Monte Carlo Methods

Direct Simulation Monte Carlo can be viewed as a Monte Carlo (i.e. random) method for solving the Boltzman Equations. Collisions are generated stochastically, based on the kinetic theory of gases. An approach like this is necessary whenever the Knudsen number is bigger than about 0.1.

DSMC consists of two basic steps. First, particles free stream for a time , positions appropriately updated, reflections etc from boundaries accounted. In the second step, a specified number of particles are selected for collision processing. Here, the interaction is assumed short-ranged, like a gas of hard spheres.

This second step required more explanation. The domain is divided into cells not larger than the mean free path of a particle. All particles in a given cell, are assumed to be candidates for collisions. The collision probablility for a particle pair is

Rather than computing this sum, the following is performed

1. Choose collision pair i and j at random
2. Accept the pair if , where is the maximum relative speed in the cell and r is a uniform deviate in (0,1).
3. If the pair is accepted, process
4. If the pair is rejected, or after processing of an accepted pair, return to 1 until the required number of candidate collisions has been processed.

Of course, evaluating is as expensive as the earlier double sum. Instead, we guess its value. note that the total number of collisions that should occur in a given cell is

where is the average of the (actual) relative velocities of all molecules ( again!). But

so we should select

candidate pairs. Thus our guess of factors into the computation in a natural way.

Collisions are processed by balancing incoming and outgoing momentum and energy.

Cell averages of density, momentum and energy can be computed at any timestep, and averaged.