Singular Solutions, Integral Transport Theory, and the S_n Method

The integral transport equation clearly indicates that the angular flux in a void is constant along each characteristic. Yet, simple arguments can be used to demonstrate that there exist angular flux solutions in voids that have a delta-function angular dependence and a nonconstant spatial dependence. Such solutions can appear to be nonconstant along a characteristic. Using a simple example problem, we demonstrate that such solutions represent the limit of a continuous sequence of nonsingular solutions, each of which is constant along every characteristic. We also show that care must be taken in applying the integral transport equation to singular problems of this type because erroneous solutions are easily obtained. Two reliable approaches for obtaining proper solutions are presented. We also show that the differential form of the transport equation in one-dimensional spherical geometry requires less care than the integral form of the transport equation for problems of this type. Finally, we discuss the applicability of the S_n method to problems in curvilinear geometries with singular solutions of this type.