Thermonuclear kinetics in astrophysics

Over the billions of years since the Big Bang, the lives, deaths and afterlives of stars have enriched the Universe in the heavy elements that make up so much of ourselves and our world. This review summarizes the methods used to evolve these nuclear abundances within astrophysical simulations. These methods fall into 2 categories; evolution via rate equations and via equilibria. Because the rate equations in nucleosynthetic applications involve a wide range of timescales, implicit methods have proven mandatory, leading to the need to solve matrix equations. Efforts to improve the performance of such rate equation methods are focused on efficient solution of these matrix equations, in particular by making best use of the sparseness of these matrices, and finding methods that require less frequent matrix solutions. 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 which assume complete equilibrium fail.