SPH without a Tensile Instability

The tensile instability in smoothed particle hydrodynamics results in a clustering of smoothed particle hydrodynamics (SPH) particles. The clustering is particularly noticeable in materials which have an equation of state which can give rise to negative pressures, but it can occur in gases where the pressure is always positive and in magnetohydrodynamics (MHD) problems. It is a particular problem in solid body computations where the instability may corrupt physical fragmentation by numerical fragmentation which, in some cases, is so severe that the dynamics of the system is completely wrong. In this paper it is shown how the instability can be removed by using an artificial stress which, in the case of fluids, is an artificial pressure. The method is analyzed by examining the dispersion relation for small oscillations in a fluid with a stiff equation of state. The short and long wavelength limits of the dispersion relation indicate appropriate parameters for the artificial pressure and, with these parameters, the errors in the long wavelength limit are small. Numerical studies of the dispersion relation for a wide range of parameters confirm the approximate analytical results for the dispersion relation. Applications to several test problems show that the artificial stress works effectively. These problems include the evolution of a region with negative pressure, extreme expansion in one dimension, and the collision of rubber cylinders. To study this latter problem the artificial pressure is generalized to an artificial stress. The results agree well with the calculations of other stable codes.