空间分数阶Cahn-Hilliard方程数值解中结构化方程组的预处理方法研究

Preconditioning Methods for Structured Systems in the Numerical Solution of Space-Fractional Cahn-Hilliard Equations

摘要: 空间分数阶Cahn-Hilliard方程能精确描述具有反常扩散、复杂界面动力学或长程关联系统中的相演化过程。由于方程的非局部性与刚性导致解析解难以获得，需建立能量稳定的离散格式。数值求解空间分数阶Cahn-Hilliard方程通常归结为求解线性方程组，构造有效预处理器可加速迭代求解。近年来，学者们基于不同离散格式下系数矩阵的结构与性质提出了不同的预处理方法，显著提高了求解效率。针对空间分数阶Cahn-Hilliard方程数值求解问题，本文分别整理并分析了一维和高维空间分数阶Cahn-Hilliard方程在不同离散情形下的相应预处理方法，并为后续研究提供了建议。

Abstract: The space-fractional Cahn-Hilliard equation provides an accurate description of phase evolution processes in systems exhibiting anomalous diffusion, complex interfacial dynamics, or long-range interactions. Due to the nonlocal nature and stiffness of the equation, analytical solutions are difficult to obtain, necessitating the development of energy-stable discretization schemes. The numerical solution of the space-fractional Cahn-Hilliard equation typically reduces to solving linear systems, and constructing effective preconditioners can accelerate iterative solutions. In recent years, researchers have proposed various preconditioning methods based on the structure and properties of coefficient matrices arising from different discretization schemes, significantly improving solution efficiency. Addressing the numerical solution of space-fractional Cahn-Hilliard equations, this paper reviews and analyzes corresponding preconditioning methods for one-dimensional and higher-dimensional space-fractional Cahn-Hilliard equations under different discretization scenarios, and provides recommendations for future research.