Slow Manifolds in the Space of Variables

In a closed system, if the simulation is started from an arbitrary point in concentration space, it will finally end up at the equilibrium point, whilst the values of conserved variables remain constant. The equilibrium point is determined by the conserved properties, which are defined by the initial state of the system. If in an isothermal system there are Ng species and Nc conserved properties, then the trajectory of the system will move on a hypersurface with dimension Ns Nc- As time elapses, active modes will collapse, with the fastest mode relating to the largest negative eigenvalue relaxing first. The trajectory then approaches a hypersurface with dimension 1. The relaxation will be approximately according to an 

When the second fastest mode relaxes, the trajectory will reach a stuface with dimension Ns—Nc—2. In a closed system, this process continues tmtil the trajectory in the space of concentrations reaches a 3D stuface, a 2D stuface (a curved plane) and a ID surface (a curved line) and finally ends up near the OD equilibrium point. Therefore, following the ideas of Roussel and Fraser, we can imagine the system collapsing onto a cascade of manifolds of decreasing dimension with the fastest modes collapsing first and the slowest last. For a non-isothermal system, temperature may also be a variable increasing the dimension of the phase space by 1, but the same principles apply. In our discussions, we denote Ns as the dimension of the full system which may include temperature as a variable. 

The trajectories are seen to be attracted to the manifold from all initial conditions although the approach is steeper from some directions. 

From the figure, it appears that the trajectories reach exactly the same manifold, but it is easy to illustrate that a trajectory always only approaches the low-dimensional surface (or even the equilibrium point), but never reaches it exactly. In principle, time is reversible in a system defined by the system of ODEs (2.9). This means that calculating the trajectory from to to fj, and then continuing the simulation backwards in time from to to, the same concentration set should be recovered. This is impossible if trajectories starting from different initial conditions end up at exactly the same point. However, for most applications, the approximate nature of the slow manifolds is not a barrier to their use in model reduction strategies, since where large separations between timescales exist, the error related to the approximation of the slow manifold should be small. 

Maas and Pope developed an approach for the calculation of slow manifolds (Maas and Pope 1992a, b, 1994 Maas 1995, 1998, 1999 Maas and Thevenin 1998) utilising the approach of Roussel and Fraser, as well as the suggestion of Lam and Goussis, that timescales should be investigated pointwise via the eigenvalue-eigenvector decomposition of the Jacobian. Their approach was to tabulate these low-dimensional slow manifolds in phase space for several reaction systems in combustion. They called the slow manifolds intrinsic low-dimensional manifolds (ILDM). 

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Several of the numerical-based methods exploit the presence of slow manifolds within chemical kinetic systems which can help to reduce the dimensionality of the system (see Sects. 6.5 and 7.7.3) whilst retaining the ability to reproduce the important system dynamics. A slow manifold is rapidly approached during a simulation as the fast system timescales collapse. Let us assume that we have identified a point in the space of variables that is on (or close to) an A -dimensional manifold. The state of the system can then be characterised by the following variable vector... 

Using the intrinsic low-dimensimial manifold (ILDM) algorithm of Maas and Pope introduced in Sect. 6.5 and detailed below, the location of the slow manifolds in the concentration space can be determined. If we denote to be the dimension of the slow manifold, then variables should be selected for its representation (Golub and Van Loan 1983 Rhodes et al. 1999), and the concentrations of the other variables will be determined as a function of these variables. 

The slow manifold is therefore defined by points in composition space where the chemical source term only has a component in the direction of the slow processes. The slow variables are projected accordingly (uito the manifold defined by Eq. (7.91) yielding... 

The system must have at least three independent variables. The folded surface labeled as the slow manifold is a region on which the system moves relatively slowly through the concentration space. If the system is in a state that does not lie on this manifold, rapid processes cause it to move swiftly onto that surface. The steady state, which lies on the lower sheet of the manifold, is stable to perturbations transverse to the manifold, but is unstable along the manifold. Thus, if the... 

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