Recent Studies on the Asymptotic Convergence of the Spatial Discretization for Two-Dimensional Discrete Ordinates Solutions

Skin effect in neutron transport theory

“…The physical character of the neutron fluxes within and around heterogeneous materials, for given sources, has been the subject of significant literature over the last 70 years [1][2][3][4][5][6][7]. Much of this literature has been devoted to unraveling the complex interactions that result from the combined effect of neutron collision and transport phenomena [2][3][4][5].…”

“…4 in Ref. [2]) or separable configurations [6], these boundary-layer transitions, which lead to unbounded gradients arbitrarily close to the interface boundaries, need to be correctly accounted for and fully recognized.…”

“…It is worthwhile noting that Eq. (1) also governs the transport of photons and other neutral particles, and it therefore impacts upon a wide range of important disciplines [2,3,6,[12][13][14][15][16][17][18][19][20][21][22][23].…”

“…[4] (see p. 40 therein) shows that different numerical differentiation schemes may lead to different numerical solutions. It has been demonstrated that, for separable geometries, a degree of accuracy can be obtained for certain spatial "region averages" of the "angular average" of the neutron flux [6,7,12]; naturally, however, general domains are not separable, and, in addition, the full neutron flux (not just such multiply averaged quantities) is generally required for detector response calculations (see p. 60, problem 1-13 in Ref. [3] and Sec.…”

“…The logarithmic spatial change of variables v = log( x 1−x ), in turn, is used to resolve the spatial boundary layer, which gives rise to points x extremely close to the boundary [without detriment to the integration process, in view of the uniformrderivative bounds (6)], leading to high-order precision in both the ξ and x variables. If the change of variables ξ = r p were not introduced, then finer and finer angular discretizations, without a bound on the number of angular discretization points used, would be necessary to yield a fixed prescribed accuracy as the spatial point x approaches a boundary point.…”