A reactive system's slow dynamic behavior is approximated well by evolution on manifolds of dimension lower than that of the full composition space. This work addresses the construction of one-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady spatially homogeneous closed reactive systems. Additionally, the relation between the systems' slow dynamics, described by the constructed manifolds, and thermodynamics is clarified. It is shown that other than identifying the equilibrium state, traditional equilibrium thermodynamic potentials provide no guidance in constructing the systems' actual slow invariant manifolds. The construction technique is based on analyzing the composition space of the reactive system. The system's finite and infinite equilibria are calculated using a homotopy continuation method. The slow invariant manifolds are constructed by calculating attractive heteroclinic orbits which connect appropriate equilibria to the unique stable physical equilibrium point. Application of the method to several realistic reactive systems, including a detailed hydrogen-air kinetics model, reveals that constructing the actual slow invariant manifolds can be computationally efficient and algorithmically easy.

Abstract. The method of slow invariant manifolds (SIMs), applied previously to model the reduced kinetics of spatially homogeneous reactive systems, is extended to systems with diffusion. Through the use of a Galerkin projection, the governing partial differential equations are cast into a finite system of ordinary differential equations to be solved on an approximate inertial manifold. The SIM construction technique of identifying equilibria and connecting heteroclinic orbits is extended by identifying steady state solutions to the governing partial differential equations and connecting analogous orbits in the Galerkin-projected space. In parametric studies varying the domain length, the time scale spectrum is shifted, and various classes of nonlinear dynamics are identified. A critical length scale is identified below which the spatially homogeneous one-dimensional SIM models the long time dynamics of the system. At this critical length scale, a bifurcation in the slow dynamics of the system is identified; additional real nonsingular steady state solutions are found which lead to a diffusion-modified one-dimensional SIM. At these longer lengths, the spectral gap in the time scales indicates that an appropriate manifold for a reduction technique is higher than one-dimensional. This is shown for two examples: a simple chemical reaction mechanism, and the Zel'dovich reaction mechanism of NO production. These examples are evaluated in the spatially homogeneous case (a one-term projection), a two-term projection capturing the coarsest effects of diffusion, and a high-order projection that is fully resolved. 1. Introduction. Numerical simulations of the partial differential equations (PDEs) that model multiscale continuum physics are prevalent across many fields of engineering. To obtain results with fidelity to the underlying continuum model, discrete simulations must generally resolve the entire range of scales present, both spatial and temporal; a large disparity in these scales is typically referred to as stiffness. The computational costs associated with these stiff simulations grow with the disparity of scales [1].In recent decades, there have been efforts in model reduction techniques to decrease the computational costs of simulating reactive flows while maintaining as much consistency with

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Slow invariant manifolds (SIM) are calculated for spatially inhomogeneous closed reactive systems to obtain a model reduction. A simple oxygen dissociation reaction-diffusion system is evaluated. SIMs are calculated using a robust method of finding the system's equilibria and integrating to find heteroclinic orbits. Diffusion effects are obtained by using a Galerkin method to project the infinite dimensional dynamical system onto a low dimensional approximate inertial manifold. This projection rigorously accounts for the coupling of reaction and diffusion processes. An analytic coupling between reaction and diffusion time scales is shown to be a function of length scale. A critical length scale is identified where reaction and diffusion time scales are equal. At this critical length scale, a supercritical pitchfork bifurcation occurs which changes the SIM.

Slow invariant manifolds (SIM) are calculated for spatially inhomogeneous closed reactive systems to obtain a model reduction. A robust method of constructing a onedimensional SIM by calculating equilibria and then integrating along heteroclinic orbits is extended to two new cases: i) adiabatic systems, and ii) spatially inhomogeneous systems with simple diffusion. The adiabatic condition can be modeled as a new algebraic constraint in the limit of unity Lewis number. Diffusion effects on the SIM are examined for systems with small characteristic lengths. A diffusion correction is obtained by using a Galerkin method to project the infinite dimensional dynamical system onto a low dimensional approximate inertial manifold. This projection rigorously accounts for the coupling of reaction and diffusion processes, and an analytic length and time scale coupling is shown. An example is demonstrated on a system of N O production which highlights the correlation between an established isothermal spatially homogeneous technique and the new techniques.