A Multi-Parameter Singular Perturbation Analysis of the Robertson Model

TL;DR

This paper provides a comprehensive asymptotic analysis of the Robertson model, a classical stiff ODE system in chemical kinetics, using geometric singular perturbation theory and blow-up techniques for multiscale regimes.

Contribution

It introduces a novel multi-parameter asymptotic analysis of the Robertson model, identifying four distinct regimes and applying advanced geometric methods to analyze its dynamics.

Findings

Identification of four regimes near the singular limit

Asymptotic solutions match numerical results

Extension of GSPT to multi-parameter problems

Abstract

The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates $k_1,\,k_2$, and $k_3$, with largely differing orders of magnitude, acting as parameters. The model has been widely used as a numerical test problem. Surprisingly, no asymptotic analysis of this multiscale problem seems to exist. In this paper we provide a full asymptotic analysis of the Robertson model under the assumption $k_1, k_3 \ll k_2$. We rewrite the equations as a two-parameter singular perturbation problem in the rescaled small parameters $(\varepsilon_1,\varepsilon_2):=(k_1/k_2,k_3/k_2)$, which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multi-parameter singular structure, we perform blow-ups in parameter- and variable space. We identify four distinct regimes in a neighbourhood of the singular limit $(\varepsilon_1,\varepsilon_2)= (0,0)$. Within these four regimes we use GSPT and additional blow-ups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics.