Application of DSMC method to astrophysical flows

The Direct Simulation Monte Carlo (DSMC) method, developed originally to calculate rarefied gas dynamical problems, is applied to continuous flow including shocks assuming that the Knudsen number is sufficiently small. In particular, we study the formation of spiral shocks in the accretion disc of a close binary system. The method involves viscosity and thermal conduction automatically, and can thus simulate turbulent viscosity. 1. The Direct Simulation Monte Carlo Method The Direct Simulation Monte Carlo method was invented by Bird (see Bird 1994). It was originally developed to treat rarefied gas flows. In DSMC, the microscopic molecular motion and their mutual collisions are calculated. Macroscopic quantities such as density, velocity and temperature of fluids are calculated by taking some mean of microscopic quantities. Molecular collisions are handled in a stochastic manner. The collision model in DSMC is such that the molecules, or more precisely, the collection of molecules considered as a particle, are assumed to be hard billiard balls. We divide space into many cells, each containing many such model molecules. Particles in a cell are assumed to have the possibility of mutual collisions. The direction of the particle motion after the collision is decided stochastically. Density can then be calculated by using the particles number in a cell. The fluid velocity in a cell is nothing but a mean velocity of molecules in the cell and we obtain the thermal velocity of a particle by subtracting the mean velocity in the cell from the particle velocity. We may then define the temperature of the gas using the thermal velocity of molecules. In the process of collision, the number and the total momentum of two particles are conserved. Total kinetic energy of two particles is conserved in an adiabatic case, but it may not conserve if we include cooling. We consider two method of cooling (see below).