In this paper we present a method for solving the equations governing timedependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection-diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Our approach to adaptive refinement uses a nested hierarchy of logically-rectangular girds with simultaneous refinement of the girds in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithms's accuracy and convergence properties, and illustrate the behavior of the method. An additional example demonstrates the performance of the method on a more realistic problem, namely, a three-dimensional variable density shear layer.

We explore the dependence on spatial dimension of the viability of the neutrino heating mechanism of core-collapse supernova explosions. We find that the tendency to explode is a monotonically increasing function of dimension, with 3D requiring ∼40−50% lower driving neutrino luminosity than 1D and ∼15−25% lower driving neutrino luminosity than 2D. Moreover, we find that the delay to explosion for a given neutrino luminosity is always shorter in 3D than 2D, sometimes by many hundreds of milliseconds. The magnitude of this dimensional effect is much larger than the purported magnitude of a variety of other effects, such as nuclear burning, inelastic scattering, or general relativity, which are sometimes invoked to bridge the gap between the current ambiguous and uncertain theoretical situation and the fact of robust supernova explosions. Since real supernovae occur in three dimensions, our finding may be an important step towards unraveling one of the most problematic puzzles in stellar astrophysics. In addition, even though in 3D we do see pre-explosion instabilities and blast asymmetries, unlike the situation in 2D, we do not see an obvious axially-symmetric dipolar shock oscillation. Rather, the free energy available to power instabilites seems to be shared by more and more degrees of freedom as the dimension increases. Hence, the strong dipolar axisymmetry seen in 2D and previously identified as a fundamental characteristic of the shock hydrodynamics may not survive in 3D as a prominent feature.

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We study the statistics of the Lyα forest in a flat ΛCDM cosmology with the N-body + Eulerian hydrodynamics code Nyx. We produce a suite of simulations, covering the observationally relevant redshift range 2 z 4. We find that a grid resolution of 20 h −1 kpc is required to produce one percent convergence of Lyα forest flux statistics, up to k = 10 h −1 Mpc. In addition to establishing resolution requirements, we study the effects of missing modes in these simulations, and find that box sizes of L > 40 h −1 Mpc are needed to suppress numerical errors to a sub-percent level. Our optically-thin simulations with the ionizing background prescription of Haardt & Madau (2012) reproduce an IGM density-temperature relation with T 0 ≈ 10 4 K and γ ≈ 1.55 at z = 2, with a mean transmitted flux close to the observed values. When using the ionizing background prescription of Faucher-Giguère et al. (2009), the mean flux is 10-15 per cent below observed values at z = 2, and a factor of 2 too small at z = 4. We show the effects of the common practice of rescaling optical depths to the observed mean flux and how it affects convergence rates. We also investigate the practice of 'splicing' results from a number of different simulations to estimate the 1D flux power spectrum and show it is accurate at the 10 per cent level. Finally, we find that collisional heating of the gas from dark matter particles is negligible in modern cosmological simulations.

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