          Astron. Astrophys. 325, 857-865 (1997)

## 3. Restrictive use of von Neumann-Richtmeyer artificial viscosity

In SPH the formation of shocks is mostly an effect of the von Neumann-Richtmeyer viscosity. To avoid the undesirable effects in the subsonic region, I propose to restict the use of it to those regions. This modification will allow shocks to form, and prevent interparticle penetration at supersonic velocities.

Consider a gas cloud in a spherical symmetric gravitational collapse. Suppose the physical viscosity is small, so that the compression can be assumed to be adiabatic. When the model has reached an equilibrium, the pressure force that prevents further gravitational compression balance the gravitational force. If the cloud is warm, the forces may balance each other even in its initial state. But if on the other hand the cloud is initially cool, it must be compressed to gain the required thermal energy density, perhaps by orders of magnitude. In many cases this is not possible with standard artificial viscosity, Eq. (6), due to poor resolution. If the cloud in the example above has a radius of unit length and is modelled with particles, the mean interparticle distances are about 0.1. In the spherical compression the particles at different radii therefore have supersonic relative velocities if the gas is cool enough. Standard artificial viscosity, Eq. (6), will decelerate and heat the gas, and prevent further compression.

Since the time derivative for any particle can be written This relation can be used to decide whether a particle follows the fluid or if artificial viscosity is necessary. Consider two particles with a separation of r moving towards each other with a speed of they follow the fluid in the neighbourhood, the neighbourhood is under compression and the particles' relative velocity will approximately satisfy If two particles are approaching each other at a velocity exceeding the sound speed, that is if artificial viscosity is necessary to prevent interparticle penetration. I therefore propose to restrict the von Neumann-Richtmeyer artificial viscosity as     © European Southern Observatory (ESO) 1997

Online publication: April 28, 1998 