Nonnegativity of the Anisotropic Scattering Source in Two-Angle (μ, φ)Transport

The use of piecewise constant functions (PCFs) in two-angle linear transport theory to represent the scattering cross sections σ(v), v ∈ [-1,1], and the angular scattering source density S(), ≡ (μ, φ) ∈ on a partition (S_N or finite element discretization, for example) of the unit sphere of directions is considered. Average oriented transition cross sections σ_tn (±,B',B) describe scattering from ≡ (, )∈ B’ ⊂  to ≡ (μ,φ)∈ B ⊂ with the constraint 0< ±(φ - φ') <π. Unit steps σ(v) = H(v —γ) and σ(v) = δ(v — γ) are pretreated on an “intrinsic” γ grid for the chosen partition. All σ_tn(±,B',B) are derived by interpolation. The invariance properties of the σ_tn’s and the permitted B'→B transition (σ_tn > 0) are identified. Then, the PCF representation of S() is obtained with a minimum of work. Angular rebalancing restores the correct zeroth- and first-order angular moments without losing the nonnegativity of σ_tn and S. The preferential domains of application of this PCF method and the classical spherical harmonics method (which may violate nonnegativity) are discussed.