Adaptive simplification of complex multiscale systems

TL;DR

This work tackles the problem of extracting a minimal, physically meaningful description from large, multiscale dissipative dynamics by adaptively constructing slow invariant manifolds (SIM) of dimension $q$ using the Relaxation Redistribution Method (RRM). RRM avoids direct numerical solution of the film equation $\dfrac{d\psi(\xi)}{dt} = f(\psi(\xi)) - P f(\psi(\xi))$ by coupling full relaxation with a redistribution step that subtracts slow motion, enabling stable, dimension-adaptive SIMs in arbitrary $q$. The method is validated on hydrogen-air autoignition with a detailed mechanism, revealing a cascade of SIMs $(q=5,4,3,2,1,0)$ that reproduce the full mechanism's species histories and temperature, while minority species like HO2 and H2O2 may require higher $q$. This approach delivers accurate, physically interpretable reduced descriptions and scales to high dimensions, with potential applicability to other dissipative systems.

Abstract

A fully adaptive methodology is developed for reducing the complexity of large dissipative systems. This represents a significant step towards extracting essential physical knowledge from complex systems, by addressing the challenging problem of a minimal number of variables needed to exactly capture the system dynamics. Accurate reduced description is achieved, by construction of a hierarchy of slow invariant manifolds, with an embarrassingly simple implementation in any dimension. The method is validated with the auto-ignition of the hydrogen-air mixture where a reduction to a cascade of slow invariant manifolds is observed.

Figures (6)

• Figure 1: (Color online). a) Model reduction techniques assume the following idea: After a fast initial transient at the time instants $t \le t_0$, the (slow) dynamics of a complex system takes place along a slow invariant manifold (SIM) on the phase-space $U$ at any future time $t>t_0$ (invariance) toward the steady state. b) The definition of a projector $P$ onto the tangent space $T$ introduces a decomposition of slow and fast motions of the field $f$. In a vicinity of the SIM, slow and fast motions are locked in the image and null space of the thermodynamic projector $P$bookCiCP2010, respectively.
• Figure 2: (Color online). a) The relaxation due to (\ref{['kin_sys']}) of a non-invariant manifold. Fast dynamics drives it toward the slow invariant manifold, whereas the concurrent action of the slow dynamics causes a shift toward the steady state (shagreen effect). On the contrary, relaxation due to the film equation (\ref{['film.dynamics']}) - (\ref{['relaxation.scheme']}) allows movements only in the fast subspace. b) Relaxation Redistribution Method. The displacement in the slow subspace, generated during relaxation, is annihilated by a redistribution step in the parameter space.
• Figure 3: (Color online). The Davis-Skodje system DS99. Two different initial grids are refined using the forward Euler scheme for the relaxation ($\delta t=10^{-2}$). Results after $50$ RRM iterations are reported (refined grids) with $\gamma=50$. Triangles show an intermediate step (after two RRM iterations) starting from the initial smooth grid.
• Figure 4: (Color online). a) Relaxation redistribution method: Local formulation. Only a small patch of the SIM is constructed. After refinement, the coordinates of the pivot provide the reduced system (\ref{['macro']}) with a closure. b) Simplexes can be conveniently adopted for a patch-wise description of the SIM in any dimension.
• Figure 5: (Color online). Heterogeneous slow invariant manifold of hydrogen-air combustion mechanism by local RRM. Three-dimensional projection of the six-dimensional phase space onto spectral variables (see text). The two-dimensional patch ("kite", triangles) is tight by a one dimensional "thread" (line) to the zero-dimensional equilibrium and merges with the three-dimensional "cloud" (tetrahedra). Legend: mass fraction of OH. Explicit Euler scheme with $\delta t=5 \times 10^{-8} [s]$ was used for the relaxation of simplex. RRM convergence criteria: $N=2000$, $\epsilon=10^{-4}$.