We have defined and analyzed a semi-Toeplitz preconditioner for time-dependent and steady-state convection-diffusion problems. Analytic expressions for the eigenvalues of the preconditioned systems are obtained. An asymptotic analysis shows that the eigenvalue spectrum of the time-dependent problem is reduced to two eigenvalues when the number of grid points go to infinity. The numerical experiments sustain the results of the theoretical analysis, and the preconditioner exhibits a robust behavior for stretched grids.
A semi-Toeplitz preconditioner for the linearized Navier-Stokes equations for compressible flow is proposed and tested. The preconditioner is applied to the linear system of equations to be solved in each time step of an implicit method. The equations are solved with flat plate boundary conditions and are linearized around the Blasius solution. The grids are stretched in the normal direction to the plate and the quotient between the time step and the space step is varied. The preconditioner works well in all tested cases and outperforms the method without preconditioning both in number of iterations and execution time.
Note: Included papers available at http://www.it.uu.se/research/publications/lic/2002-007/2002-007-paperA.pdf, http://www.it.uu.se/research/publications/lic/2002-007/2002-007-paperA.ps.gz, http://www.it.uu.se/research/publications/lic/2002-007/2002-007-paperB.ps.gz, and http://www.it.uu.se/research/publications/lic/2002-007/2002-007-paperB.pdf
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