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

We study the flatland (two dimensional) linear transport equation, under an angular  2π2πperiodicity   assumption both on particle density function ψ(x,y,θ)ψ(x,y,θ) and on the differential scattering σs(θ)σs(θ). We consider the beam problem, with a forward peaked source on phase-space, and derive P1P1 approximation with a diffusion coefficient of 1/2σtr, (versus 1/3σtr of the three dimensional problem), where σtr is the transport cross section. Further assumptions as peaked σs(θ)σs(θ) near θ=0θ=0 (small angle of scattering), and small angle of flight   (θ≈0θ≈0) yield Fokker–Planck and Fermi approximations with the diffusion coefficients σtr (rather than σtr/2 of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples.