The need for using larger, more complete thermonuclear reaction networks in multi-dimensional astrophysics simulations, driven by the need to compare these simulations to the detailed nucleosynthesis revealed by observations, creates a need for more efficient ways to solve systems of equations. Numerical stiffness, the computational manifestation of the wide range of physical timescales active in these systems, greatly restricts the available solution methods. Most commonly, implicit solution methods centered on matrix solutions are used. The result is a highly non-linear increase in computational cost with increasing number of nuclear species included in the network. As a result, typical multi-dimensional simulations in many areas of stellar astrophysics utilize small (often too small) reaction networks. Our group has several investigations whose goal is reducing the computational cost of adequately sized reaction networks in order to replace the current too small reaction networks frequently used.

Quasi-Equilibrium Reduced Network (Parete-Koon, Hix, Chertkow with Thielemann)

For her dissertation project, Parete-Koon has completed development of more efficient numerical methods for nucleosynthesis in supernovae, methods based on Nuclear Statistical Equilibrium and Quasi-Equilibrium (QSE). In 2007 we published a paper documenting factors of 5 or more decrease in the computational cost of the network during the most computationally expensive phases, by assuming 3 fixed QSE groups and effectively suppressing the matrix solution which otherwise dominates the calculation. We have recently generalized this method to allow adaptive groups to be chosen as changing conditions warrant, resulting in 15-20 times faster solution than the conventional network with sufficient accuracy in the energy generation and dominant species. Alternately, the adaptivity of the groups can be used to achieve high accuracy for less abundant species, at a still considerable five-fold or larger increase in speed. For part of his PhD dissertation, Chertkow will be implementing this method in CHIMERA to enable self-consistent nucleosynthesis that is fully coupled to turbulent, neutrino-driven hydrodynamic flow.

Thermonuclear Reaction Networks without matrix solutions (Guidry, Feger, jay, Chupryna, Hix)

The ultimate way to improve the scaling of computational cost with reaction network size is to remove the matrix solution. A number of such methods have proven useful in solving the stiff sets of equations present in other fields, but to date such methods have shown little utility for the extremely stiff systems in thermonuclear reaction networks. For part of his PhD dissertation, Feger is undertaking a comprehensive study of stiff solution methods for thermonuclear reaction networks. We are also exploring a homegrown stiff solution method, the Flux-Constrained Forward Differencing method.