Infinite-Medium Solutions of the Transport Equation, SNDiscretization Schemes, and the Diffusion Approximation
Edward W. Larsen · Transport Theory and Statistical Physics · 2003
In the beginning of this article, we construct a family of infinite-medium solutions of the linear transport equation. This family consists of angular fluxes that (i) vary linearly (or quadratically) in space and angle and (ii) are driven by isotropic sources that vary linearly (or quadratically) in space. Each angular flux in this family satisfies Fick's Law; thus, the corresponding scalar flux satisfies the familiar diffusion equation. Then, we show that (i) only certain discretization schemes for the transport equation preserve the “linear” infinite-medium solutions, and (ii) these schemes are more accurate in diffusive problems. More precisely, we show that the “quadratic” solutions of these discrete schemes are much more accurate for problems in which the cell width is not optically thin. The overall goal of this article is to demonstrate why it is advantageous for discretization schemes to preserve the “linear” (as well as the “flat”) infinite-medium solutions of the transport equation.