Discontinuous finite element discretizations for the SN neutron transport equation in problems with spatially varying cross sections
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We examine the accuracy of arbitrary degree discontinuous finite element spatial discretizations of the SN particle transport equations with cross sections that are continuous functions of space. For cell-wise constant cross section problems in purely absorbing materials, it has previously been shown that using arbitrary degree DFEM with quadrature-based mass matrix lumping techniques can result in fully accurate schemes with strictly positive angular flux outflows in 1-D slab geometry. We adapt these quadrature-based, or "self-lumping", schemes to problems with arbitrarily varying spatial cross sections and incorporate the cross-section spatial dependence into the definition of the mass matrices. We compare two approaches to deal with the cross-section spatial dependence. The first approach approximates the true cross section as a cell-wise constant that preserves the cell-average cross-section value. The second method uses self-lumping quadrature to evaluate the exact cross section at quadrature points. Regardless of DFEM trial space degree, approximating a spatially varying cross section with a cell-wise constant cross section results in schemes that are at most second-order accurate in space. Additionally, we demonstrate that assuming a cell-wise constant cross section generates interaction rate profiles that have highly non-physical, non-monotonic discontinuities. Using the self-lumping technique to account for cross-section spatial variation with Gauss or Lobatto quadrature yields fully accurate DFEM schemes for problems with spatially varying cross section. Self-lumping schemes that evaluate the cross section at quadrature points do not exhibit non-physical interaction rate profiles. Unfortunately, only a self-lumping linear DFEM with Lobatto quadrature is guaranteed to produce strictly positive angular flux outflow in a pure absorber when accounting for cross-section spatial variation. Robustness and convergence order comparisons are first carried out using a pure absorber test problem. The cross-section spatial variation is chosen such that an analytical solution can be obtained. Next, we study the spatial effects of isotopic concentration changes during fuel depletion. The depletion problem confirms the accuracy results of the pure-absorber problem. 2014 Elsevier Ltd. All rights reserved.
