Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized Relaxation Redistribution Method

TL;DR

Addresses the challenge of reducing large multiscale dissipative ODE systems by constructing a slow invariant manifold ($SIM$) using a linearized Relaxation Redistribution Method (RRM) that produces a stiff ODE system mimicking the fast relaxation toward the SIM. The method yields governing equations for a fictitious evolution of the state $Y$ and a slow-subspace mapping $F^{k}(\xi)=A^{k}\xi+l^{k}$, along with a parallel fast-subspace refinement and an adaptive dimension $q$. Initialization via a quasi-equilibrium manifold defined by a Lyapunov function $G$ and tangent-space projections complements the framework, and the approach is validated on hydrogen–air combustion and a Chapman–Enskog benchmark, achieving high accuracy against exact invariance solutions. The work offers a simple, robust route to constructing high-dimensional SIMs with standard stiff solvers and suggests avenues for higher-order, multi-steady-state extensions.

Abstract

In this paper, we introduce a fictitious dynamics for describing the only fast relaxation of a stiff ordinary differential equation (ODE) system towards a stable low-dimensional invariant manifold in the phase-space (slow invariant manifold - SIM). As a result, the demanding problem of constructing SIM of any dimensions is recast into the remarkably simpler task of solving a properly devised ODE system by stiff numerical schemes available in the literature. In the same spirit, a set of equations is elaborated for local construction of the fast subspace, and possible initialization procedures for the above equations are discussed. The implementation to a detailed mechanism for combustion of hydrogen and air has been carried out, while a model with the exact Chapman-Enskog solution of the invariance equation is utilized as a benchmark.

Figures (9)

• Figure 1: (Color online). Schematic representation of the basic idea behind the Relaxation Redistribution Method (RRM). In a small neighborhood of the pivot $\bar{Y}^k$ (large circles), a linear approximation of the SIM at the iteration $k$ is considered. With the aim of driving the pivot towards the SIM, RRM ChKa2011CKPRE2010CAV2011 prescribes an updating rule $\bar{Y}^k \rightarrow \bar{Y}^{k+1}$ as schematically sketched above (small circles represent neighbors of the pivot). Here, an ODEs system (\ref{['Y.evol']}) whose dynamics approximates the latter updating rule is suggested and tested.
• Figure 2: (Color online). Rationale behind the refinement process of the fast subspace: In a neighborhood of the SIM (slow subspace), the anti-parallel dynamics $-f$ reacts with a torque if (\ref{['Y.fast']}) does not span the fast subspace. As a result, the latter subspace is the stable stationary solution of Eqs. (\ref{['A.fast.updated']}) and (\ref{['A.fast.dynamical']}). Small circles denote neighbors of a pivot (large circle).
• Figure 3: (Color online). Pictorial representation of the notion of quasi-equilibrium manifold (QEM) (\ref{['QEM.def']}), $H$ and $\bar{B}$ being the second derivative matrix of the Lyapunov function $G$ and the null space of the full set of constraints in (\ref{['QEM.def']}), respectively.
• Figure 4: (Color online). Slow invariant manifold with respect to the dynamical system (\ref{['ODEben']}) with (\ref{['choice']}), $\omega=3$ and $\epsilon=0.025$. Starting from the initial conditions (\ref{['initial.conditions']}), the governing equations of the linearized RRM (\ref{['Y.evol']}) with $n=4$, $q=2$ and $\tau=3 \times 10^{-10}$ are solved by means of the stiff numerical scheme ode15s readily available in Matlab$\textsuperscript{\textregistered}$ode15s. Here, the steady state is reached after an integration time $T_f=1$. At steady state, the solution trajectory (dots) finally lands on the SIM.
• Figure 5: (Color online). An array of initial states has been refined by means of the linearized RRM. Stationary states of the (\ref{['Y.evol']}) are reported (circles). For a comparison, the exact Chapman-Enskog solution and a detailed solution trajectory of the system (\ref{['ODEben']}) are also shown. Here we use $\omega=3$, $\tau=3 \times 10^{-10}$, while integration of (\ref{['Y.evol']}) is performed for an integration time $T_f=1$ at any point. The computational time required to refine the entire array, composed by $(21 \times 21)$ points, was $2.5$ minutes using a Matlab code on a single processor with 1.73 GHz.