发表机构
University of Twente（特温特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非均匀平板辐射传输方程，证明扩散合成加速迭代的谱半径一致有界，并推广到变分离散化，确保预处理共轭梯度快速收敛。
AI 中文摘要
扩散合成加速（DSA）源迭代是辐射传输方程的标准求解器。利用傅里叶分析，对于无限均匀介质，已建立了收敛速率 \\(\rho_\infty(c)\le0.2247\\,c\\)，其中 \\(c\\) 是散射截面与总截面的最大比值。对于具有流入边界条件和任意有界截面的平板几何，我们证明了 DSA 迭代的谱半径至多为 \\(\rho_\infty(c)\\)。我们证明该收敛速率适用于 DSA 迭代的变分离散化，只要角因子包含常数的任何相容张量积 Galerkin 空间。该分析基于谱半径的精确最小-最大特征。由此我们推导出一个可检验的充分条件，用于给定速率。我们通过一个合适的能量稳定投影来验证该条件，该投影是一个加权角平均，其权重选择使得该条件以速率 \\(\rho_\infty(c)\\) 成立。由于该投影将离散空间映射到离散空间，该论证逐字适用于离散迭代。因此，预处理系统的条件数一致地以 \\(1+0.29\\,c\\) 为界，这意味着预处理共轭梯度方法快速收敛。
英文摘要
The diffusion synthetic accelerated (DSA) source iteration is a standard solver for the radiative transfer equation. Using Fourier analysis, a convergence rate \(ρ_\infty(c)\le0.2247\,c\) with \(c\) the maximum ratio of scattering to total cross section has been established for an infinite homogeneous medium. For slab geometry with inflow boundary conditions and arbitrary bounded cross sections we prove that the spectral radius of the DSA iteration is at most \(ρ_\infty(c)\). We show that this convergence rate carries over to a variational discretization of the DSA iteration on every conforming tensor-product Galerkin space whose angular factor contains the constants. The analysis rests on an exact min--max characterization of the spectral radius. From it we derive a checkable sufficient condition for a given rate. We verify this condition using a suitable energy-stable projection, a weighted angular average whose weight is chosen so that the condition holds with the rate \(ρ_\infty(c)\). Since the projection maps discrete spaces into discrete spaces, the argument applies verbatim to the discrete iteration. As a consequence, the condition number of the preconditioned system is uniformly bounded by \(1+0.29\,c\), which implies rapid convergence of the preconditioned conjugate gradients method.
