# Interdisciplinary Applied Mathematics

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Number of time steps in DSMC: The selection of the number of DSMC time steps, Nstep, during each coupled iteration is important for the efficiency of the coupled method. Two issues need to be considered in selecting Nstep. The first issue deals with the noise in the DSMC solution. The noise considerations that were discussed in connection with particle weight also apply for the selection of Nstep. In the simulations reported here, the particle weight, wp, is reduced proportionally as Nstep is decreased. The second issue that needs to be considered in selecting Nstep is the time-dependent nature of the DSMC solution computed during the coupled iterations. Starting from an initial state, the multiscale coupled algorithm will take a certain number of iterations to compute a converged solution. During each iteration, the flow in the DSMC subdomain will evolve in a time-accurate manner toward a steady-state solution determined by the boundary conditions. If Nstep is large enough during each coupled iteration, a steady-state solution can be reached. However, there is no need to compute steady-state solutions during each coupled iteration because the boundary conditions enforced on the DSMC subdomain are not necessarily steady-state boundary conditions. Since the goal is to compute a steady-state solution for the entire system (including both Stokes and DSMC subdomains), Nstep can be selected shorter, and the boundary conditions can be updated in an iterative manner until a steady-state solution is reached.

To investigate the effect of Nstep, the filter geometry is simulated by keeping Nstep/wp constant while Nstep is changed. This keeps the noise in the DSMC estimates at the same level. Nstep values of 200, 1000, 5000, and 25,000 are investigated. The convergence of the pressure boundary condition in the input section was investigated (see (Aktas and Aluru, 2002), for convergence plots). We observed that for Nstep = 200 and 1000, a larger number of coupling iterations are needed for convergence when compared to Nstep = 5000, and 25,000. A comparison of the total simulated DSMC iterations until convergence shows that for Nstep = 200 and 1000, the total DSMC iterations are approximately equal. However, for the other two cases, the DSMC iterations until convergence are much larger. Thus, we can conclude that if Nstep x Ncpi (where Ncpi is the number of coupling iterations until convergence) is longer than the number of iterations the DSMC subdomain takes to reach a steady state, convergence is determined by the properties of the coupling method, whereas if Nstep x Ncpl is smaller, then the DSMC subdomain will evolve in a quasi-static manner and Ncpl will be increased. For an efficient implementation, Nstep x Ncpl should be close to the time constant of the DSMC subdomain.

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