Research on the Iterative Method for Solving Neutron Diffusion Equation
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摘要: 为提升求解中子扩散方程特征值的计算效率，本文针对幂迭代方法和基于Krylov子空间思想的特征值迭代算法展开深入研究。首先，在幂迭代方法中，提出了裂变源归一的初始值设定方式，并与传统的初始值设定进行了对比；接着，对幂迭代方法和基于Krylov子空间的迭代方法的迭代次数和计算时间进行了比较；最后，对基于Krylov子空间迭代方法预处理技术进行了研究，比较了雅克比预处理、不完全LU分解预处理、代数多重网格预处理对迭代次数和计算时间的影响。对国际原子能机构（IAEA） 3D基准题的计算结果表明，Davidson方法结合不完全LU分解预处理技术具有较高计算效率。对于一个具有200万单元的问题，该方法使用单核可以在1 min内完成计算，与传统幂迭代方法相比，计算效率提升了约25倍。该成果大幅提高了中子扩散方程特征值问题的计算效率，显著降低了中子学计算的时间成本。
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关键词:
- 中子扩散方程 /
- 有限元方法 /
- Krylov子空间方法 /
- Davidson方法 /
- 预处理技术
Abstract: To improve the computational efficiency of solving the eigenvalue of the neutron diffusion equation, this paper conducts an in-depth research on the power iteration method and the eigenvalue iteration algorithm based on the Krylov subspace idea. Firstly, in the power iteration method, an initial value setting based on fission source normalization is proposed and compared with the traditional method. Then, the number of iterations and computation time of the power iteration method and Kroylov-based iteration algorithm are compared. Finally, the preconditioning techniques of Kroylov-based iterative method are studied, including the Jacobian preconditioning, incomplete LU decomposition preconditioning, and algebraic multigrid preconditioning, and their impact on the number of iterations and computation time are analyzed. Through the calculation of the IAEA 3D benchmark problem, the results show that the Davidson type iterative algorithm combined with incomplete LU decomposition preconditioning has superior computational efficiency. For a problem with two million elements, the calculation can be completed within one minute by using a single core. Compared with traditional power iteration methods, the computational efficiency is improved by about 25 times. This achievement has significantly enhanced the computational efficiency of eigenvalue problems of the neutron diffusion equation and remarkably reduced the time cost of neutronics calculations. -
图 4 带预处理的不同迭代算法迭代次数
NONE—未采用预处理；JACOBI—雅克比预处理；ILU—不完全LU分解预处理；AMG—代数多重网格预处理。
Figure 4. Number of Iterations of Different Iterative Algorithms with Preconditioning
图 5 带预处理的不同迭代算法计算时间
Figure 5. Computation Time of Different Iterative Algorithms with Preconditioning
表 1 迭代方法对比（无预处理）
Table 1. Comparison of Iterative Algorithms (without Preconditioning)
方法 迭代次数 计算时间/s 幂迭代方法 132 1334.7 幂迭代方法
（裂变源归一）93 980.4 Krylov-Schur方法 4 290.4 广义Davidson方法 390 108.2 Jacobi-Davidson方法 14 100.5 
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