Multi-dimensional analytical benchmarks for neutral particle transport.

Abstract

The linear Boltzmann equation for the transport of neutral particles is investigated with the objective of generating benchmark-quality calculations for homogeneous infinite and semi-infinite media. In all cases, the problems are stationary, of one energy group, and the scattering is isotropic. In the transport problems considered, the scalar flux is generally the quantity of interest. The scalar flux will have one-, two-, or three-dimensional variation, based on the nature of the medium and source. The solutions are obtained through the use of Fourier and Laplace transform methods. For the multi-dimensional problems, the transformed transport equation is formulated in a form that can be related to a one-dimensional pseudo problem, thus providing some analytical leverage for the inversions. The numerical inversions use standard numerical techniques such as Gauss-Legendre quadrature, summation of infinite series, and Euler-Knopp acceleration. Consideration of the suite of benchmarks in infinite homogeneous media begins with the standard one-dimensional problems: an isotropic point source, an isotropic planar source, and an isotropic infinite line source. The physical and mathematical relationships between these source configurations is investigated. The progression of complexity then leads to multi-dimensional problems with sources which also emit particles isotropically: the finite line source, the disk source, and the rectangular source. It is noted that a finite isotropic disk will have a two-dimensional variation in the scalar flux and a finite rectangular source will have a three-dimensional variation in the scalar flux. Next, sources which emit particles anisotropically are considered. The most basic such source is the point-beam, or Green's function source. The Green's function source holds an interesting place in the suite of infinite medium benchmarks as it is the most fundamental of sources yet may be constructed from the isotropic point source solution. Finally, the anisotropic plane and anisotropically emitting infinite line sources are considered. Many of the mathematical techniques used to generate results for the anisotropic line are of use in the three-dimensional searchlight problem. Thus, a firm theoretical and numerical basis is established for benchmarks which are most appropriate in infinite homogenous media. Attention is then turned to a homogeneous semi-infinite medium. The final problem which is investigated is the three-dimensional searchlight problem for a half-space. The primary feature is a canted incident beam at the center of the free surface. For the three-dimensional problem, the surface scalar flux and current are obtained, and the interior scalar flux is obtained with significant additional computational effort.