This paper describes ZEUS-MP, a multi-physics, massively parallel, message-passing implementation of the ZEUS code. ZEUS-MP differs significantly from the thoroughly documented ZEUS-2D code, the completely undocumented (in peer-reviewed literature) ZEUS-3D code, and a marginally documented "version 1" of ZEUS-MP first distributed publicly in 1999. ZEUS-MP offers an MHD algorithm which is better suited for multidimensional flows than the ZEUS-2D module by virtue of modifications to the Method of Characteristics scheme first suggested by Hawley & Stone (1995). This MHD module is shown to compare quite favorably to the TVD scheme described by Ryu et al. (1998). ZEUS-MP is the first publicly-available ZEUS code to allow the advection of multiple chemical (or nuclear) species. Radiation hydrodynamic simulations are enabled via an implicit flux-limited radiation diffusion (FLD) module. The hydrodynamic, MHD, and FLD modules may be used, singly or in concert, in one, two, or three space dimensions. Additionally, so-called "1.5-D" and "2.5-D" grids, in which the "half-D" denotes a symmetry axis along which a constant but non-zero value of velocity or magnetic field is evolved, are supported. Self gravity may be included either through the assumption of a GM/r potential or a solution of Poisson's equation using one of three linear solver packages (conjugategradient, multigrid, and FFT) provided for that purpose. Point-mass potentials are also supported.Because ZEUS-MP is designed for large simulations on parallel computing platforms, considerable attention is paid to the parallel performance characteristics of each module in the code. Strong-scaling tests involving pure hydrodynamics (with and without self-gravity), MHD, and RHD are performed in which large problems (256 3 zones) are distributed among as many as 1024 processors of an IBM SP3. Parallel efficiency is a strong function of the amount of communication required between processors in a given algorithm, but all modules are shown to scale well on up to 1024 processors for the chosen fixed problem size.
The Padoan and Nordlund model of the stellar initial mass function (IMF ) is derived from low-order statistics of supersonic turbulence, neglecting gravity (e.g., gravitational fragmentation, accretion, and merging). In this work, the predictions of that model are tested using the largest numerical experiments of supersonic hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence to date ($1000 3 computational zones) and three different codes (Enzo, Zeus, and the Stagger code). The model predicts a power-law distribution for large masses, related to the turbulenceenergy power-spectrum slope and the shock-jump conditions. This power-law mass distribution is confirmed by the numerical experiments. The model also predicts a sharp difference between the HD and MHD regimes, which is recovered in the experiments as well, implying that the magnetic field, even below energy equipartition on the large scale, is a crucial component of the process of turbulent fragmentation. These results suggest that the stellar IMF of primordial stars may differ from that in later epochs of star formation, due to differences in both gas temperature and magnetic field strength. In particular, we find that the IMF of primordial stars born in turbulent clouds may be narrowly peaked around a mass of order 10 M , as long as the column density of such clouds is not much in excess of 10 22 cm À2 .
Many astrophysical applications involve magnetized turbulent flows with shock waves. Ab initio star formation simulations require a robust representation of supersonic turbulence in molecular clouds on a wide range of scales imposing stringent demands on the quality of numerical algorithms. We employ simulations of supersonic super-Alfvénic turbulence decay as a benchmark test problem to assess and compare the performance of nine popular astrophysical MHD methods actively used to model star formation. The set of nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER, and ZEUS. These applications employ a variety of numerical approaches, including both split and unsplit, finite difference and finite volume, divergence preserving and divergence cleaning, a variety of Riemann solvers, a range of spatial reconstruction and time integration techniques. We present a comprehensive set of statistical measures designed to quantify the effects of numerical dissipation in these MHD solvers. We compare power spectra for basic fields to determine the effective spectral bandwidth of the methods and rank them based on their relative effective Reynolds numbers. We also compare numerical dissipation for solenoidal and dilatational velocity components to check for possible impacts of the numerics on small-scale density statistics. Finally, we discuss convergence of various characteristics for the turbulence decay test and impacts of various components of numerical schemes on the accuracy of solutions. The nine codes gave qualitatively the same results, implying that they are all performing reasonably well and are useful for scientific applications. We show that the best performing codes employ a consistently high order of accuracy for spatial reconstruction of the evolved fields, transverse gradient interpolation, conservation law update step, and Lorentz force computation. The best results are achieved with divergence-free evolution of the magnetic field using the constrained transport method, and using little to no explicit artificial viscosity. Codes which fall short in one or more of these areas are still useful, but they must compensate higher numerical dissipation with higher numerical resolution. This paper is the largest, most comprehensive MHD code comparison on an application-like test problem to date. We hope this work will help developers improve their numerical algorithms while helping users to make informed choices in picking optimal applications for their specific astrophysical problems.
The most accurate measurements of magnetic fields in star-forming gas are based on the Zeeman observations analyzed by Crutcher et al. (2010). We show that their finding that the 3D magnetic field scales approximately as density 0.65 can also be obtained from analysis of the observed line-of-sight fields. We present two large-scale AMR MHD simulations of several thousand M of turbulent, isothermal, self-gravitating gas, one with a strong initial magnetic field (Alfvén Mach number M A,0 = 1) and one with a weak initial field (M A,0 = 10). We construct samples of the 100 most massive clumps in each simulation and show that they exhibit a power-law relation between field strength and density (n H ) in excellent agreement with the observed one. Our results imply that the average field in molecular clumps in the interstellar medium is B tot (n H ) ≈ 42n 0.65 H, 4 µG. Furthermore, the median value of the ratio of the line-ofsight field to density 0.65 in the simulations is within a factor of about (1.3, 1.7) of the observed value for the strong and weak field cases, respectively. The median value of the mass-to-flux ratio, normalized to the critical value, is 70 per cent of the line-of-sight value. This is larger than the 50 per cent usually cited for spherical clouds because the actual mass-to-flux ratio depends on the volume-weighted field, whereas the observed one depends on the mass-weighted field. Our results indicate that the typical molecular clump in the ISM is significantly supercritical (∼ factor of 3). The results of our strong-field model are in very good quantitative agreement with the observations of Li et al. (2009), which show a strong correlation in field orientation between small and large scales. Because there is a negligible correlation in the weak-field model, we conclude that molecular clouds form from strongly magnetized (although magnetically supercritical) gas, in agreement with the conclusion of Li et al. (2009).
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