Properties of new numerical approximations to the transport equation

Abstract

Two new numerical methods are formulated for the solution of the Boltzmann transport equation. Both methods are designed to reduce distortion of angular flux distributions produced by conventional S N or discrete ordinates formulations. For the first method, in which the distribution function is represented by a set of discrete, discontinues, straight lines, a general formulation is given; and, as an example, finite difference equations are derived for two-dimensional rectangular geometry. The relationship to connected straight line and step function representations is shown, and an estimate of the relative truncation error of these schemes is given, showing that the new scheme has the potential of suppressing spatial oscillations without loss of second-order accuracy. By Fourier analysis of the line source problem, it is shown that the lowest order of the new approximation ( CS 2) compares favorably with the usual S 4 discrete ordinates approximation in the suppression of ray effects. This result is confirmed by numerical comparisons. The second new method involves the conversion of two (or higher) dimensional discrete ordinates equations to angular moments equations to eliminate the directional streaming properties of discrete ordinates equations. Three approaches are examined for these conversions. In the first, the general formula is given for the artificial source needed to truncate the spherical harmonics moments equations derived from discrete ordinates equations in ( x,y) rectangular geometry. While it is shown that the moments in these equations may not be the coefficients in the polynomial expansion of the discrete ordinates flux, numerical results from this conversion show greatly smoothed scalar fluxes. A second conversion formula, in which an artificial source is introduced to make the highest order spherical harmonics moments vanish, also produces smooth fluxes in numerical examples, but requires more iteration than the first method. In the third approach, artificial source corrections to convert discrete ordinates equations to angular moments equations with exact equivalence are given.