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

TL;DR

This paper presents a new method to efficiently approximate slow and fast dynamics in multiscale systems by linearizing the Relaxation Redistribution Method, simplifying the construction of slow invariant manifolds in stiff ODEs.

Contribution

The paper introduces a fictitious dynamics approach for fast relaxation in stiff ODEs, enabling easier construction of slow invariant manifolds using existing stiff numerical schemes.

Findings

Successfully applied to hydrogen-air combustion mechanism

Achieved accurate approximation of slow and fast dynamics

Validated against Chapman-Enskog invariance solution

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.