Fast-slow vector fields of reaction-diffusion systems

TL;DR

This paper introduces a geometrically invariant framework for analyzing reaction-diffusion systems using fast-slow vector fields, providing a theoretical basis for the reaction-diffusion manifolds (REDIM) algorithm and demonstrating its practical numerical application.

Contribution

It extends the concept of singularly perturbed vector fields to reaction-diffusion systems and offers a robust computational approach for model decomposition and analysis.

Findings

Transport terms can be neglected in fast subsystems under physical assumptions.

Theoretical justification for the REDIM algorithm is established.

Practical numerical methods for reaction-diffusion systems are demonstrated.

Abstract

A geometrically invariant concept of fast-slow vector fields perturbed by transport terms (describing molecular diffusion processes) is proposed in this paper. It is an extension of our concept of singularly perturbed vector fields to reaction-diffusion systems. Fast-slow vector fields can be represented locally as "singularly perturbed systems of PDE". The paper focuses on possible ways of original models decomposition to fast and slow subsystems. We demonstrate that transport terms can be neglected (under reasonable physical assumptions) for fast subsystem. A method of analysis of slow subsystem as an evolution of singularly perturbed profiles along slow invariant manifolds was discussed in our previous work \cite{BCMGM2016}. This paper is motivated by an algorithm of reaction-diffusion manifolds (REDIM) \cite{BM2007,BM2009,MB2011}. It can be considered as its theoretical justification extending it from a practical algorithm to a robust computational procedure under some reasonable physical assumptions. A practical application of the proposed algorithm for numerical treatment of reaction-diffusion systems is demonstrated.