We study the sine-transform-based splitting preconditioning technique for the linear systems arising in the numerical discretization of time-dependent one dimensional and two dimensional Riesz space fractional diffusion equations. Those linear systems are Toeplitz-like. By making use of diagonal-plus-Toeplitz splitting iteration technique, a sine-transform-based splitting preconditioner is proposed to accelerate the convergence rate efficiently when the Krylov subspace method is implemented. Theoretically, we prove that the spectrum of the preconditioned matrix of the proposed method is clustering around 1. In practical computations, by the fast sine transform the computational complexity at each time level can be done in \({{\mathcal {O}}}(n\log n)\) operations where n is the matrix size. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.

我们研究了线性正弦系统基于正弦变换的分裂预处理技术，该技术是在时间相关的一维和二维Riesz空间分数扩散方程的数值离散化中产生的。那些线性系统类似于Toeplitz。通过利用对角加Toeplitz分裂迭代技术，提出了一种基于正弦变换的分裂预处理器，以在实现Krylov子空间方法时有效地加快收敛速度​​。从理论上讲，我们证明了该方法的预处理矩阵的谱在1附近聚集。在实际计算中，通过快速正弦变换，可以在\（{{\ mathcal {O}} }（n \ log n）\）个操作，其中n是矩阵大小。数值例子说明了所提算法的有效性。