On block-circulant preconditioners for high-order compact approximations of convection-diffusion problems

We study some properties of block-circulant preconditioners for high-order compact approximations of convection-diffusion problems. For two-dimensional problems, the approximation gives rise to a nine-point discretisation matrix and in three dimensions, we obtain a nineteen-point matrix. We derive analytical expressions for the eigenvalues of the block-circulant preconditioner and this allows us to establish the invertibility of the preconditioner in both two and three dimensions. The eigenspectra of the preconditioned matrix in the two-dimensional case is described for different test cases. Our numerical results indicate that the block-circulant preconditioning leads to significant reduction in iteration counts and comparisons between the high-order compact and upwind discretisations are carried out. For the unpreconditioned systems, we observe fewer iteration counts for the HOC discretisation but for the preconditioned systems, we find similar iteration counts for both finite difference approximations of constant-coefficient two-dimensional convection-diffusion problems.