Truelove, Klein, Mc Kee, et al. (‘98). ​'Self-Gravitational hydrodynamics with 3d AMR: methodolgy and applications to molecular cloud collapse and fragmentation.'

Summary

Develops methods for AMR, dynamic grids that allot finer resolution over many length scales that is imperative for studying problems of gravitational collapse. The criteria for refinement is crucial, and their’s is the jeans length. This sets the resolution smaller than the local Jean’s length, allowing new benchmarks in the probing of the dynamics. They find uniformly rotating spherical clouds to collapse along the equatorial plane. When perturbed, these form ‘filamentary singularties’ that don’t fragment when isothermal.

Methodology

This paper goes through in extensive detail on the 3 components of their code methodology: the hyperbolic solvers that employ the Gudonov method for solution of the hydro equations, elliptic solvers that utilize AMR multigrid method to solve Poisson’s equation, and finally these two methods operation within an AMR framework. The use of stencils as different layers of cell-centered quantities used for averaging the node centered quantities (for self - gravity) is detailed. Paper refers to Almgren for discussion on AMR multigrid cycle procedures.

AMR

Paper refers to Pember et al. (‘98) for a summary of the procedure for advancing grids. Their methods are based on Berger and Oliger (‘84), Berger and Collela (‘89), and Bell et al. (‘94).

Truelove, Klein, McKee, et al. (‘97). The Jeans Condition: A new constraint on spatial resolution in simulations of isothermal self-gravitational hydrodynamics. ​http://adsabs.harvard.edu/abs/1997ApJ...489L.179T

Poisson's boundary conditions

The boundary condition they employ for self-gravity is periodic. They describe this as mimicking an infinite region of repeating boxes. Thus, when they model a cloud in the box using these boundary conditions, the solver for the gravity within this box, takes into account the gravitational effects of an infinite number of other spheres in boxes. To minimize the effect of these other spheres, they suggest making the distance between the center-to-center images d=4R. By increasing the d=8R, results in less gravitational pull by the mirrored clouds that hastens the collapse by only 1%.