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

Journal article, 2008

We study the flatland (two dimensional) linear transport equation, under an angular 2π periodicity assumption both on particle density function ψ(x,y,θ) and on the differential scattering σs(θ). We consider the beam problem, with a forward peaked source on phase-space, and derive P1 approximation with a diffusion coefficient of , (versus of the three dimensional problem), where is the transport cross section. Further assumptions as peaked σs(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 (rather than of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples.