  In this paper we present the analytic solutions for two test problems
involving two-fluid mixtures of dust and gas in an astrophysical context. The
solutions provide a means of benchmarking numerical codes designed to simulate
the non-linear dynamics of dusty gas. The first problem, dustybox, consists of
two interpenetrating homogeneous fluids moving with relative velocity
difference. We provide exact solutions to the full non-linear problem for a
range of drag formulations appropriate to astrophysical fluids (i.e., various
prescriptions for Epstein and Stokes drag in different regimes). The second
problem, dustywave consists of the propagation of linear acoustic waves in a
two-fluid gas-dust mixture. We provide the analytic solution for the case when
the two fluids are interacting via a linear drag term. Both test problems are
simple to set up in any numerical code and can be run with periodic boundary
conditions. The solutions we derive are completely general with respect to both
the dust-to-gas ratio and the amplitude of the drag coefficient. A stability
analysis of waves in a gas-dust system is also presented, showing that sound
waves in an astrophysical dust-gas mixture are linearly stable.
