Time-dependent, Multifluid, Magnetohydrodynamic Shock Waves with Grain Dynamics. I. Formulation and Numerical Tests

This is the first in a series of papers on the effects of dust on the formation, propagation, and structure of nonlinear MHD waves and MHD shocks in weakly ionized plasmas. We model the plasma as a system of nine interacting fluids, consisting of the neutral gas, ions, electrons, and six types of grain fluids comprised of very small grains or polycyclic aromatic hydrocarbons and classical grains in different charge states. We formulate the governing equations for perpendicular shocks under approximations appropriate for dense molecular clouds, protostellar cores, and protoplanetary disks. We describe a code that obtains numerical solutions using a finite difference method, and we establish its accuracy by comparing numerical and exact solutions for special cases.