abstract

Many common kinetic model reduction approaches are explicitly based on inherent multiple time scales and often assume and directly exploit a clear time scale separation into fast and slow reaction processes. They approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones. The corresponding restrictive assumption of full relaxation of fast modes often renders the resulting approximation of slow attracting manifolds inaccurate as a representation of the reduced model and makes the numerical solution of the nonlinear "reduction equations" particularly difficult in many cases where the gap in intrinsic time scales is not large enough. We demonstrate that trajectory optimization approaches can avoid such severe restrictions by computing numerical solutions that correspond to "maximally relaxed" dynamical modes in a suitable sense. We present a framework of trajectory-based optimization for model reduction in chemical kinetics and a general class of reduction criteria characterizing the relaxation of chemical forces along reaction trajectories. These criteria can be motivated geometrically exploiting ideas from differential geometry and fundamental physics and turn out to be highly successful in example applications. Within this framework, we provide results for the computational approximation of slow attracting low-dimensional manifolds in terms of families of optimal trajectories for a six-component hydrogen combustion mechanism.

Entropy-Related Extremum Principles for Model Reduction of Dissipative Dynamical Systems

“…In this section we consider a small test mechanism, which has already been used for model reduction purposes in [31,36,37]. It consists of six chemical species involved in six elementary reactions constrained by two element mass conservation relations for hydrogen and oxygen: …”

“…The successive relaxation of chemical forces causes curvature in the reaction trajectories (in the sense of velocity change along the trajectory). Therefore, in [31,32] Φ…”

“…In the original work [21] the entropy production rate (1) has been chosen as a criterion Φ in (2a) and following the fundamental idea to search for an entropy-related extremum principle that characterizes trajectories on or near slow attracting manifolds we discuss geometric criteria in [31]. In Section 4. we show that the latter are superior to the previously used entropy production rate and yield accurate approximations of slow invariant manifolds for an example model of chemical kinetics.…”

Reaction Kinetics Path Based on Entropy Production Rate and Its Relevance to Low-Dimensional Manifolds

“…However, for chemical reaction systems, there is no example of deriving the equation that describes a chemical change directly based on entropy production or on a Lyapunov function that does not use a chemical kinetics equation. Gorban et al [18][19][20] and Lebiedz [21][22][23] utilized the second law of thermodynamics to determine the so-called slow invariant manifold (SIM) or the so-called low-dimensional manifold (LDM). Gorban et al [18][19][20] utilized the basis orthonormal with respect to their "entropic" scalar product that uses the Hessian of a Lyapunov function (the free energy of a perfect gas in a constant volume at constant temperature) because their almost orthogonal "projector" that defines SIM helps convergence to determine SIM [20].…”

The Role of Thermodynamics in Model Reduction When Using Invariant Grids

“…Therefore, much effort has been devoted to setting up automated model reduction procedures based (explicitly or implicitly) on the notion of SIM: The method of invariant grids (MIG) [6,7], the computational singular perturbation (CSP) method [29][30][31], the intrinsic low dimensional manifold (ILDM) [27,28], the invariant constrained equilibrium edge preimage curve method (ICE-PIC) [32], the equation-free approaches [33], the method of minimal entropy production trajectories (MEPT) [35] and minimum curvature [36], the constrained runs algorithm in [34] and the finite-time Lyapunov analysis in [37] are some representative examples.…”