  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.
