Direct simulation Monte Carlo (DSMC) method uses probabilistic Monte Carlo simulation to solve the Boltzmann equation for finite Knudsen number fluid flows.
The DSMC method was proposed by Graeme Bird,[1][2][3] emeritus professor of aeronautics, University of Sydney.
DSMC is a numerical method for modeling rarefied gas flows, in which the mean free path of a molecule is of the same order (or greater) than a representative physical length scale (i.e. the Knudsen number Kn is greater than 1).
The DSMC method has been extended to model continuum flows (Kn < 1) and the results can be compared with Navier Stokes solutions.
The DSMC method models fluid flows using probabilistic simulation molecules to solve the Boltzmann equation.
Molecules are moved through a simulation of physical space in a realistic manner that is directly coupled to physical time such that unsteady flow characteristics can be modeled.
[6] Currently, the DSMC method has been applied to the solution of flows ranging from estimation of the Space Shuttle re-entry aerodynamics to the modeling of microelectromechanical systems (MEMS).
The direct simulation Monte Carlo algorithm is like molecular dynamics in that the state of the system is given by the positions and velocities of the particles,
Unlike molecular dynamics, each particle in a DSMC simulation represents
molecules in the physical system that have roughly the same position and velocity.
This allows DSMC to rescale length and time for the modeling of macroscopic systems (e.g., atmospheric entry).
As a rule of thumb there should be 20 or more particles per cubic mean free path for accurate results.
[citation needed] The evolution of the system is integrated in time steps,
At each time step all the particles are moved and then a random set of pairs collide.
In the absence of external fields (e.g., gravity) the particles move ballistically as
After all the particles have moved, they are sorted into cells and some are randomly selected to collide.
based on probabilities and collision rates obtained from the kinetic theory of gases.
After the velocities of all colliding particles have been reset, statistical sampling is performed and then the process is repeated for the next time step.
Typically the dimension of a cell is no larger than a mean free path.
All pairs of particles in a cell are candidate collision partners, regardless of their actual trajectories.
In the hard spheres model, the collision probability for the pair of particles,
Because of the double sum in the denominator it can be computationally expensive to use this collision probability directly.
Instead, the following rejection sampling scheme can be used to select collision pairs: This procedure is correct even if the value of
Writing the relative velocity in terms of spherical angles,
these angles are selected by a Monte Carlo process with distributions given by the collision model.
Note that by conservation of linear momentum and energy the center of mass velocity and the relative speed are unchanged in a collision.
, given by kinetic theory the total number of hard sphere collisions in a cell during a time
The number of collision candidates selected in a cell over a time step
This approach for determining the number of collisions is known as the No-Time-Counter (NTC) method.
is set excessively high then the algorithm processes the same number of collisions (on average) but the simulation is inefficient because many candidates are rejected.