# Interdisciplinary Applied Mathematics

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As far as frequency of coupling is concerned, the time step restrictions on the Navier-Stokes side are dictated by the Courant number (CFL; AtU/Ax) and most probably the diffusion number (Atv/Ax2) for the very low Reynolds number we consider, whereas on the DSMC side the time step is controlled by the mean collision time /c. This suggests that coupling should take place at almost every time step if the diffusion time step constraint dominates or a subcycling procedure should be followed, in which one Navier-Stokes step is followed by many DSMC steps if the convective time step dominates.

To understand the various algorithmic and implementation issues, we consider the example of multiscale analysis of microfilters. The specific algorithmic issues are elaborated below and in the next section, and the results are presented in the subsequent section. A high-level description of the multiscale method is presented in Algorithm 1. Given an arbitrary initial state and a set of boundary conditions along the overlapping interfaces, a Schwarz technique is implemented to find a self-consistent solution to the Stokes (or Navier-Stokes) and the DSMC subdomains. Self-consistency is determined by a convergence check that requires that the DSMC noise as well as the updates to the solution (for example, pressure and the velocities) be less than a specified tolerance value. Self-consistency also ensures that the boundary conditions at the interface have converged to the specified tolerance. To further reduce the DSMC noise in the solution, a postprocessing step is performed after the initial convergence check is satisfied. In the postprocessing step, several coupling iterations between Stokes (or Navier-Stokes) and DSMC subdomains are performed. The final results in the DSMC subdomain are obtained as an average of the samples collected during the postprocessing step. As a last step, the results from the DSMC subdomains are used as boundary conditions to find the solution in the continuum subdomains. The Schwarz algorithm and the interpolation between the Stokes (or Navier-Stokes) and the DSMC domains is discussed next.

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