ERROR DUE TO ANGULAR DISCRETIZATION IN THE DISCRETE ORDINATES APPROXIMATION OF THE TRANSPORT EQUATION IN TWO-DIMENSIONAL CARTESIAN GEOMETRY

摘要

Asymptotic convergence of the solution to the discrete ordinates approximation of the neutron transport equation with increasing quadrature order is examined. Four angular quadrature types with increasing number of angles are considered: level symmetric, Legendre-Chebyshev quadrangular, Legendre-Chebyshev triangular, and quadruple range. These quadratures integrate polynomials of the angle cosines with increasing accuracy as the quadrature order is raised, hence Madsen's Theorem indicates the corresponding solution should also converge with angular refinement. A simple test problem is solved using the Arbitrarily High Order Transport method of the Nodal type and 0th order spatial expansion (AHOT-N0) on a successively refined mesh. The cell-wise solution is then used to determine region-averaged scalar fluxes that were found to converge quadratically to a reference solution with mesh refinement. The spatially extrapolated reference solutions for all quadrature types are found to approach a common limit that represents the best available approximation for the region-averaged scalar flux computed by the one-speed transport equation. However, verifying asymptocity of this angular truncation error's convergence trend and translating it to the cell-averaged scalar flux level remain open questions.