Multiple time-scale approximations for the time dependent P1 and P3 equations

This work presents the derivation of a closed form expression as well for a three time-scales approximation of the point kinetics equations, as the one-group time-dependent P1- and P3- equations, for a homogeneous multiplying medium, in planar geometry. The results produced by the three-scale approximation for the point kinetics equations are shown to be practically as accurate as the numerical results produced by the Kaganove-type algorithms used in production codes, yet at significantly less costs in computational time and resources. For the P1 equations, closed-form three-scale approximations have been derived for the neutron flux and its higher-order spherical harmonics, as well as for the precursors for two groups of delayed neutrons. The development of these three-scale approximations did not rely on imposing separation of space and time; furthermore, they offer the possibility of a direct investigation of the differences between the P1-, P3-, and diffusion equations, respectively. In particular, significant differences are observed regarding the time-behavior of the various approximations for the neutron current, smaller differences are observed regarding the differences in the time-behavior of the various expressions for the neutron flux, while the distinctions between the various approximations for the precursor distributions are negligible. Furthermore, the three-scale approximations for the P1- and P3-equations have been shown to be superior to the results offered by the improved quasistatic methods described in the literature, particularly for the shortest time scales in the presence of large reactivity insertions. In summary, the method of multiple-scale expansion offers a new way for developing very accurate and effective approximations for solving typical stiff-systems, such as the time-dependent neutron transport and/or diffusion equations with delayed neutrons. (authors)