We construct near-optimal quadratures for the sphere that are invariant under the icosahedral rotation group. These quadratures integrate all (N + 1) 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree N . The nodes of these quadratures are nearly uniformly distributed, and the number of nodes is only marginally more than the optimal (N + 1) 2 /3 nodes. Using these quadratures, we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of Lagrange interpolation on the sphere. This representation uses function values at the quadrature nodes. In addition, the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature, thus providing a much better localization than spherical harmonic expansions. We show that this representation may be localized even further. We also describe two algorithms of complexity O(N 3 ) for using these grids and representations. Finally, we note that our approach is also applicable to other discrete rotation groups.

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The classical S n equations of Carlson and Lee have been a mainstay in multi-dimensional radiation transport calculations. In this paper, an alternative to the S n equations, the "Lagrange Discrete Ordinate" (LDO) equations are derived. These equations are based on an interpolatory framework for functions on the unit sphere in three dimensions. While the LDO equations retain the formal structure of the classical S n equations, they have a number of important differences. The LDO equations naturally allow the angular flux to be evaluated in directions other than those found in the quadrature set. To calculate the scattering source in the LDO equations, no spherical harmonic moments are needed-only values of the angular flux. Moreover, the LDO scattering source preserves the eigenstructure of the continuous scattering operator. The formal similarity of the LDO equations with the S n equations should allow easy modification of mature 3D S n codes such as PARTISN or PENTRAN to solve the LDO equations. Numerical results are shown that demonstrate the spectral convergence (in angle) of the LDO equations for smooth solutions and the ability to mitigate ray effects by increasing the angular resolution of the LDO equations.

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