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 it is the jeans condition. This sets the resolution smaller than the local Jean’s condition, 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 and refinement criteria

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).

To trigger refinement from level 0 to level 1, the code employs a density criterion to cells that contain gas of the original cloud. Such cells are identified if they contain a density greater than or equal to ½ the original cloud density at the outer edge. This results in the level 0 cells effectively removing the boundary of the computational volume from the surface of the cloud. Jean's condition is than used for refinement to 2nd and higher levels (discussed below).

The Jeans Condition/TrueLove Condition

Jeans' analysis of the linearized 1D gravitohydrodynamic (GHD) equations of a medium of infinite extent lead to the expression for the jeans length:

\lambda_j = (\frac{\pi c_s^2}{G \rho})^{1/2}$

which showed that perturbations that are larger than this are gravitationally unstable and collapse. Truelove ('97) showed that these instabilities can be triggered from numerical errors of the GHD solvers. Errors introduced on the cell scale at coarser grids can be transmitted to finer levels and can lead to 'artificial fragmentation'. A way to avoid this is to maintain resolution of the local jeans length. By defining:

J = \frac{\triangle x}{\lambda_j} $

Truelove ('97) showed that keeping

J \leqq 0.25$

prevented artificial fragmentation of an isothermal cloud spanning 7 decades of density. It is expected to be necessary albeit not necessarily sufficient for isothermal collapse. By not having the resolution of gradients can trigger artificial viscosity (used for numerical stability), which can lead to an incorrect formulation of the problem originally deemed inviscid.

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%.