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DUSTYBOX and DUSTYWAVE: Two test problems for numerical simulations of two fluid astrophysical dust-gas mixtures

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.