AI 中文总结
针对带热扩散项的三维非线性稳态辐射输运方程，该研究基于离散纵标法构建三种不同耦合近似的块预条件子，结合Newton-Krylov求解器，经数值实验验证其网格无关性与鲁棒性。
AI 中文摘要
在本研究中，基于离散纵标法，我们针对带有热扩散项的三维非线性稳态辐射输运方程提出了一种鲁棒的块预处理策略。温度方程的扩散项的存在使得无法将其消去为仅含辐射强度的单一方程。为克服这一困难，所有物理变量被组装成一个整体线性系统。热通量和温度在混合H(div)协调有限元格式中被视为独立变量。辐射强度方程采用带有迎风流的间断有限元方法进行离散，其中使用矢量有限元空间来耦合每个单元内不同方向的辐射强度。随后，我们构建了一个Newton-Krylov迭代求解器来求解非线性方程，其核心部分是高效的预处理。为加速Krylov方法的收敛，我们构造了三种块预条件子，对应于温度与辐射强度之间耦合的不同近似水平：$P_{Schur}$保留全部耦合；$P_{Split}$去掉辐射块的传导贡献；$P_{BJ}$忽略辐射到温度的耦合，仅保留温度到辐射的耦合。数值实验验证了所提出预条件子的网格无关性和鲁棒性。
英文摘要
In this work, based on the discrete ordinate method, we propose a robust block preconditioning strategy for the 3D nonlinear steady-state radiation transport equation with heat diffusion term. The presence of the diffusive term of the temperature equation prevents its elimination into a single equation for the radiation intensity. To overcome this difficulty, all physical variables are assembled into a single monolithic linear system. The heat flux and temperature are treated as independent variables in a mixed $H(\mathrm{div})$-conforming finite element formulation. The equation for radiation intensity is discretised by a discontinuous Galerkin method with upwind flux, where a vectorial finite element space is used to couples the radiation intensity in different directions within each element. We then construct a Newton-Krylov iterative solver to solve the nonlinear equations, for which the core part is efficient preconditioning. To accelerate the convergence of Krylov's method, three block preconditioners are constructed, corresponding to different levels of approximation of the coupling between the temperature and radiation intensity. $P_{\mathrm{Schur}}$ retains the full coupling. $P_{\mathrm{Split}}$ drops the conductive contribution to the radiation block. $P_{\mathrm{BJ}}$ neglects the radiation-to-temperature coupling, retaining only the temperature-to-radiation coupling. Numerical experiments demonstrate the mesh independence and robustness of the proposed preconditioners.
