Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners
We propose preconditioning as a viable acceleration scheme for the inner iterations of transport calculations in slab geometry. In particular we develop Adjacent-Cell Preconditioners (AP) that have the same coupling stencil as cell-centered diffusion schemes. For lowest order methods, e.g., Diamond Difference, Step, and 0-order Nodal Integral Method (0NIM), cast in a Weighted Diamond Difference (WDD) form, we derive AP for thick (KAP) and thin (NAP) cells that for model problems are unconditionally stable and efficient. For the First-Order Nodal Integral Method (1NIM) we derive a NAP that possesses similarly excellent spectral properties for model problems, [Note that the order of NIM refers to the truncated order of the local expansion of the cell and edge fluxes in Legendre series.] The two most attractive features of our new technique are: (1) its cell-centered coupling stencil, which makes it more adequate for extension to multidimensional, higher order situations than the standard edge-centered or point-centered Diffusion Synthetic Acceleration (DSA) methods; and (2) its decreasing spectral radius with increasing cell thickness to the extent that immediate pointwise convergence, i.e., in one iteration, can be achieved for problems with sufficiently thick cells. We implemented these methods, augmented with appropriate boundary conditions and mixing formulas for material heterogeneities, in the test code AP1D that we use to successfully verify the analytical spectral properties for homogeneous problems. Furthermore, we conduct numerical tests to demonstrate the robustness of the KAP and NAP in the presence of sharp mesh or material discontinuities. We show that the AP for WDD is highly resilient to such discontinuities, but for 1NIM a few cases occur in which the scheme does not converge; however, when it converges, AP greatly reduces the number of iterations required to achieve convergence.
neutral particle transport adjacent-cell preconditioner preconditioning spectral analysis