A discontinuous Bubnov-Galerkin spectral element spatial discretisation of the first-order form of the neutron transport equation using a discrete ordinate angular discretisation with multidimensional anisotropic dispersion and dissipation analysis

Abstract

© 2026 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. http://creativecommons.org/licenses/by/4.0/In this paper, the discontinuous Bubnov-Galerkin spectral element method (DBG-SEM) has been used to spatially discretise the first order form of the neutron transport (FONT) equation. A discrete ordinate (SN[jls-end-space/]) approximation is also used to discretise the angular variables and the multigroup approximation has been used to discretise the energy variable. The DBG-SEM has been compared against the discontinuous Bubnov-Galerkin finite element method (DBG-FEM) using two benchmark verification test cases. This is the first time that a detailed comparison of DBG-SEM and DBG-FEM has been completed in the context of the FONT equation. The method of manufactured solutions (MMS) has been used to investigate the convergence rate of the proposed discretisation schemes. This demonstrates that both the DBG-SEM and DBG-FEM yield the same rate of convergence with increasing polynomial order. Both the DBG-SEM and DBG-FEM were used to solve the OECD/NEA two-dimensional (2D) C5G7 nuclear reactor physics benchmark verification test case. Again both methods yield a similar level of numerical accuracy using both straight-sided and curvilinear elements. This analysis of the local element matrix condition number demonstrates that the DBG-SEM yields a lower condition number than the DBG-FEM. Finally, multidimensional anisotropic dispersion and dissipation analysis (MA-DDA) was utilised to understand the dispersion and dissipation errors of the DBG-SEM and DBG-FEM. This is the first time that DDA has been utilised for analysing the dispersive and dissipative properties of spatial discretisation methods for the FONT equation in multidimensions. This analysis demonstrated that the DBG-SEM and DBG-FEM have similar dispersion and dissipative behaviour when exact element integration was used, which is an expected result.

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