Adaptive Discrete-Ordinates Quadratures Based on Discontinuous Finite Elements Over Spherical Quadrilaterals

We present a new family of discrete-ordinates (Sn) angular quadratures based on discontinuous finite elements (DFEM) in angle. The angular domain is divided into spherical quadrilaterals (SQs) on the unit sphere surface. Linear and quadratic discontinuous finite element (LDFE and QDFE) basis functions in the direction cosines are defined over each SQ, producing LDFE-SQ and QDFE-SQ angular quadratures, respectively. The new angular quadratures demonstrate more uniform direction and weight distributions than previous DFEM-based angular quadratures, local refinement capability, strictly positive weights, generation to large numbers of directions, and 4th-order accurate high-degree spherical harmonics (SH) integration. Results suggest that particle-conservation errors due to inexact high-degree SH integration rapidly diminish with quadrature refinement, and tend to be orders of magnitude smaller than other discretization errors affecting the solution. Results also demonstrate that the performance of the new angular quadratures without local refinement is on par with or better than that of traditional angular quadratures for various radiation transport problems. The performance of the new angular quadratures can be further improved by using local refinement, especially within an adaptive Sn algorithm. The effectiveness of DFEM-based angular quadratures in adaptive Sn algorithms is limited by the accuracy of the mapping algorithms required for passing the angular flux solution between spatial regions with different angular quadrature refinement. An “optimal" mapping algorithm should preserve both the shape and the angular moments of interest from the incoming solution. We present a new mapping algorithm which is nearly \optimal" for mapping sufficiently smooth solutions away from octant boundaries. If the mapped solution contains over- and under-shoots, we apply a fix-up algorithm that uses multi-objective optimization to ensure that the mapped solution remains within prescribed bounds, and exactly preserves the 0th angular moment of the incoming solution. We have demonstrated the use of the new mapping and fix-up algorithms for mapping increasing-degree SH functions, and nearly discontinuous angular flux solutions. The new angular quadratures along with the new mapping and fix-up algorithms provide the necessary tools for using DFEM-based angular quadratures in adaptive Sn algorithms. Future work should include testing various adaptive Sn algorithms, and their efficient parallel implementation