Introduction

We apply the technique of smoothed particle hydrodynamics (SPH), a gas-dynamical Lagrangian numerical scheme, to analyse the non-linear response of a gaseous disc to an imposed potential. For the first time, we compare SPH and semi-analytical results for a galaxy model with small spiral pitch angles. Density amplitudes, phases and general density profile shapes, including the subtle effects of the n = 2 ultra-harmonic resonance, are in good agreement throughout the disc. We therefore establish the applicability of SPH to wide range of problems involving density waves in galactic discs.

The SPH scheme, which is based on kernel estimates of the physical parameters of the gas, has been described elsewhere (e. g. Gingold & Monaghan 1982, Monaghan & Lattanzio 1985). In SPH, gradients of physical quantities, such as pressure, are expressed as gradients of the kernel alone through an integration by parts. The resulting SPH fluid equations contain bulk flow parameters, such as the gas sound speed and viscosity, which are explicitly specified. This helps constrain the problem and permits a stringent test of the technique. SPH has some major advantages over the familiar grid-based schemes: first, there is no need to solve the continuity equation separately; second, the convective terms in the momentum equations are represented exactly; third, the mesh is adaptive – it becomes dense wherever the gas density becomes large – which is important in the highly non-linear flows often found in galaxies.