Uniform convergence of diffusion synthetic acceleration for heterogeneous slab transport

Abstract: The diffusion synthetic accelerated (DSA) source iteration is a standard solver for the radiative transfer equation. Using Fourier analysis, a convergence rate (ρ\infty(c)\le0.2247\,c) with (c) the maximum ratio of scattering to total cross section has been established for an infinite homogeneous medium. For slab geometry with inflow boundary conditions and arbitrary bounded cross sections we prove that the spectral radius of the DSA iteration is at most (ρ\infty(c)). We show that this convergence rate carries over to a variational discretization of the DSA iteration on every conforming tensor-product Galerkin space whose angular factor contains the constants. The analysis rests on an exact min--max characterization of the spectral radius. From it we derive a checkable sufficient condition for a given rate. We verify this condition using a suitable energy-stable projection, a weighted angular average whose weight is chosen so that the condition holds with the rate (ρ_\infty(c)). Since the projection maps discrete spaces into discrete spaces, the argument applies verbatim to the discrete iteration. As a consequence, the condition number of the preconditioned system is uniformly bounded by (1+0.29\,c), which implies rapid convergence of the preconditioned conjugate gradients method.