SP ₂-Asymptotic P ₁ Equivalence

This article is devoted to analyzing the physical features of the time-dependent P ₂ model in plane geometry and the simplified P ₂ (SP ₂) model for general geometry. The relationships that can be established with the P ₁; model (which gives rise to the telegrapher's equation) are discussed in detail. In particular, the propagation properties are considered, showing that the signal is characterized by propagation velocity that is times the correct particle velocity. This result is also obtained by direct observation of the equivalent three discrete ordinates equations (S ₃) in a slab geometry. In addition, a consistent asymptotic approach is carried out on the SP ₂ equations, as can be done in a P ₁ formulation. We find that in the diffusion limit, the SP ₂ approximation yields the same asymptotic behavior as the asymptotic P ₁ approximation that is derived from the exact time-dependent Boltzmann equation. In the absorbing limit, the asymptotic behavior of the SP ₂ equations tends to be the exact behavior of SP ₂; i.e., with a propagation velocity that is times the particle velocity, which is lower than that for the asymptotic P ₁ approximation that tends to the exact particle velocity. These observations are supported by numerical results for a simple problem in a one-dimensional transport problem. The asymptotic P ₁ and P ₂ approximations are much closer to the exact transport solution than the classic P ₁ approximation.