A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell

Abstract

A numerically stable version of the spherical harmonics (PN) method for solving the one-speed neutron transport equation with Lth order anisotropic scattering in a spherical shell is developed. Implementing a stable PN solution for this problem is a challenging task for which no satisfactory answer has been given in the literature. The approach used in this work follows and generalizes a previous work on a problem whose domain is defined by the exterior of a sphere. First, a transformation is used to reduce the original transport equation in spherical geometry to a plane-geometry-like transport equation, where the angular redistribution term in spherical geometry is treated as a source. Then, a PN solution in plane geometry given by a combination of the solution of the associated homogeneous equation and a particular solution is developed. This is followed by a post-processing step which is very effective in improving the PN solution. An additional amount of work with respect to that required for solving problems in plane geometry occurs in the form of a system of N+1 Volterra integral equations of the second kind that has to be solved for the coefficients of the particular solution. The proposed approach has, in any case, the merit of avoiding the ill-conditioning caused by the presence of modified spherical Bessel functions in the standard PN solution, as demonstrated by numerical results tabulated for some test cases.