# Interdisciplinary Applied Mathematics

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(a)

 1 Si Di S2 D2 i i i i !

(b)

FIGURE 15.5. Decomposition of a sample geometry into (a) two overlapping subdomains, and (b) four subdomains.

boundary conditions are employed. The alternating Schwarz method can be modified to use nonoverlapping domains and Neumann-type boundary conditions, as explained in the previous section using a relaxation technique; see also (LeTallec and Mallinger, 1997), for a description on coupling time-dependent Navier-Stokes with DSMC using a nonoverlapped Schwarz coupling and Robin-type (mixed) boundary conditions. Several variations of the basic Schwarz technique for elliptic partial differential equations are presented in (Smith et al., 1996).

The alternating Schwarz method as described above is a serial technique. In Algorithm 1, the serial alternating Schwarz method and two parallel implementations of the Schwarz technique for coupling Stokes (or Navier-

Stokes) and DSMC subdomains are shown. To understand the implementation of various Schwarz techniques, consider the geometry and its decomposition shown    in    Figure 15.5(b).    Here    Si    denotes    the    Stokes    (or    the

Navier-Stokes) subdomain, and Di denotes the DSMC subdomain. In the serial alternating Schwarz method, each subdomain is solved sequentially, i.e., Si,    followed    by Di,    followed    by    S2,    and    so    on.    In a    colored Schwarz

method, the subdomains are divided into groups (i.e., colored), and the subdomains in each group are solved concurrently, while each group is solved sequentially. The optimal coloring depends on the geometry and its decomposition. For example, for the subdomains shown in Figure 15.5(b), all the Di,s are assigned one color, and all the Si,s are assigned a different color. All the Di,s    are    solved    at    once,    followed by the    solution    of    all    the    Si’s. In

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