空间分数阶扩散方程数值解中结构化方程组的预处理方法研究
A Research of Preconditioning Method for Structured Systems in Numerical Solutions of Spatial Fractional Diffusion Equations
摘要: 空间分数阶扩散方程能有效地描述众多科学领域中的反常扩散现象。而大多数FDE难以得到解析解，需通过建立离散格式以获得高精度的数值解。数值求解FDE通常归结为线性方程组的求解，而预处理技术则是加速迭代求解的关键。近年来，基于不同离散格式下系数矩阵的结构和性质，学者们研究了高效的预处理方法，显著地降低了计算成本。针对数值求解空间分数阶扩散方程问题，本文整理和分析了方程不同形式下的离散情形和预处理方法，并为预处理进一步的研究提供思路参考。
Abstract: Spatial fractional diffusion equations effectively describe anomalous diffusion phenomena in various scientific fields. However, most FDEs are difficult to solve analytically, necessitating the establishment of discrete schemes to obtain high-precision numerical solutions. Numerical solutions of FDEs typically reduce to solving linear systems, where preconditioning techniques are crucial for accelerating iterative solvers. In recent years, scholars have investigated efficient preconditioning methods based on the structure and properties of coefficient matrices under different discretization schemes, significantly reducing computational costs. For the numerical solution of spatial fractional diffusion equations, this paper organizes and analyzes the discrete scenarios and preconditioning methods for various forms of spatial fractional diffusion equations, providing insights and references for further research in preconditioning.
文章引用：康帅娟, 张旋, 王超杰, 李婷, 邵圣烨. 空间分数阶扩散方程数值解中结构化方程组的预处理方法研究[J]. 应用数学进展, 2025, 14(4): 453-463. https://doi.org/10.12677/aam.2025.144176

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