A Comparison of Three Quadrature Schemes for Discrete-Ordinates Calculations of Neutral Particle Transport in Ducts

The use of three different quadrature schemes in discrete-ordinates calculations of neutral particle transport in ducts of arbitrary but uniform cross-sectional geometry is studied. The considered schemes are: (i) the single-quadrature scheme defined by the standard Gauss-Chebyshev quadrature generated by the measure (1-μ²)^1/2dμ on [-1, 1], (ii) the double-quadrature scheme based on the nonstandard Gaussian quadrature generated by the measure (1μ²)^1/2dμ on [0, 1] and its counterpart on [-1, 0], and (iii) the double-quadrature scheme based on the standard Gauss-Legendre quadrature mapped onto [-1, 0] and [0, 1]. To implement the discrete-ordinates method, an approximate model of transport in ducts derived from a weighted residual procedure with two basis and two weight functions is adopted. The main conclusion of this work is that the double-quadrature scheme based on the nonstandard measure is the best of the three. However, for those who do not wish to deal with the special techniques of the constructive theory of orthogonal polynomials that are required to generate this scheme, the standard double Gauss-Legendre scheme constitutes a viable alternative. (author)