Interdisciplinary Applied Mathematics

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15.1 Direct Simulation Monte Carlo (DSMC) Method

The direct simulation Monte Carlo method was first developed by G. Bird, its inventor and pioneer (Bird, 1994). Several review articles about the DSMC method are also available (Bird, 1978; Muntz, 1989; Cheng and Emmanuel, 1995; Oran et al., 1998). Most of these articles present reviews of the DSMC method for low-pressure rarefied gas flow applications, with the exception of (Oran et al., 1998), where microfluidic applications are also addressed. The reader is referred to these books and articles for technical algorithmic details and sample DSMC programs.

Under standard conditions in a volume of 10 p,m x 10 p,m x 10 p,m there are about 1010 molecules. A molecular-based simulation model that can compute the motion and interactions of all these molecules is not possible. The typical DSMC method employs hundreds of thousands or even millions of “simulated” molecules that mimic but do not follow exactly the motion of real molecules.

DSMC is based on the splitting of the molecular motion and intermolecular collisions by choosing a time step smaller than the mean collision time, and tracking the evolution of this molecular process in space and time. For efficient numerical implementation, the space is divided into cells similar to the finite volume method. The DSMC cells are chosen proportional to the mean free path A. In order to resolve large gradients in flow with realistic    (physical)    viscosity    values,    the    average    cell    size    Дхс    is about    A/3

(Oran et al., 1998). The time- and cell-averaged molecular quantities are presented as the macroscopic values at cell centers.

DSMC involves four main steps:

   Motion of the particles.

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