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 is 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 (Guidry and Harris 2013 Comput. Sci. Disc. 6 015002) examines quasi-steady-state methods as an alternative to asymptotic methods in systems well away from equilibrium and a third paper (Guidry et al 2013 Comput. Sci. Disc. 6 015003) 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.