Unconditionally Stable and Robust Adjacent-Cell Diffusive Preconditioning of Weighted-Difference Particle Transport Methods Is Impossible

Abstract

We construct a particle transport problem for which there exists no preconditioner with a cell-centered diffusion coupling stencil that is unconditionally stable and robust. In particular we consider an asymptotic limit of the periodic horizontal interface (PHI) configuration wherein the cell height in both layers approaches zero like σ 2 while the total cross section vanishes like σ in one layer and diverges like σ −1 as σ→0 in the other layer. In such cases we show that the conditions for stability and robustness of the flat eigenmodes of the iteration residual imply instability of the modes flat in the y-dimension and rapidly varying in the x-dimension. Two assumptions are made in the proof. (i) Only cell-centered adjacent-cell preconditioners (AP) are considered; nevertheless numerical experiments with face-centered preconditioners of the diffusion synthetic acceleration (DSA) type on problem configurations with sharp material discontinuities suffer similar deterioration in spectral properties. (ii) The spatial weights of the arbitrarily high-order transport method of the nodal type and zeroth order (AHOT-N0) are used in the analysis; nevertheless similar results are expected for alternative spatial approximation methods as long as their spatial weights continuously approach the correct asymptotic limits: 0 and 1 for thin and thick cells, respectively. The result of the proof is verified by solving a finite approximation of PHI with three existing codes, two of which are not constrained by these assumptions. The spectral radii reported by the three codes behave as predicted by our analysis, i.e., reaching values that far exceed the maximum spectral radii determined via homogeneous model configuration analysis. This constitutes preliminary evidence that our conclusions might extend to a wider class of transport methods and acceleration schemes than is considered in the analysis.