Spectral Green’s function nodal method for multigroup S N problems with anisotropic scattering in slab-geometry non-multiplying media Welton A. Menezes, Hermes Alves Filho, Ricardo C. Barros ⇑ Programa de Pós-graduação em Modelagem Computacional, Instituto Politécnico – IPRJ, Universidade do Estado do Rio de Janeiro – UERJ, P.O. Box 97282, 28610-974 Nova Friburgo, RJ, Brazil article info Article history: Received 21 July 2013 Accepted 11 September 2013 Available online 7 November 2013 Keywords: Neutral particle transport Discrete ordinates Spectral nodal method Energy multigroup model Anisotropic scattering abstract A generalization of the spectral Green’s function (SGF) method is developed for multigroup, fixed-source, slab-geometry discrete ordinates (S N ) problems with anisotropic scattering. The offered SGF method with the one-node block inversion (NBI) iterative scheme converges numerical solutions that are completely free from spatial truncation errors for multigroup, slab-geometry S N problems with scattering anisotropy of order L, provided L < N. As a coarse-mesh numerical method, the SGF method generates numerical solutions that generally do not give detailed information on the problem solution profile, as the grid points can be located considerably away from each other. Therefore, we describe in this paper a tech- nique for the spatial reconstruction of the coarse-mesh solution generated by the multigroup SGF method. Numerical results are given to illustrate the method’s accuracy. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction In realistic neutral particle transport calculations, it is necessary to consider an approximation of the energy-dependent transport equation in which the energy variable is discretized into contigu- ous groups, giving rise to the classic multigroup approximation. In this paper, we generalize the spectral Green’s function (SGF) methods (Barros and Larsen, 1990, 1991) to obtain numerical solu- tions of multigroup, slab-geometry, fixed-source discrete ordinates (S N ) problems with anisotropic scattering that are free from spatial truncation errors. That is, the offered SGF method generates numerical values for the node-edge and node-average angular fluxes that exactly agree with those of the analytical solution of the S N equations for an arbitrary number G of energy groups in the multigroup model and a given order L of anisotropy of the dif- ferential scattering macroscopic cross section in the direction-of- motion variable, provided L < N, with N being the order of the angular quadrature set used in the S N model. There are four essential ingredients in the multigroup SGF method: i. A complete set of basis functions to the kernel of the S N dif- ferential operator is constructed in each spatial node of the discretization grid. We remark that these basis functions differ from one node to the next if the cross sections are different; otherwise they are the same. ii. A system of algebraic linear equations is formulated and is composed of the conventional multigroup discretized S N spatial balance equations together with auxiliary equations that express each group node-average angular flux in terms of the node-edge fluxes entering the node in all energy groups. We remark that each equation is satisfied by every basis function that we referred to in part i., as the auxiliary equations have parameters that are to be determined to pre- serve the complete set of basis functions. iii. The system of equations obtained in part ii is solved itera- tively, using ‘‘one-node block inversions’’; i.e., all the esti- mated multigroup incident fluxes on each node of the spatial grid are used to calculate new exiting fluxes, which are then used to improve the estimated multigroup incident fluxes for the contiguous nodes, until a prescribed stopping criterion is satisfied. iv. With the converged multigroup node-edge angular fluxes, we use the expression of the local general solution to solve for the unknown constants and determine the angular flux at each point inside the discretization node. As with the first ingredient, we clearly can express the multigroup S N solu- tion in each discretization node as a linear combination of these G  N basis functions and assign boundary and conti- nuity conditions to obtain a system of G  N  J equations, where J is the total number of discretization nodes set up on the domain. Solving this system yields the constants of the local general solutions, which then yields the exact S N solution at each point of the slab. Instead of following this procedure, we implement parts ii, iii and iv above, due to 0306-4549/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2013.09.023 ⇑ Corresponding author. Tel.: +55 21 78745906. E-mail addresses: walvesm@gmail.com (W.A. Menezes), halves@iprj.uerj.br (H.A. Filho), rcbarros@pq.cnpq.br (R.C. Barros). Annals of Nuclear Energy 64 (2014) 270–275 Contents lists available at ScienceDirect Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene