Unconditional effectiveness of cell-centered weighted-difference preconditioners

Determine whether cell-centered preconditioners for weighted-difference schemes possess unconditional effectiveness for heterogeneous slab transport problems, analogous to the unconditional effectiveness established in the paper for the variational diffusion synthetic acceleration iteration.

Background

The paper proves uniform convergence bounds for the variational diffusion synthetic acceleration source iteration and its Galerkin discretization in heterogeneous slab geometry. Specifically, the spectral radii are bounded by the infinite-medium rate, independently of slab thickness, spatial mesh, and angular resolution; the authors describe this as establishing unconditional effectiveness in the terminology of the cited literature.

The conclusion identifies an unresolved counterpart: Azmy had left open whether cell-centered preconditioners applied to weighted-difference transport schemes enjoy the same type of unconditional effectiveness. This question concerns a different class of discretizations and preconditioners than the variational method analyzed in the paper.