Skin effect in neutron transport theory

We identify a neutron-flux “skin effect” in the context of neutron transport theory. The skin effect, which emerges as a boundary layer at material interfaces, plays a critical role in a correct description of transport phenomena. A correct accounting of the boundary-layer structure helps bypass computational difficulties reported in the literature over the last several decades, and should lead to efficient numerical methods for neutron transport in two and three dimensions.

Figure 1

One-dimensional finite “slab” geometry: $\xi =cos\left(\theta \right)$

Figure 2

Scattering-free solution [Eq. (5)]. The boundary layers in the $\xi$ and $x$ directions are clearly emphasized by the superimposed blue (along the $\xi$ direction) and red (along the $x$ direction) coordinate curves.

Figure 3

Error of numerical against analytic integral, $E\left(M\right)={\text{max}}_{0\le x\le 1}|{\sum }_{i=1}^M{w}_{i}u(x,{\xi }_i)-I^{\text{an}}\left(x\right)|$, for $u$ given by Eq. (5). The errors are presented for the quadrature rule (7) (circles and diamonds) and the plain Gauss-Legendre (GL) quadrature (triangles).

Figure 4

Convergence properties of the proposed algorithm for (a) the time discretization, (b) the spatial discretization, and (c) the number of discrete ordinates employed. In circles, the error $E(\mathrm{\Delta }v,M,\mathrm{\Delta }t)={\text{max}}_{x,\xi }|u(x,\xi )-u^{\text{c}}(x,\xi )|$ is displayed for various grids, where $u^{\text{c}}(x,\xi )$ denotes the converged solution.

Figure 5

Boundary layers near $x=0$, obtained by solving Eq. (8) with ${\mu }_a={\mu }_s=q=1$ for various values of $\xi$ and $x$, with $0.84>{\xi }_i>{\xi }_{i+1}>{10}^{-7}$ and ${10}^{-2}>x_i>x_{i+1}>{10}^{-6}$. (a) shows the boundary layer along the $x$ variable for various values of $\xi$, and (b) shows the boundary layers along the $\xi$ variable for various values of $x$. The neutron flux $u(x,\xi )$ does not vanish on the boundary for the outgoing directions $-1\le \xi <0$, and, therefore, no boundary layer exists at $x=0$ for such directions (not shown).

Figure 6

Boundary layers obtained by solving Eq. (8), with ${\mu }_a=q=1$ and different values of ${\mu }_s$. Solutions for the direction $\xi ={\xi }_{\text{min}}\simeq {10}^{-6}$ are shown. Note the large ${\mu }_t$-dependent slopes of the transport solution as $x\to 0^{+}$ [cf. Eq. (4), where ${\mu }_t={\mu }_s+{\mu }_a$].

Figure 7

DD and FC-DOM approximations of $u(x,{\xi }_{15})$ for ${\mu }_t={\mu }_s=1000$, $q=0.1$, and ${\xi }_{\text{15}}\simeq {10}^{-3}$. The boundary-layer oscillations resulting from the DD scheme are clearly visible. $N=400$ and $N=10000$ discrete points were used in the spatial variable for the FC and DD methods, respectively. Note the high near-boundary resolution that results, in view of the exponential spatial change of variables used, from the $N=400$ FC-DOM discretization.