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

Max-Planck Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85740 Garching, Germany e-mail: emueller@mpa-garching.mpg.de

Received: 9 October 2009
Accepted: 3 March 2010

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.