# Structure Approximation-Based Preconditioning for Solving Tempered Fractional Diffusion Equations
###### Abstract  
Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank-Nicolson method and the tempered weighted and shifted Grunwald formula are used to discretize the tempered fractional diffusion equations. 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. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the \(\tau\) matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. t 2025 18 307 14 April 2025 2020 21 May 2025 2025 18 307 14 April 2025 2020 20 May 2  
and  
\[{}_{x}D_{b}^{\beta,\lambda}u(x)=\mathrm{e}^{\lambda x}{}_{x}D_{b}^{\beta}\left( \mathrm{e}^{-\lambda x}u(x)\right)=\frac{\mathrm{e}^{\lambda x}}{\Gamma(2-\beta )}\frac{\partial^{2}}{\partial x^{2}}\int_{x}^{b}\frac{\mathrm{e}^{-\lambda s}u (s)}{(s-x)^{\beta-1}}ds.\]  
Tempered FDEs are a crucial class of equations widely applied in fields such as biology, geophysics, and finance [2; 3; 4]. These equations are characterized by incorporating tempered fractional derivatives, modifying standard fractional diffusion equations by introducing an exponential tempering factor. This modification allows tempered FDEs to better simulate processes with finite propagation speed, accurately describing actual physical phenomena. However, analytical solutions to tempered FDEs and FDEs are generally difficult to obtain, leading to extensive research on the numerical solutions of FDEs in recent years [5; 6; 7; 8; 9; 10; 11]. For tempered fractional derivatives, stability of the resulting finite difference schemes can only be ensured if the spatial step size is sufficiently small when using standard fractional derivative approximations directly [12]. A novel shifted approximation method for tempered fractional derivatives was proposed in [13], successfully developing an unconditionally stable numerical scheme for one-dimensional (1D) tempered fractional diffusion equations with constant coefficients. Additionally, a Crank-Nicolson scheme was proposed [14] for solving initial boundary value problems of a class of variable coefficient tempered fractional diffusion equations, providing the discretization method used in this paper. The inherent non-locality of tempered fractional operators manifests in the algebraic structure of discretized systems, producing coefficient matrices with dense or fully populated configurations. The system's coefficient matrix assumes the configuration \(I+DT\), with \(I\) denoting the identity matrix, \(D\) a diagonal matrix, and \(T\) an SPD Toeplitz matrix. Traditional direct solution methods, such as Gaussian elimination, require large computational cost and storage space. The Krylov subspace methods are typically used to solve these kinds of linear system. Meanwhile, effective preconditioning strategies can significantly accelerate the convergence rate of the Krylov subspace methods [15; 16; 17; 18]. For 1D and 2D tempered FDEs, [PERSON], [PERSON] and [PERSON] [19] proposed a fast-iterative method and a fast-direct method. And they used the circulant preconditioner to accelerate the iteration process. In [20], a robust preconditioner was developed for the discretized scheme of non-linear tempered FDEs. In [21], a circulant and skew-circulant splitting iteration method was used to solve the discretized system of tempered FDEs.