In this paper, we present an efficient fully spectral approximation scheme for exploring the one-dimensional steady-state neutron transport equation. Our methodology integrates the spectral-(Petrov-)Galerkin scheme in the spatial dimension with the Legendre-Gauss collocation scheme in the directional dimension. The directional integral in the original problem is discretized with Legendre-Gauss quadrature. We furnish a rigorous proof of the solvability of this scheme and, to our best knowledge, conduct a comprehensive error analysis for the first time. Notably, the order of convergence is optimal in the directional dimension, while in the spatial dimension, it is suboptimal and, importantly, non-improvable. Finally, we verify the computational efficiency and error characteristics of the scheme through several numerical examples.

翻译：本文提出了一种高效的全谱近似方案，用于求解一维稳态中子输运方程。我们的方法在空间维度上采用谱（Petrov-）Galerkin格式，在方向维度上采用Legendre-Gauss配置点格式。原始问题中的方向积分通过Legendre-Gauss求积公式进行离散化。我们严格证明了该格式的可解性，并据我们所知首次进行了全面的误差分析。值得注意的是，收敛阶在方向维度上是最优的，而在空间维度上是次优的，并且重要的是，该阶次无法进一步提升。最后，我们通过多个数值算例验证了该格式的计算效率与误差特性。