A one-d simplified discrete-ordinates method with no spatial truncation error

Abstract

A simplified Discrete-Ordinates (S N) method completely free from all spatial truncation errors is proposed for the solution of one-group and isotropic source plane-geometry transport problems with an arbitrary anisotropic scattering of order L (≤ N−1). The method is based on the expansion of the angular flux in spherical harmonic (P N−1) solutions. The analytic expression for the angular flux for each discrete-ordinates direction depends on the exponential functions, arbitrary constants and interior source. Two iterative methods are described to obtain the solution of heterogeneous S N problems. In the first method, the arbitrary constants as well as the outgoing angular fluxes in each spatial cell are iteratively computed by using the external boundary or cell-interface conditions. The solution of the S N equations for homogeneous problems is obtained without any iteration. In the second method, the outgoing angular fluxes in each spatial cell for each discrete-ordinates direction are defined in terms of a Green's function, the incoming angular fluxes from all directions and the interior source. In this case, the cell-interface fluxes are iteratively calculated. Both methods result in nonstandard spatially discretized S N equations for the outgoing angular fluxes where the scattering source does not appear. Numerical results with various quadrature sets such as the Gauss-Legendre, Double Gauss-Legendre and Level Symmetric quadrature sets are given to demonstrate the accuracy of the method and compared with the previous studies.