A Spectral Nodal Method for One-Group X,Y-Geometry Discrete Ordinates Problems

A new nodal method is developed for the solution of one-group discrete ordinates (S_N) problems with linearly anisotropic scattering in x,y-geometry. In this method, the “spectral Green’s function” (SGF) scheme, originally developed for solving S_N problems in slab geometry with no spatial truncation error, is generalized to solve the one-dimensional transverse-integrated S_N nodal equations with the “constant” approximation for the transverse leakage terms. The resulting “SGF-constant nodal” (SGF-CN) method is more accurate than conventional coarse-mesh methods for deep penetration problems because it treats the scattering source terms implicitly and exactly; the only approximation involves the transverse leakage terms. In conventional S_N nodal methods, the transverse leakage terms and scattering source are both approximated. We solve the SGF-CN equations using the one-node block inversion iterative scheme, which uses the best available estimates for the node-entering fluxes to evaluate the node-exiting fluxes in the directions that constitute the incoming fluxes for the adjacent nodes as the equations are swept across the system. Finally, we give numerical results that illustrate the accuracy of the SGF-CN method for coarse-mesh calculations.