A Quantitative Theory of Angular Truncation Errors in Three-Dimensional S_N Calculations

A quantitative theory of angular truncation errors is developed for three-dimensional discrete-ordinates (S_N) particle transport calculations. The theory is based on an analysis of a special problem: a localized radially symmetric source in an infinite homogeneous scattering medium, with an arbitrary scattering ratio c satisfying 0 < c < 1. For both the linear Boltzmann equation and the S_N equations, we construct and compare analytic solutions of this problem that are asymptotically valid far from the source region. Comparing these analytic solutions, we find that the relative error in the S_N solution increases without bound for large distances from the source region but decreases at each fixed spatial point as the scattering ratio or N (the order of the quadrature set) increases. Also, the S_N error patterns conform to classic ray effects for small c but not for larger c. We present numerical results that test and validate the theoretical predictions.