The fast solution of linear equations has always been one of the hot spots in scientific computing.A kind of the diagonal matrix splitting iteration methods are provided,which is different from the classical matrix sp...The fast solution of linear equations has always been one of the hot spots in scientific computing.A kind of the diagonal matrix splitting iteration methods are provided,which is different from the classical matrix splitting methods.Taking the decomposition of the diagonal elements for coefficient matrix as the key point,some new preconditioners are constructed.Taking the tri-diagonal coefficient matrix as an example,the convergence domains and optimal relaxation factor of the new method are analyzed theoretically.The presented new iteration methods are applied to solve linear algebraic equations,even 2D and 3D diffusion problems with the fully implicit discretization.The results of numerical experiments are matched with the theoretical analysis,and show that the iteration numbers are reduced greatly.The superiorities of presented iteration methods exceed some classical iteration methods dramatically.展开更多
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...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。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.展开更多
基金The National Natural Science Foundations of China (12202219)the Natural Science Foundations of Ningxia (2024AAC02009, 2023AAC05001)the Ningxia Youth Top Talents Training Project。
文摘The fast solution of linear equations has always been one of the hot spots in scientific computing.A kind of the diagonal matrix splitting iteration methods are provided,which is different from the classical matrix splitting methods.Taking the decomposition of the diagonal elements for coefficient matrix as the key point,some new preconditioners are constructed.Taking the tri-diagonal coefficient matrix as an example,the convergence domains and optimal relaxation factor of the new method are analyzed theoretically.The presented new iteration methods are applied to solve linear algebraic equations,even 2D and 3D diffusion problems with the fully implicit discretization.The results of numerical experiments are matched with the theoretical analysis,and show that the iteration numbers are reduced greatly.The superiorities of presented iteration methods exceed some classical iteration methods dramatically.
文摘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。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.
