# Interdisciplinary Applied Mathematics

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Indexing and cross-referencing of particles.

Simulation of collisions.

Sampling of macroscopic properties of the flow field.

The basic steps of a DSMC algorithm are given in Figure 15.1 and are summarized below.

The first step involves motion of the simulated molecules during a time interval of At. Since the molecules will go through intermolecular collisions, the overall time step for simulation is chosen smaller than the mean collision time ДА. Once the molecules are advanced in space, some of them will have gone through wall collisions or will have left the computational domain through the outflow boundaries. Hence, the boundary conditions must be enforced at this level, and the macroscopic properties along the solid surfaces must be sampled. This is done by modeling the surface molecule interactions by application of conservation laws on individual molecules, rather

FIGURE 15.1. Typical steps for a DSMC method. (Courtesy of E. Oran.)

than using a velocity distribution function (commonly used in Boltzmann equation algorithms; see sections 15.4 and 15.5). This approach allows inclusion of many other physical processes, such as the chemical reactions, radiation effects, three-body collision, and ionized flow effects, without major modifications to the basic DSMC procedure. However, a priori knowledge of the accommodation coefficients must be used in this process (see

Section 2.2.2), and this constitutes a weakness of the DSMC method, similar to the Navier-Stokes-based slip and even Boltzmann-equation-based simulation models. This issue is discussed in detail in the following section.

The second step is indexing and tracking of the particles. This is necessary, since during the first stage the molecules might have moved to a new cell. The new cell locations of the molecules are indexed; hence the intermolecular collisions and flow field sampling can be handled accurately. This is a very crucial step in an efficient DSMC algorithm. The indexing, molecule tracking, and data structuring algorithms should be carefully designed for specific computing platforms. Dietrich and Boyd presented DSMC calculations for more than 100 million simulated particles on 400-node IBM SP2 computer with 90% parallel efficiency (Dietrich and Boyd, 1996). Parallel efficiency of DSMC algorithms requires very effective load balancing methods based on the number of molecules, because the computational work of a DSMC method is proportional to the number of simulated molecules.

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