We show that several diffusion-based approximations (classical diffusion or SP 1 , SP 2 , SP 3 ) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors -no truncation errors.

We present numerical schemes for the P1-moment and M1-moment approximations of a non-classical transport equation modeling radiative transfer in atmospheric clouds. In contrast to classical radiative transfer, the photon path-length is introduced as an additional variable and serves as pseudo-time in this model. Because clouds may have optically thick regions, we introduce a diffusive scaling and show that the diffusion limits of the moment models and the original equations agree. Furthermore, we show that the numerical schemes also preserve the diffusion asymptotics as well as the set of admissible and realizable states, both for the explicit and the implicit discretization of the pseudo-time variable. A source iteration-like method is proposed, and we observe that it converges slowly in the optical thick case, but a suitable initialization can help to overcome this problem. We validate our method in 1D and present simulation results in the 2D-case for real cloud data.In case of a collision event, 0 < c < 1 denotes the probability that the photon is scattered. Then, a photon with direction v changes its direction to v 0 with a probability given by the scattering kernel .v, v 0 /. With a probability of 1 c, the particle is absorbed. After scattering, the particle travels a certain distance until the next collision event occurs.A probability density p for the distance between two collisions can be defined in terms of the scattering frequency † t and vice versaSolving the full linear Boltzmann equation is numerically costly because of its high dimensionality. Hence, we derive moment models for Equation (1) in order to reduce complexity. Introducing the first three angular moments of function given by

We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path-length s), and models particle transport taking place in homogenized random media in which a particle's distance-to-collision is not exponentially distributed. To solve the nonclassical equation one needs to know the s-dependent ensemble-averaged total cross section, Σ t (µ, s), or its corresponding path-length distribution function, p(µ, s). We consider a 1-D spatially periodic system consisting of alternating solid and void layers, randomly placed in the x-axis. We obtain an analytical expression for p(µ, s) and use this result to compute the corresponding Σ t (µ, s). Then, we proceed to numerically solve the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions µ = ±1. To assess the accuracy of these solutions, we produce "benchmark" results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely-used atomic mix model in most problems.

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In this paper, we study the sensitivities of electron dose calculations with respect to stopping power and transport coefficients. We focus on the application to radiotherapy simulations. We use a Fokker-Planck approximation to the Boltzmann transport equation. Equations for the sensitivities are derived by the adjoint method. The Fokker-Planck equation and its adjoint are solved numerically in slab geometry using the spherical harmonics expansion ($P_N$) and an Harten-Lax-van Leer finite volume method. Our method is verified by comparison to finite difference approximations of the sensitivities. Finally, we present numerical results of the sensitivities for the normalized average dose deposition depth with respect to the stopping power and the transport coefficients, demonstrating the increase in relative sensitivities as beam energy decreases. This in turn gives estimates on the uncertainty in the normalized average deposition depth, which we present.