This review concentrates on the two principle methods used to evolve nuclear
abundances within astrophysical simulations, evolution via rate equations and
via equilibria. Because in general the rate equations in nucleosynthetic
applications form an extraordinarily stiff system, implicit methods have proven
mandatory, leading to the need to solve moderately sized matrix equations.
Efforts to improve the performance of such rate equation methods are focused on
efficient solution of these matrix equations, by making best use of the
sparseness of these matrices. Recent work to produce hybrid schemes which use
local equilibria to reduce the computational cost of the rate equations is also
discussed. Such schemes offer significant improvements in the speed of reaction
networks and are accurate under circumstances where calculations with complete
equilibrium fail.Comment: LaTeX2e with graphicx, 40 Pages with 5 embedded figures. To be
published in Computational Astrophysics, The Journal of Computational and
Applied Mathematics, eds. H. Riffert, K. Werne