  In contrast to the prevailing view in the literature, it is shown that even
extremely stiff sets of ordinary differential equations may be solved
efficiently by explicit methods if limiting algebraic solutions are used to
stabilize the numerical integration. The stabilizing algebra differs
essentially for systems well-removed from equilibrium and those near
equilibrium. Explicit asymptotic and quasi-steady-state methods that are
appropriate when the system is only weakly equilibrated are examined first.
These methods are then extended to the case of close approach to equilibrium
through a new implementation of partial equilibrium approximations. Using
stringent tests with astrophysical thermonuclear networks, evidence is provided
that these methods can deal with the stiffest networks, even in the approach to
equilibrium, with accuracy and integration timestepping comparable to that of
implicit methods. Because explicit methods can execute a timestep faster and
scale more favorably with network size than implicit algorithms, our results
suggest that algebraically-stabilized explicit methods might enable integration
of larger reaction networks coupled to fluid dynamics than has been feasible
previously for a variety of disciplines.
