An axis-free overset grid in spherical polar coordinates for simulating 3D self-gravitating flows

Abstract

Aims. Three dimensional explicit hydrodynamic codes based on spherical polar coordinates using a single spherical polar grid suffer from a severe restriction of the time step size due to the convergence of grid lines near the poles of the coordinate system. More importantly, numerical artifacts are encountered at the symmetry axis of the grid where boundary conditions have to be imposed that flaw the flow near the axis. The first problem can be eased and the second one avoided by applying an overlapping grid technique. Methods. A type of overlapping grid in spherical coordinates is adopted. This so called "Yin-Yang" grid is a two-patch overset grid proposed by Kageyama and Sato for geophysical simulations. Its two grid patches contain only the low-latitude regions of the usual spherical polar grid and are combined together in a simple manner. This property of the Yin-Yang grid greatly simplifies its implementation into a 3D code already employing spherical polar coordinates. It further allows for a much larger time step in 3D simulations using high angular resolution ( < ∼ 1 • ) than that required in 3D simulations using a regular spherical grid with the same angular resolution. Results. The Yin-Yang grid is successfully implemented into a 3D version of the explicit Eulerian grid-based code PROMETHEUS including self-gravity. The modified code successfully passed several standard hydrodynamic tests producing results which are in very good agreement with analytic solutions. Moreover, the solutions obtained with the Yin-Yang grid exhibit no peculiar behaviour at the boundary between the two grid patches. The code has also been successfully used to model astrophysically relevant situations, namely equilibrium polytropes, a Taylor-Sedov explosion, and Rayleigh-Taylor instabilities. According to our results, the usage of the Yin-Yang grid greatly enhances the suitability and efficiency of 3D explicit Eulerian codes based on spherical polar coordinates for astrophysical flows.

Parametrized 3D models of neutrino-driven supernova explosions

“…The simulation code employs an axis-free overlapping "Yin-Yang" grid (Kageyama & Sato 2004) in spherical polar coordinates, which was recently implemented into Prometheus, for spatial discretization (Wongwathanarat et al 2010a). The YinYang grid relaxes the CFL-timestep condition and avoids numerical artifacts near the polar axis.…”

“…This choice of parameters implies R ib ≈ 19 km at 1.3 s for models W15 and N20, and R ib ≈ 30 km at 1.4 s for the L15 models. Hydrostatic equilibrium is assumed at the inner radial grid boundary R ib , which is thus a Lagrangian (co-moving) position, while a free outflow boundary condition is employed at the outer radial grid boundary (for more details, see Wongwathanarat 2011;Wongwathanarat et al 2010a). …”

“…We have analyzed a set of 3D simulations (Wongwathanarat 2011;Wongwathanarat et al 2010b, and in prep.) based on two 15 M progenitor models (W15 and L15), and a 20 M progenitor model (N20).…”

Sasi Activity in Three-Dimensional Neutrino-Hydrodynamics Simulations of Supernova Cores

“…Excising the high-density core and replacing it by a contracting inner boundary condition, at which neutrino luminosities with chosen values could be imposed, allowed us to control the supernova core conditions. In these simulations we studied a 25 M ⊙ progenitor and made use of an axis-free Yin-Yang grid (Wongwathanarat et al 2010a) instead of a polar grid. These simulations did not only show that the discussed SASI phenomenon can play a role in different progenitor stars and is not dependent on the choice of a particular computational grid, they also confirmed that SASI mass motions in 3D can be triggered by the same conditions that are known to be favorable in 2D.…”

3D Core-Collapse Supernova Simulations: Neutron Star Kicks and Nickel Distribution

“…Details regarding the numerical implementation of the Yin-Yang grid for astrophysical self-gravitating systems are given in Wongwathanarat et al (2010a).…”