Composite spatial grid spectral nodal method for one-speed discrete ordinates deep penetration problems in X, Y geometry

Abstract

Computer modeling of radiation deep penetration problems is historically based on the discrete ordinates ( S N) formulation. For efficiency reasons, besides accuracy, coarse-mesh spatial discretization is desirable. The spectral Green's function (SGF) methods form a class of accurate coarse-mesh numerical methods as they use polynomial approximations only for the node-edge transverse leakage terms; the scattering source terms are treated analytically in the numerical algorithm. Therefore, algebraic work and the computational algorithms of the spectral nodal methods are rather complicated. To alleviate this negative feature, we offer in this paper a composite spatial grid SGF nodal method for the numerical solution of one-speed deep penetration S N problems with isotropic scattering in X, Y geometry. This method uses a rectangular coarse spatial grid, that is coincident with the material region distribution within the shielding structure. We first transverse integrate the S N equations separately in the x- and y-coordinate directions inside each material region, and then we introduce flat approximations for the transverse leakage terms. Furthermore, we use a fine spatial grid to discretize each set of “one-dimensional” S N nodal equations. As the spatial directions are coupled by the transverse leakage terms, we use an explicit alternate direction technique to converge the numerical solution. In order to verify the offered method's accuracy, we present numerical results for typical model problems. Moreover, we compare the computing performance of this method with the conventional SGF-constant nodal method.