Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners

We propose preconditioning as a viable acceleration scheme for the inner it erations of transport calculations in slab geometry. In particular we devel op Adjacent-Cell Preconditioners (AP) that have the same coupling stencil a s cell-centered diffusion schemes. For lowest order methods, e.g., Diamond Difference, Step, and 0-order Nodal Integral Method (0NIM), cast in a Weigh ted Diamond Difference (WDD) form, we derive AP for thick (KAP) and thin (N AP) cells that for model problems are unconditionally stable and efficient. For the First-Order Nodal Integral Method (1NIM) we derive a NAP that poss esses similarly excellent spectral properties for model problems, [Note tha t 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 featu res of our new technique are: (1) its cell-centered coupling stencil, which makes it more adequate for extension to multidimensional, higher order sit uations than the standard edge-centered or point-centered Diffusion Synthet ic Acceleration (DSA) methods; and (2) its decreasing spectral radius with increasing cell thickness to the extent that immediate pointwise convergenc e, i.e., in one iteration, can be achieved for problems with sufficiently t hick cells. We implemented these methods, augmented with appropriate bounda ry conditions and mixing formulas for material heterogeneities, in the test code AP1D that we use to successfully verify the analytical spectral prope rties 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 resilie nt to such discontinuities, but for 1NIM a few cases occur in which the sch eme does not converge; however, when it converges, AP greatly reduces the n umber of iterations required to achieve convergence.