Explicit Integration of Extremely-Stiff Reaction Networks: Quasi-Steady-State Methods

A preceding paper demonstrated that explicit asymptotic methods generally work much better for extremely stiff reaction networks than has previously been shown in the literature. There we showed that for systems well removed from equilibrium explicit asymptotic methods can rival standard implicit codes in speed and accuracy for solving extremely stiff differential equations. In this paper we continue the investigation of systems well removed from equilibrium by examining quasi-steady-state (QSS) methods as an alternative to asymptotic methods. We show that for systems well removed from equilibrium, QSS methods also can compete with, or even exceed, standard implicit methods in speed, even for extremely stiff networks, and in many cases give somewhat better integration speed than for asymptotic methods. As for asymptotic methods, we will find that QSS methods give correct results, but with non-competitive integration speed as equilibrium is approached. Thus, we shall find that both asymptotic and QSS methods must be supplemented with partial equilibrium methods as equilibrium is approached to remain competitive with implicit methods.

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

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.

Explicit Integration of Extremely-Stiff Reaction Networks: Asymptotic Methods

We show that, even for extremely stiff systems, explicit integration may compete in both accuracy and speed with implicit methods if algebraic methods are used to stabilize the numerical integration. The required stabilizing algebra depends on whether the system is well-removed from equilibrium or near equilibrium. This paper introduces a quantitative distinction between these two regimes and addresses the former case in depth, presenting explicit asymptotic methods appropriate when the system is extremely stiff but only weakly equilibrated. A second paper examines quasi-steady-state methods as an alternative to asymptotic methods in systems well away from equilibrium and a third paper extends these methods to equilibrium conditions in extremely stiff systems using partial equilibrium methods. All three papers present systematic evidence for timesteps competitive with implicit methods. Because an explicit method can execute a timestep faster than an implicit method, algebraically-stabilized explicit algorithms might permit integration of larger networks than have been feasible before in various disciplines.