Abstract
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 stabilizing algebra differs for systems well removed from equilibrium and those 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 [1] examines quasi-steady-state methods as an alternative to asymptotic methods in systems well away from equilibrium and a third paper [2] 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 explicit methods can execute a timestep faster than an implicit method, our results imply that algebraically stabilized explicit algorithms may offer a means to integration of larger networks than have been feasible previously in various disciplines.
