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In contrast to the prevailing view in the literature, it is shown that evenextremely stiff sets of ordinary differential equations may be solvedefficiently by explicit methods if limiting algebraic solutions are used tostabilize the numerical integration. The stabilizing algebra differsessentially for systems well-removed from equilibrium and those nearequilibrium. Explicit asymptotic and quasi-steady-state methods that areappropriate when the system is only weakly equilibrated are examined first.These methods are then extended to the case of close approach to equilibriumthrough a new implementation of partial equilibrium approximations. Usingstringent tests with astrophysical thermonuclear networks, evidence is providedthat these methods can deal with the stiffest networks, even in the approach toequilibrium, with accuracy and integration timestepping comparable to that ofimplicit methods. Because explicit methods can execute a timestep faster andscale more favorably with network size than implicit algorithms, our resultssuggest that algebraically-stabilized explicit methods might enable integrationof larger reaction networks coupled to fluid dynamics than has been feasiblepreviously for a variety of disciplines.

Algebraic Stabilization of Explicit Numerical Integration for Extremely Stiff Reaction Networks