Integration of Nuclear Reaction Networks for Stellar Hydrodynamics

Methods for solving the stiÜ system of ordinary diÜerential equations that constitute nuclear reaction networks are surveyed. Three semi-implicit time integration algorithms are examined; a traditional —rst- orderaccurate Euler method, a fourth-orderaccurate Kaps-Rentrop method, and a variable-order Bader-Deu—hard method. These three integration methods are coupled to eight diÜerent linear algebra packages. Four of the linear algebra packages operate on dense matrices (LAPACK, LUDCMP, LEQS, GIFT), three of them are designed for the direct solution of sparse matrices (MA28, UMFPACK, Y12M), and one uses an iterative method for sparse matrices (BiCG). The scaling properties and behav- ior of the 24 combinations (3 time integration methods times 8 linear algebra packages) are analyzed by running each combination on seven diÜerent nuclear reaction networks. These reaction networks range from a hardwired 13 isotope a-chain and heavy-ion reaction network, which is suitable for most multidi- mensional simulations of stellar phenomena, to a 489 isotope reaction network, which is suitable for determining the yields of isotopes lighter than technetium in spherically symmetric models of Type II supernovae. Each of the time integration methods and linear algebra packages are capable of generating accurate results, but the efficiency of the various methodsevaluated across several diÜerent machine architectures and compiler optionsdiÜer dramatically. If the execution speed of reaction networks that contain less than about 50 isotopes is an overriding concern, then the variable-order Bader-Deu—hard time integration method coupled with routines generated from the GIFT matrix package or LAPACK with vendor-optimized BLAS routines is a good choice. If the amount of storage needed for any reaction network is a concern, then any of the sparse matrix packages will reduce the storage costs by 70%¨90%. When a balance between accuracy, overall efficiency, and ease of use is desirable, then the variable-order Bader-Deu—hard time integration method coupled with the MA28 sparse matrix package is a strong choice. Subject headings: hydrodynamicsmethods: numerical ¨ nuclear reactions, nucleosynthesis, abundancesstars: interiors