Interdisciplinary Applied Mathematics

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a parallel alternating Schwarz method, all the subdomains are solved at once; i.e., the subdomains S1, S2, D1, and D2 are all solved concurrently.

15.2.2 Interpolation Between Domains

In general, the position of the continuum nodes and the DSMC particles do not match in the overlapped regions. When a solution is computed in the DSMC subdomain, the solution for the continuum nodes in the DSMC subdomain can be obtained using an interpolation scheme. This interpolation scheme is referred to as the DSMC-to-continuum interpolation. Similarly, when a solution in the Stokes subdomain is computed, the solution for the DSMC particles (or DSMC cell centers) in the Stokes subdomain can again be computed by an interpolation scheme. This interpolation scheme will be referred to as the continuum-to-DSMC interpolation. Both continuum-to-DSMC and DSMC-to-continuum interpolation schemes can be implemented by using the meshless interpolation techniques described in Chapter 14. We explain the two interpolation techniques next.

DSMC-to-Continuum Interpolation

Let us denote by ci the DSMC cells that will be used to compute the solution at the continuum boundary nodes nj. Let xci denote the position of the DSMC cell center and xnj denote the position of a continuum boundary node at which the solution needs to be interpolated. Let uci denote the nodal    parameters    for the    solution    at    the    DSMC    cell    centers    (the    so

lution that needs to be interpolated can be pressure, velocity, etc.) and u(xnj) be    the    interpolated    solution    at    the    continuum    boundary    node    nj.

To compute u(xnj), a kernel or a weighting function is first centered at the position xnj. The kernel centered at xnj defines the cloud Hj and the number of    DSMC cell    centers,    NCj,    that lie    within    the    cloud    Hj    (see Figure    15.6(a)    for    the definition    of    the    cloud and    the    cells    that    fall    within

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