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# 8.4 Direct Simulation Monte Carlo / Bird method

Originally developed for dilute gas flow in engineering and in space science [BIRD 94], [NANBU 83].
Recent extensions for dense gases: [ALEXANDER 95]

Basic idea of the DSMC method for a dilute gas of hard spheres:
1. - Divide the sample into cells of volume , each with particles with given positions and velocities.
- The side length of the cells should be smaller than but of the order of the mean free path.
- Boundary conditions appropriate to the problem are defined, usually specular (reflecting wall) and/or periodic boundaries.
- A time step smaller than the typical intercollision time is assumed.
2. Translate all particles according to , applying the given boundary conditions.
3. Within each cell, draw pairs of particles that are candidates for a collision:
1. Let the probability of a pair to collide depend only on their relative speed and not to their positions. The argument for this is that all particles in one cell are within free path range of each other. The probability for the pair to collide is thus simply proportional to the relative speed: .
Recalling the rejection method of Section we draw pairs in accordance with this probability density: assuming the maximum of for all pairs in the cell to be known, draw a random number from a uniform distribution in and compare it to .
Calculating would amount to the expensive scanning of all pairs of particles in the cell. Therefore, use an estimated value of . If that value is larger than the actual , the density is still sampled correctly but with a slightly lower efficiency.
2. The total number of collision pairs to be sampled in a cell during one time step is determined as follows. For a gas of hard spheres with diameter the average number of pair collisions within the cell is
 (8.89)

where is the kinetic collision rate per unit volume, is the number density, and is the average relative speed. In order to have trial pairs survive the rejection procedure of step 3a we have to sample
 (8.90)

collisions.
4. Now perform the collision for the pair . Since the post-collision velocities are determined by the impact parameter which is unknown, they must be sampled in a physically consistent way. In the hard sphere case this is most easily done by assuming an isotropic distribution of the relative velocity after the collision. Since the relative speed remains unchanged, the problem is reduced to sampling a uniformly distributed unit vector. Marsaglia's recipe may be used for this (see Figure 3.0.2).