The second-order upwind difference residual operators and implicit preconditioners are analyzed using the linearized 2-D Navier-Stokes equations. The purpose of alternating direction implicit preconditioner is to cluster residual eigenvalues tightly for high frequency Fourier error modes so that optimal-damping of the multistage relaxation can be obtained in the classical full-coarsening multigrid strategy. This study concentrats on the behaviors of the upwind schemes in the high frequency region and the preconditioning capability of the alternating direction implicit preconditioners.
A classical second-order upwind scheme and an alternate candidate of the second-order upwind scheme using the multi-dimensional stencil commonly used in the unstructured grid solver are analyzed in the Fourier domain. It is presented that the the second-order upwind scheme using the multi-dimensional stencil approaches to the first-order upwind scheme in the high frequency region. This means that the second-order scheme using the multi-dimensional stencil can be preconditioned better than the classical second-order upwind scheme by the implicit preconditioners based on the first-order upwind scheme, especially in the high frequency region. The preconditioning aspect of the upwind schemes are confirmed by using the Fourier footprints and convex hulls of the preconditioned residual eigenvalues.
Two kinds of alternating direction implicit preconditioners, ADI and DDADI preconditioners are compared on the aspect of eigenvalue clustering using the Fourier footprints and convex hulls. While the ADI scheme shows more stable than the other, DDADI preconditioner shows better preconditioning capability than ADI scheme.
To verify the present analysis numerical tests are performed for an inviscid transonic flow and turbulent flows past an airfoil. The DDADI preconditioned MUSCL-type linear reconstruction scheme shows best convergence on both cases and linear convergence characteristic from initial to machine accuracy when combined with the multigrid method. The Baldwin-Lomax turbulence model and the Spalart-Allmaras model are compared in the multigrid context to show that the latter has better convergence.