Conditions for justifying single-fluid approximation for charged and neutral dust fluids and a smoothed particle magnetohydrodynamics method for dust-gas mixture

We describe a numerical scheme for magnetohydrodynamics simulations of dust-gas mixture by extending smoothed particle magnetohydrodynamics. We employ the single-species particle approach to describe dust-gas mixture with several modifications from the previous studies. We assume that the charged and neutral dusts can be treated as single-fluid and the electro-magnetic force acts on the gas and that on the charged dust is negligible. The validity of these assumption in the context of protostar formation is not obvious and is extensively evaluated. By investigating the electromagnetic force and electric current with terminal velocity approximation, it is found that as the dust size increases, the contribution of dust to them becomes smaller and negligible. We conclude that our assumptions of the electro-magnetic force on the dusts is negligible are valid for the dust size with a d & 10{\mu}m. On the other hand, they do not produce the numerical artifact for the dust a d . 10{\mu}m in envelope and disk where the perfect coupling between gas and dusts realizes. However, we also found that our assumptions may break down in outflow (or under environment with very strong magnetic field and low density) for the dust a d . 10{\mu}m. We conclude that our assumptions are valid in almost all cases where macroscopic dust dynamics is important in the context of protostar formation. We conduct numerical tests of dusty wave, dusty magnetohydrodynamics shock, and gravitational collapse of magnetized cloud core with our simulation code. The results show that our numerical scheme well reproduces the dust dynamics in the magnetized medium.