A Modified P₃-Like Approximation to the Transport Equation

Standard P_N theory is well developed as an approximation to the neutron transport equation. However, this theory contains no physics in the sense that it simply represents the angular flux as a sum of polynomials in angle. Thus, standard P_N theory (with N finite) cannot qualitatively predict correct asymptotic transport behavior except in the limit of pure scattering. In this paper‚ we modify standard P_N theory by incorporating certain transport physics, namely, the Case discrete modes, into a modified P_N expansion of the angular flux. The theory resulting from using this modified P_N-like expansion predicts the exact transport asymptotic growth/decay length, since it contains the discrete Case eigenvalue. Such modified P₃-like equations and associated boundary conditions are derived in planar geometry according to a recently introduced variational calculus. Analyses and numerical calculations reveal that this modified P₃-like theory possesses the following features: (a) It reduces to standard P₃ theory in the limit of pure scattering; (b) it conserves neutrons but exhibits a scalar flux discontinuity at a material interface; (c) it is shown numerically to be exceedingly accurate, much more accurate than standard P₃ theory, in predicting various transport theory behavior for homogeneous problems; and (d) for heterogeneous problems, it is necessary that each material region in the system be sufficiently large for this theory to predict better results than standard P₃ theory.