Abstract

The iterative solution of systems of equations arising from partial differential equations (PDEs) governing boundary layer flow for large Reynolds numbers is studied. The PDEs are discretized using finite volume or finite difference approximations on tensor product grids. We consider a convergence acceleration technique, where a semicirculant approximation of the spatial difference operator is employed as preconditioner. A relevant model problem is derived, and the spectrum of the preconditioned coefficient matrix is analyzed. It is proved that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of gridpoints and the Reynolds number Re. The same type of result is also derived for finite size grids, where the solution fulfills a given accuracy requirement. By linearizing the Navier--Stokes equations around an approximate solution, we form a system of linear PDEs with variable coefficients. When utilizing the semicirculant preconditioner for this problem, which has properties very similar to the full nonlinear equations, the results show that the favorable convergence properties hold also here. We compare the semicirculant preconditioner to a multigrid scheme. The number of iterations and the arithmetic complexities are considered, and it clear that semicirculant method is much more efficient for problems where the Reynolds number is large. The number of iterations for the multigrid method grows like $\sqrt{Re}$, while the convergence rate for the scheme using semicirculant preconditioning is independent of Re. Also, the multigrid scheme is very sensitive to the level of artificial dissipation, while the method using semicirculant preconditioning is not.

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