Smoothed Particle Magnetohydrodynamics – I. Algorithm and tests in one dimension

In this paper we show how the Smoothed Particle Hydrodynamics (SPH) equations for ideal Magnetohydrodynamics (MHD) can be written in conservation form with the positivity of the dissipation guaranteed. We call the resulting algorithm Smoothed Particle Magnetohydrodynamics (SPMHD). The equations appear to be accurate, robust and easy to apply and do not suffer from the instabilities known to exist previously in formulations of the SPMHD equations. In addition we formulate our MHD equations such that errors associated with non-zero divergence of the magnetic field are naturally propagated by the flow and should therefore remain small. In this and the companion paper we present a wide range of numerical tests in one dimension to show that the algorithm gives very good results for one-dimensional flows in both adiabatic and isothermal MHD. For the one-dimensional tests the field structure is either twoor threedimensional. The algorithm has many astrophysical applications and is particularly suited to star-formation problems.