Tempered fractional diffusion equations are a crucial class of equations widely applied in many physical fields. In this paper, the Crank-Nicolson method and the tempered weighted and shifts Gr\"unwald formula are firstly applied to discretize the tempered fractional diffusion equations. We then obtain that the coefficient matrix of the discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite(SPD) Toeplitz matrix. Based on the properties of SPD Toeplitz matrices, we use $\tau$ matrix approximate it and then propose a novel approximate inverse preconditioner to approximate the coefficient matrix. The $\tau$ matrix based approximate inverse preconditioner can be efficiently computed using the discrete sine transform(DST). In spectral analysis, the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Finally, numerical experiments demonstrate the effectiveness of the proposed preconditioner.

翻译：缓变分数阶扩散方程是一类在众多物理领域广泛应用的关键方程。本文首先应用Crank-Nicolson方法与缓变加权位移Grünwald公式对缓变分数阶扩散方程进行离散化处理。随后我们得到离散化系统的系数矩阵具有单位矩阵与对角矩阵乘以对称正定Toeplitz矩阵之和的结构。基于对称正定Toeplitz矩阵的特性，我们采用τ矩阵对其进行近似，进而提出一种新型的近似逆预处理子来逼近系数矩阵。该基于τ矩阵的近似逆预处理子可利用离散正弦变换进行高效计算。谱分析表明，预处理后系数矩阵的特征值聚集在1附近，这保证了采用新预处理子的Krylov子空间方法具有快速收敛性。最终，数值实验验证了所提预处理子的有效性。