192 1997 Sverker Holmgren, Henrik Brandén and Erik Sterner sverker@tdb.uu.se Convergence acceleration for the Navier-Stokes equations using optimal semicirculant approximations
Abstract
The iterative solution of systems of equations arising from partial
differential equations (PDE) governing boundary layer flow for large
Reynolds numbers is studied. We consider a convergence acceleration technique,
where an optimal 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. 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 PDE with variable coefficients. When utilizing the
semicirculant (SC) 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 results for the SC method to those for a multigrid (MG)
scheme. The number of iterations and the arithmetic complexities are
considered, and it clear that SC method is much more efficient for
problems where the Reynolds number is large. The number of iterations
for the MG method grows like the square root of the Reynolds number,
while the convergence rate for the SC method is independent of this
parameter. Also, the MG scheme is very sensitive to the level of
artificial dissipation, while the SC method is not.