Abstract

When the transport equation is solved by the discrete ordinate method, the problem arises of constructing quadrature formulas on a sphere characterized by the required accuracy and making it possible to use the quadrature nodes to approximate the transport equation in \(r,\;\vartheta ,\;z\) geometry, in which quadrature nodes are simultaneously used to approximate the derivative with respect to the azimuth angle \(\varphi \) of the transport equation, that is, must be located in levels on the sphere with the same values of the polar angle \(\theta \). An algorithm is considered to construct quadrature formulas of the needed form that are characterized by regular prism (dihedron) symmetry and exact for all spherical polynomials of degree not exceeding some maximal value \(L\). This study is a development of the work of A.N. Kazakov and V.I. Lebedev (1994). The constructed family of quadratures, unlike that in the above work, does not contain nodes with \(\varphi = 0,{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2},\pi ,{{3\pi } \mathord{\left/ {\vphantom {{3\pi } 2}} \right. \kern-0em} 2}\), at the poles \(\theta = \pm {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}\), and on the equator \(\theta = 0\) of the sphere. It is shown that this family ensures a significant computational gain when radiation transport problems are solved in three-dimensional geometry.

中文翻译：

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具有正则棱柱对称性刻画结点的球面的高斯型求积公式

摘要

当通过离散纵坐标法求解运输方程时，会出现以下问题：在具有所需精度特征的球面上构造正交公式，并可能使用正交节点来近似\（r，\; \ vartheta中的运输方程，，;; z \）几何图形，其中同时使用正交节点来近似相对于输运方程的方位角\（\ varphi \）的导数，也就是说，必须位于相同的球面上的水平极角\（\ theta \）的值。考虑一种算法来构造所需形式的正交公式，该公式具有正则棱柱（二面体）对称性，并且对于度不超过某个最大值\（L \）的所有球形多项式都是精确的。这项研究是对AN Kazakov和VI Lebedev（1994）的工作的发展。与上面的工作不同，构造的正交族不包含\（\ varphi = 0，{\ pi \ mathord {\ left / {\ vphantom {\ pi 2}} \ right。\ kern-0em } 2}，\ pi，{{3 \ pi} \ mathord {\ left / {\ vphantom {{3 \ pi} 2}} \ right。\ kern-0em} 2} \），在两极\（\ theta = \ pm {\ pi \ mathord {\ left / {\ vphantom {\ pi 2}} \ right。\ kern-0em} 2} \），在赤道\（\ theta = 0 \）球体。结果表明，当在三维几何中解决辐射传输问题时，该族确保了显着的计算增益。