Explicit Integration of Extremely-Stiff Reaction Networks: Partial Equilibrium Methods

TL;DR

This paper introduces a partial equilibrium method that enhances explicit integration schemes for extremely stiff reaction networks, enabling accurate and fast solutions even near equilibrium, potentially replacing implicit methods.

Contribution

The paper develops a novel partial equilibrium approach that, combined with existing explicit methods, effectively handles the approach to equilibrium in stiff reaction networks.

Findings

Explicit methods with partial equilibrium are competitive with implicit methods.

The new approach extends the feasible size of networks for explicit integration.

Explicit schemes can accurately handle the stiffest networks near equilibrium.

Abstract

In two preceding papers we have shown that, when reaction networks are well-removed from equilibrium, explicit asymptotic and quasi-steady-state approximations can give algebraically-stabilized integration schemes that rival standard implicit methods in accuracy and speed for extremely stiff systems. However, we also showed that these explicit methods remain accurate but are no longer competitive in speed as the network approaches equilibrium. In this paper we analyze this failure and show that it is associated with the presence of fast equilibration timescales that neither asymptotic nor quasi-steady-state approximations are able to remove efficiently from the numerical integration. Based on this understanding, we develop a partial equilibrium method to deal effectively with the approach to equilibrium and show that explicit asymptotic methods, combined with the new partial equilibrium methods, give an integration scheme that plausibly can deal with the stiffest networks, even in the approach to equilibrium, with accuracy and speed competitive with that of implicit methods. Thus we demonstrate that such explicit methods may offer alternatives to implicit integration of even extremely stiff systems, and that these methods may permit integration of much larger networks than have been possible before in a number of fields.