Linear and Quadratic Hexahedral Wavelets on the Sphere for Angular Discretizations of the Boltzmann Transport Equation

This paper presents two new methods for discretizing the angular dimension of the Boltzmann transport equation that describes the transport of neutral particles such as neutrons and photons. Our methods represent the direction of particle travel using linear and quadratic varying approximations over a quadrilateral partitioning of the unit sphere's surface (which is used to represent a particle's direction), which is similar to the approximations provided by a finite element expansion. However, our approximations are generated using a second generation spherical wavelet technique. This method generates hierarchical sets of compactly supported basis functions that are important properties for our future work in applying adaptive resolution in the transport equation's angular dimension. These new wavelet methods are applied to five monoenergetic transport problems to demonstrate their capabilities to efficiently represent the angular flux. Particular emphasis is placed on their ability to approximate particle transport in problems involving extreme material cross sections, namely, particle streaming through voids and their transport through highly scattering media. We are able to show that the methods work well against the common methods S_N and P_N when used within established radiation transport codes.