Low-Order Gauss--Christoffel Quadrature.

The generation of arbitrary order, non-classical, Gauss-type quadrature for transport applications

“…The non-classical quadrature proposed in [4] and [8] is based upon the assumption that the function f (α, β) can be approximated by various polynomials in x, ŷ and ẑ, or in other words, polynomials in the trigonometric functions.…”

“…The original idea for this work was presented by Shumays in [4], and more recently in [8]. However, and as is well known, the direct method of calculation used in [4,8] suffers from inherent problems preventing high order quadrature sets 1 from being determined.…”

“…However, and as is well known, the direct method of calculation used in [4,8] suffers from inherent problems preventing high order quadrature sets 1 from being determined. 2 The method presented here, allows the calculation of non-classical quadrature sets to arbitrary order, offering further potential improvements in transport calculation accuracy.…”

“…We conclude by describing a stable method that enables the calculation of quadrature sets for integrands that have non-classical weight functions, and then introduce the non-classical quadrature rules proposed by Abu-Shumays [4,8]. Finally we go on to apply the method to the so-called octant-range quadrature rules proposed by Abu-Shumays, showing how these can be reproduced to any order.…”

“…The method is then used to generate a number of quadrature sets, to arbitrary order, which are primarily aimed at deterministic transport calculations. The quadrature rules and sets detailed include arbitrary order reproductions of those presented by Abu-Shumays in [4,8] (known as the QR sets, but labelled QRA here), in addition to a number of new rules and associated sets; these are generated in a similar way, and we label them the QRS quadrature sets. The method presented here shifts the inherent difficulty (encountered by Abu-Shumays) associated with solving the non-linear moment equations, particular to the required quadrature rule, to one of the determination of non-classical weight functions and the subsequent calculation of various associated inner products.…”