Nonnegativity of the anisotropic scattering source in two-angle (μ,φ) transport

In this paper, the use of piecewise constant functions (PCFs) in two-angle linear transport theory to represent the scattering cross sections (σ(ν), ν element-of[-1,1], and the angular scattering source density SΩ, Ω≡(μpw)element-of U, on a partition (SN or finite element discretization, for example) of the unit sphere U of directions is considered. Average oriented transition cross sections σtn(±, B',B) describe scattering from Ω'≡μ',pw '(equivalent of)B'contained-in U to Ω≡μ, (pw)(equivalent of)B contained-in U with the constraint 0<±(pw-pw') < π. Unit steps σν = Hν-γ) and σν = δν-γ are pretreated on an intrinsic γ grid for the chosen partition. All σtn(±,B'), B are derived by interpolation