Solution algorithms for a P N-1 - Equivalent S N angular discretization of the transport equation in one-dimensional spherical coordinates
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abstract
Angular discretizations of the S N transport equation in curvilinear coordinate systems may result in a streaming-plus-removal operator that is dense in the angular variable or that is not lower-triangular. We investigate numerical solution algorithms for such angular discretizations using relationships given by Chandrasekhar to compute the angular derivatives in the one-dimensional S N transport equation in spherical coordinates with Gauss quadrature. This discretization makes the S N transport equation P N-1- equivalent, but it also makes the sweep operator dense at every spatial point because the N angular derivatives are expressed in terms of the N angular fluxes. To avoid having to invert the sweep operator directly, we must work with the angular fluxes to solve the equations iteratively. We show how we can use approximations to the sweep operator to precondition the full P N-1 equivalent S N equations. We show that these preconditioners affect the operator enough such that convergence of a Krylov iterative method improves.
