Interdisciplinary Applied Mathematics

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15.2.1 The Schwarz Algorithm


To understand the Schwarz technique, consider two overlapping subdomains as shown in Figure 15.5(a). An alternating Schwarz method for this geometry can be summarized as follows:


Begin    :    n = 0; w^lri =    initial    condition


Repeat    :{    n = n + 1


Solve Luin) = f1    in    Q1    with BC uin) = uiS    ^    on Г1


Solve Luis’1 = f2    in    with BC uiS^ = u1n)    on Г2


} until convergence


(n)


where n is the iteration number, ui    is the solution in domain Qi at it


eration n, L is the partial differential operator describing the governing equations, and fi are forcing functions of position in domain Qi. Here BC refers to the operator imposing the boundary conditions. In the alternating Schwarz method, the subdomains are overlapped, and Dirichlet-type


Algorithm 1 Description of DSMC/Stokes (or Navier-Stokes (NS)) coupling in various overlapping Schwarz methods.



* Convergence requires the convergence of coupling iterations and the reduction of DSMC noise below a specified tolerance.

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