An Extended Linear Discontinuous Method for One-group Fixed Source Discrete Ordinates Problems with Isotropic Scattering in Slab Geometry

ABSTRACT Nowadays, the obtainment of an accurate numerical solution of fixed source discrete ordinates problems is relevant in many areas of engineering and science. In this work, we extend the hybrid Finite Element Spectral Green’s Function method (FEM-SGF), originally developed to solve eigenvalue diffusion problems, for fixed source problems using as a mathematical model, the discrete ordinates formulation in one energy group with isotropic scattering in slab geometry. This new method, Extended Linear Discontinuous Discrete Ordinates (ELD-SN), is based on the use of neutron balance equations and the construction of a hybrid auxiliary equation. This auxiliary equation combines a linear discontinuous approximation and spectral parameters to approximate the neutron angular flux inside the cell. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of our methodology. ELD-SN method is free from spatial truncation errors in S 2 quadrature, and generate good results in the other quadrature sets. This method is more accurate than the conventional Diamond Difference (DD) and Linear Discontinuous (LD) methods, but surpassed by the Spectral Green’s Function (SGF) method, for quadrature order greater than two.

摘要：当前，固定源离散纵标问题精确数值解的获取在诸多工程与科学领域中具有重要应用价值。本研究将原本用于求解本征扩散问题的混合有限元谱格林函数（Finite Element Spectral Green’s Function, FEM-SGF）方法进行扩展，以平板几何下单能群各向同性散射的离散纵标公式作为数学模型，将其适配至固定源问题求解。该新方法被命名为扩展线性间断离散纵标（Extended Linear Discontinuous Discrete Ordinates, ELD-SN）方法，其基于中子平衡方程的应用与混合辅助方程的构建。该辅助方程结合线性间断近似与谱参数，用以近似计算计算单元内的中子角通量。文中给出了基准问题的数值结果，以验证所提方法的精度与计算性能。ELD-SN方法在S₂求积方案中无空间截断误差，且在其他求积集中均可获得优异的计算结果。当求积阶数大于2时，该方法的精度优于传统的菱形差分（Diamond Difference, DD）与线性间断（Linear Discontinuous, LD）方法，但仍不及谱格林函数（Spectral Green’s Function, SGF）方法。