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Integration of Nuclear Reaction Networks for Stellar Hydrodynamics (2000)
In this article, methods for solving the stiff system of ordinary differential equations that constitute nuclear reaction networks are surveyed. Three semiimplicit time integration algorithms are examined; a traditional firstorderaccurate Euler method, a fourthorderaccurate KapsRentrop method, and a variableorder BaderDeuflhard method. These three integration methods are coupled to eight different 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 behavior of the 24 combinations (3 time integration methods times 8 linear algebra packages) are analyzed by running each combination on seven different nuclear reaction networks. These reaction networks range from a hardwired 13 isotope $\alpha$chain and heavyion reaction network, which is suitable for most multidimensional 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 methods  evaluated across several different machine architectures and compiler options  differ dramatically. If the execution speed of reaction networks that contain less than about 50 isotopes is an overriding concern, then the variableorder BaderDeuflhard time integration method coupled with routines generated from the GIFT matrix package or LAPACK with vendoroptimized 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 variableorder BaderDeuflhard time integration method coupled with the MA28 sparse matrix package is a strong choice.
An Inexpensive Nuclear Energy Generation Network for Stellar Hydrodynamics (2000) In this article, we compare the nuclear energy generation rate and abundance levels given by an $\alpha$chain nuclear reaction network that contains only seven isotopes with a standard 13 isotope αchain reaction network. The energy generation rate of these two small networks are also compared to the energy generation rate given by a 489 isotope reaction network with weak reactions turned on and off. The comparison between the seven isotope and αchain reaction networks indicate the extent to which one can be replaced by the other, and the comparison with the 489 isotope reaction network roughly indicates under what physical conditions it is safe to use the seven isotope and $\alpha$chain reaction networks. The seven isotope reaction network reproduces the nuclear energy generation rate of the standard $\alpha$chain reaction network to within 30%, but often much better, during hydrostatic and explosive helium, carbon, and oxygen burning. It will also provide energy generation rates within 30% of an $\alpha$chain reaction network for silicon burning at densities less than 10$^{7}$ g cm$^{3}$. Provided there remains an equal number of protons and neutrons (Y$_e$ = 0.5) over the course of the evolution, and that flows through $\alpha$ particle channels dominate, then both of the small reaction networks return energy generation rates that are compatible with the energy generation rate returned by the 489 reaction network. If Y$_e$ is significantly different from 0.5, or if there are substantial flows through neutron and protons channels, then it is not generally safe to employ any $\alpha$chain based reaction network. The relative accuracy of the 7 isotope reaction network, combined with its reduction in the computational cost, suggest that it is a suitable replacement for $\alpha$chain reaction networks for parameter space surveys of a wide class of multidimensional stellar models.
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