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The Continued Fraction Approximation: A Time-Dependent Transport Method

William A. Yingling, Charles J. Bridgman

Nuclear Science and Engineering

Volume 53 / Number 2 / February 1974 / Pages 123-136


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A new approximation based on continued fractions is defined that yields simple closed-form solutions to the single-velocity time-dependent Boltzmann equation in a homogeneous, isotropic infinite medium. The approximation is developed for an isotropic Green’s function source with both absorption and scatter. The method is based on the development of the complete continued fraction solution of the infinite set of time-dependent P-N equations in transform space. The approximation then consists of truncating the continued fraction after a number of terms, which is shown to be equivalent to the standard P-N approximation; then, unlike the standard P-N approximation, the discarded portion of the continued fraction is replaced with a closed function. For low-order approximations, the result can be successfully inverted, yielding useful closed-form approximate solutions which demonstrate excellent temporal and spatial resolution, particularly near the wave front. Both spherically symmetric and one-dimensional plane geometries are treated. In spherical geometry, the approximation offers a closed-form solution for the time-dependent flux emanating from a point source in a scattering medium such as is of current interest in atmospheric transport studies. In an example presented in this paper, a low-order continued fraction approximation does exhibit a wave front and compares well with a time-dependent numerical calculation (TDA). In plane geometry, the method offers closed-form approximate solutions which may be of interest in the study of neutron waves. An example is presented and compared to a numerical evaluation of an exact solution by Erdmann. The continued fraction approximation compares favorably with Erdmann’s data and can be easily evaluated at positions other than the spatial origin. Finally, in the case of reduction to steady state, the continued fraction approximation predicts fluxes which closely approximate the asymptotic portion of an exact solution presented years ago by Case, de Hoffmann, and Placzek.

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