a Spectral-Nodal Method for the Solution of Discrete Ordinates Problems in One and Two-Dimensional Cartesian Geometry

A "spectral" nodal method is developed for the solution of discrete ordinates (S_{ rm N}) problems in slab and x, y-geometries. This method is referred to as the Spectral Green's Function (SGF) method. The SGF method is applied to one-group and multigroup S_{rm N} problems in slab geometry, generating a numerical solution that is completely free from all spatial truncation errors. The SGF method is also used to improve the accuracy of the standard nodal methods currently applied to two-dimensional S_{rm N} problems in Cartesian geometry. We use the SGF-method to solve the one-dimensional transverse integrated S_ {rm N} equations with flat approximation for the transverse leakage terms, so we refer to this new x, y-geometry spectral nodal method as the SGF-Constant Nodal method (SGF-CN). The iterative scheme used in our method is the One-Cell (or Node) Block Inversion (CBI or NBI) iterative scheme. This iterative scheme uses the best available estimates for all the cell-edge incoming angular fluxes to evaluate the outgoing angular fluxes in the directions of the transport sweeps. Therefore, the convergence rate of the CBI (or NBI) iterative scheme is faster than the convergence rate of the standard Source Iteration (SI) scheme. Numerical results are given to illustrate the accuracy of the SGF-method for coarse-mesh calculations, as well as the fast convergence rate of the unaccelerated CBI (or NBI) iterative scheme as compared to the unaccelerated SI-scheme.