# Interdisciplinary Applied Mathematics

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FIGURE 15.8. Comparison of pressure along the midline of the channel obtained from the DSMC only and coupled simulations (hc = 0.2 pm, dov = 0 pm).

x-velocity

FIGURE 15.9. Comparison of velocity along the midline of the channel obtained from the DSMC only and coupled simulations (hc = 0.2 pm, dov = 0 pm).

the other dov cases.

The convergence results are summarized in Figures 15.10 and 15.11. Figure 15.10    shows    the    convergence    of    the    pressure    boundary    condition    at

the input section (see (Aktas and Aluru, 2002) for convergence of velocity boundary condition). In Figure 15.10, the pressure boundary condition

FIGURE 15.10. Convergence of pressure boundary condition transferred from the continuum side to the DSMC subdomain for different overlaps. Nstep = 5000 is used.

is plotted    by    computing    the    average    of    the    boundary    condition    imposed

on all the buffer cells. The figure indicates that the convergence is weakly dependent on    the    overlap    size    for    this    problem,    with    larger    overlap size

exhibiting slightly better convergence. In order to further investigate the dependence of the convergence characteristics on the overlap size, we show the absolute error in the pressure boundary condition transferred from the Stokes subdomain to the DSMC subdomain in Figure 15.11. Absolute error is defined as the absolute deviation between the computed value and an exact value, which is determined from the DSMC-only simulation. Again, the values plotted are the average of all the DSMC buffer cells. The results in Figure 15.11 were obtained for Nstep = 5000. This result shows that the number of iterations until convergence is weakly dependent on the overlap size. A mathematical analysis of the Schwarz method for elliptic problems reveals that a faster convergence should be obtained for a larger overlap (Smith et al., 1996). However, the result in Figure 15.11 indicates clearly that the number of coupling iterations is only weakly dependent on the overlap size. The observed behavior leads to the conclusion that the Stokes and the DMSC subdomains are weakly coupled for this example.

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