On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

Figures & data

Figure 1. Solutions z₁(t_f = 0; t₀) of the boundary value formulation (27) applied to the linear model (1) with γ = 0.2 visualized by the rhombi in comparison with the analytical error (33) (dashed line) as a function of t_0.

Figure 2. Solutions z₁(t_f = 0; t₀) of the boundary value problem (27) applied to the linear model (1) with γ = 2.0 visualized by the rhombi in comparison with the analytical error (33) (dashed line) as a function of t₀.

Figure 3. Solutions z₂(t_f = 0; Z₀) of the boundary value formulation (27) applied to the Davis–Skodje test problem (2) with γ = 1.2 visualized by the rhombi in comparison with the analytical error (39) (dashed line) as a function of t₀.

Figure 4. Solutions z₂(t_f = 0; t₀) of the boundary value formulation (27) applied to the Davis–Skodje test problem (2) with γ = 3.0 visualized by the rhombi in comparison with the analytical error (39) (dashed line) as a function of t₀.

Figure 5. POIs (crosses) resulting from optimization problem (49) applied to a simplified six species mechanism (see [45]) with z(H₂O) as RPV fixed at 80 different values between $z_j^{t_{\mathrm{f}}}={10}^{-4}$ and $z_j^{t_{\mathrm{f}}}=4.0$. As can be seen, the POIs form an invariant manifold attracting several reaction trajectories (dashed curves) starting from arbitrarily chosen initial values.

Figure 6. Visual demonstration why a reverse mode formulation works more accurately than the corresponding local method. A polyhedron restricts the feasible area with the consequence that t₀ can only be chosen as small as possible within the feasibility constraints.

Figure 7. The black dots represent the equilibrium point, the solid curves the SIM and the dashed curves are solution trajectories concerning the underlying model equations. (a) Linear model. (b) Davis–Skodje model.

M. Heitel, Comparison of numerical optimization techniques for a variational problem formulation of manifold-based model reduction, Master thesis, Ulm University, Ulm, Germany, 2015.

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