Linear transport equations in flatland with small angular diffusion and their finite element approximations

Abstract

We study the flatland (two dimensional) linear transport equation, under an angular 2 π periodicity assumption both on particle density function ψ ( x , y , θ ) and on the differential scattering σ s ( θ ) . We consider the beam problem, with a forward peaked source on phase-space, and derive P 1 approximation with a diffusion coefficient of 1 / 2 σ tr , (versus 1 / 3 σ tr of the three dimensional problem), where σ tr is the transport cross section. Further assumptions as peaked σ s ( θ ) near θ = 0 ( small angle of scattering), and small angle of flight ( θ ≈ 0 ) yield Fokker–Planck and Fermi approximations with the diffusion coefficients σ tr (rather than σ tr / 2 of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples.