Smoothed Particle Hydrodynamics for astrophysical flows

In this paper we describe the application of the Smoothed Particle Hydrodynamics (SPH) method to the dynamics of protostellar discs, and in particular to the problem of characterising gravitational instabilities in such discs. Initially, we briefly describe the basic features of SPH, a Lagrangian mesh-free method of solving the equations of fluid dynamics which is widely used in astrophysics. We highlight those features of SPH that should be particularly kept in mind when simulating accretion discs -these include both physical and artificial viscosity- that directly affect the transport of angular momentum in the code. We then describe some recent advances in our understanding of the development and non-linear behaviour of gravitational instabilities in accretion discs, paying particular attention to the problem of numerical convergence within SPH simulations of self-gravitating discs.