  We present a stable and convergent method for studying a system of gas and
dust, coupled through viscous drag in both non-stiff and stiff regimes. To
account for the effects of dust drag in the update of the fluid quantities, we
employ a fluid description of the dust component and study the modified
gas-dust hyperbolic system following the approach in Miniati & Colella (2007).
In addition to two entropy waves for the gas and dust components, respectively,
the extended system includes three waves driven partially by gas pressure and
partially by dust drift, which, in the limit of vanishing coupling, tend to the
two original acoustic waves and the unhindered dust streaming. Based on this
analysis we formulate a predictor step providing first order accurate
reconstruction of the time-averaged state variables at cell interfaces, whence
a second order accurate estimate of the conservative fluxes can be obtained
through a suitable linearized Riemann solver. The final source term update is
carried out using a one-step, second order accurate, L-stable, predictor
corrector asymptotic method (the alpha-QSS method suggested by Mott et. al.
2000). This procedure completely defines a two-fluid method for gas-dust
system. Using the updated fluid solution allows us to then advance the
individual particle solutions, including self-consistently the time evolution
of the gas velocity in the estimate of the drag force. This is done with a
suitable particle scheme also based on the alpha-QSS method. A set of benchmark
problems shows that our method is stable and convergent. When dust is modeled
as a fluid (two-fluid) second order accuracy is achieved in both stiff and
non-stiff regimes, whereas when dust is modeled with particles (hybrid) second
order is achieved in the non-stiff regime and first order otherwise.
