A fast and explicit algorithm for simulating the dynamics of small dust grains with smoothed particle hydrodynamics

We describe a simple method for simulating the dynamics of small grains in a dusty gas, relevant to micron-sized grains in the interstellar medium and grains of centimetre size and smaller in protoplanetary discs. The method involves solving one extra diffusion equation for the dust fraction in addition to the usual equations of hydrodynamics. This ` diffusion approximation for dust' is valid when the dust stopping time is smaller than the computational timestep. We present a numerical implementation using smoothed particle hydrodynamics that is conservative, accurate and fast. It does not require any implicit timestepping and can be straightforwardly ported into existing 3D codes.