Analysis of a Controlled Approximation for Explicit Integrations of Stiff Thermonuclear Networks

The current standard method to solve stiff coupled differential equations relies on implicit integration methods. Explicit methods are generally avoided due to the extremely small and limiting timesteps they allow when the equations are stiff. However, implicit methods are computationally expensive because of the complex calculations that need to be done at each time step. An explicit integration method can do these calculations quicker and, if allowed to take comparable timesteps to the implicit ones, would allow the entire calculation to be done faster. Previous work by Dr. Guidry, Dr. Endeve, Dr. Hix and Dr. Billings has shown that, in principle, explicit integration can take larger timesteps than normally allowed when certain approximations are used. The speed up in the calculations from implementing algebraic approximations comes at the expense of the accuracy. However, unlike other approximations typically introduced for coupling networks to hydrodynamical simulations, these approximations can be controlled by the user and allow for a quantifiable restraint on the error. The concept of a controlled approximation is introduced by providing a quantifiable way to show the trade off of accuracy for speed when using algebraic approximations for explicit integrations of stiff thermonuclear reaction networks coupled to fluid dynamics.