In the present study, we consider the numerical method for Toeplitz-like linear systems arising from the $d$-dimensional Riesz space fractional diffusion equations (RSFDEs). We apply the Crank-Nicolson (CN) technique to discretize the temporal derivative and apply a quasi-compact finite difference method to discretize the Riesz space fractional derivatives. For the $d$-dimensional problem, the corresponding coefficient matrix is the sum of a product of a (block) tridiagonal matrix multiplying a diagonal matrix and a $d$-level Toeplitz matrix. We develop a sine transform based preconditioner to accelerate the convergence of the GMRES method. Theoretical analyses show that the upper bound of relative residual norm of the preconditioned GMRES method with the proposed preconditioner is mesh-independent, which leads to a linear convergence rate. Numerical results are presented to confirm the theoretical results regarding the preconditioned matrix and to illustrate the efficiency of the proposed preconditioner.

翻译：本文研究由$d$维Riesz空间分数阶扩散方程（RSFDEs）所导出的Toeplitz型线性系统的数值求解方法。我们采用Crank-Nicolson（CN）格式对时间导数进行离散，并应用拟紧致有限差分方法对Riesz空间分数阶导数进行离散。对于$d$维问题，相应的系数矩阵可表示为（块）三对角矩阵与对角矩阵乘积与$d$层Toeplitz矩阵之和。我们发展了一种基于正弦变换的预处理子以加速GMRES方法的收敛性。理论分析表明，采用所提预处理子的预处理GMRES方法的相对残差范数上界具有网格无关性，从而实现线性收敛速度。文中给出了数值结果，验证了关于预处理矩阵的理论分析，并展示了所提预处理子的计算效率。