The outstanding problem in star formation is the origin of the inhomogeneities that eventually collapse to form stars. The
observations tell us that magnetic fields are important on all scales since the magnetic pressure is generally significantly larger than the thermal pressure. On small scales, the magnetic Reynolds number associated with ambi-polar diffusion becomes small enough to place a severe restriction on the timestep for explicit schemes. The generation of structure on the smallest scales and the final collapse to form stars is dominated by self-gravity.
A code that can tackle these problems must therefore include MHD, self-gravity and ambi-polar diffusion. It must also be able to cope with changes in length scale of at least three orders of magnitude, which means that uniform grid Eulerian codes are of limited use.
As yet there have not been any calculations that include all these ingredients, but various aspects of the problem have been studied using two rather different approaches. The first uses Eulerian adaptive mesh grid (AMR) codes which automatically refine the grid in regions where the solution varies rapidly. The other is smoothed particle hydrodynamics (SPH), which is Lagrangian and therefore concentrates the resolution in regions with high density.
In this talk I will compare these methods and show that although the most recent versions of SPH might be competitive with AMR, most SPH simulations in the literature are chronically under resolved. This is true even when the numerical Jeans criterion is satisfied. I will also show that it is possible to construct a robust and efficient implicit algorithm for ambi-polar diffusion and the Hall effect that can be incorporated into an AMR code. As an illustration of the capabilities of AMR codes, I will present some calculations that demonstrate how thermal instability and the magnetic field can collaborate to produce density inhomogeneities.
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