# Structure Approximation-Based Preconditioning for Solving Tempered Fractional Diffusion Equations
## 2 Discretization of Tempered FDEs  
Let \(N\) and \(M\) be positive integers, representing the number of spatial and temporal partitions, respectively. We define the spatial step size \(h=\frac{a}{N+1}\) and the time step size \(\Delta t=\frac{T}{M}\). The spatial and temporal grids are given by \(x_{i}=ih\) for \(i=0,1,\ldots,N+1\) and \(t_{j}=j\Delta t\) for \(j=0,1,\ldots,M\). We review that the tempered fractional derivatives in (1) can be approximated using the tempered weighted and shifted Grunwald formula (tempered WSGD) [14]  
\[{}_{a}D_{x}^{\beta,\lambda}u\big{(}x_{i},t_{j}\big{)}-\lambda^{\beta}u\big{(} x_{i},t_{j}\big{)}=\frac{1}{h^{\beta}}\sum_{k=0}^{i+1}g_{k}^{(\beta)}u\big{(}x_{i- k+1},t_{j}\big{)}-\frac{1}{h^{\beta}}\rho_{\beta}u\big{(}x_{i},t_{j}\big{)}+O \Big{(}h^{2}\Big{)},\]  
and  
\[{}_{x}D_{b}^{\beta,\lambda}u\big{(}x_{i},t_{j}\big{)}-\lambda^{\beta}u\big{(} x_{i},t_{j}\big{)}=\frac{1}{h^{\beta}}\sum_{k=0}^{N-i+2}g_{k}^{(\beta)}u\big{(}x_{i+ k-1},t_{j}\big{)}-\frac{1}{h^{\beta}}\rho_{\beta}u\big{(}x_{i},t_{j}\big{)}+O \Big{(}h^{2}\Big{)},\]  
where \(\rho_{\beta}=\Big{(}\gamma_{1}e^{h\lambda}+\gamma_{2}+\gamma_{3}e^{-h\lambda }\Big{)}\Big{(}1-e^{-h\lambda}\Big{)}^{\beta}\); the weights \(g_{k}^{(\beta)}\) are given by  
\[g_{k}^{(\beta)}= \begin{cases}\gamma_{1}w_{0}^{(\beta)}e^{h\lambda},&k=0,\\ \gamma_{1}w_{1}^{(\beta)}+\gamma_{2}w_{0}^{(\beta)},&k=1,\\ \Big{(}\gamma_{1}w_{k}^{(\beta)}+\gamma_{2}w_{k-1}^{(\beta)}+\gamma_{3}w_{k-2 }^{(\beta)}\Big{)}e^{-(k-1)h\lambda},&k\geq 2,\end{cases}  \tag{2}\]  
where \(w_{0}^{(\beta)}=1,w_{k}^{(\beta)}=\Big{(}1-\frac{1+\beta}{k}\Big{)}w_{k-1}^{( \beta)},\text{ for }k\geq 1\), and \(\gamma_{1},\gamma_{2},\gamma_{3}\) satisfy the linear system  
\[\left\{ \begin{array}{l}\gamma_{1}+\gamma_{2}+\gamma_{3}=1,\\ \gamma_{1}-\gamma_{3}=\beta/2.\end{array} \right. \tag{3}\]  
Let \(u_{i}^{j}\approx u\big{(}x_{i},t_{j}\big{)},f_{i}^{j+\frac{1}{2}}=f\Big{(}x_{ i},t_{j+\frac{1}{2}}\Big{)},t_{j+\frac{1}{2}}=\frac{1}{2}\big{(}t_{j}+t_{j+1} \big{)}\) and \(u_{i}^{j+\frac{1}{2}}=\frac{1}{2}\Big{(}u_{i}^{j}+u_{i}^{j+1}\Big{)}\) for \(i=1,\ldots,N,j=0,1,\ldots,M\). At the mesh point \(\Big{(}x_{i},t_{j+\frac{1}{2}}\Big{)}\), we consider the Crank-Nicolson technique for time discretization and the tempered WSGD approximation for the tempered fractional derivatives to discretize the tempered FDEs in (1). Under the truncation error-free assumption [23; 24], the second-order finite difference scheme is formulated as follows:  
\[ \begin{split}\frac{u_{i}^{j+1}-u_{i}^{j}}{\Delta t}=& \frac{d_{i}}{2}\left(\frac{1}{h^{\beta}}\sum_{k=0}^{i+1}g_{k}^{( \beta)}u_{i-k+1}^{j+1}-\frac{1}{h^{\beta}}\rho_{\beta}u_{i}^{j+1}+\frac{1}{h^{ \beta}}\sum_{k=0}^{N-i+2}g_{k}^{(\beta)}u_{i+k-1}^{j+1}-\frac{1}{h^{\beta}} \rho_{\beta}u_{i}^{j+1}\right)\\ &+\frac{d_{i}}{2}\left(\frac{1}{h^{\beta}}\sum_{k=0}^{i+1}g_{k}^{ (\beta)}u_{i-k+1}^{j}-\frac{1}{h^{\beta}}\rho_{\beta}u_{i}^{j}+\frac{1}{h^{ \beta}}\sum_{k=0}^{N-i+2}g_{k}^{(\beta)}u_{i+k-1}^{j}-\frac{1}{h^{\beta}}\rho_ {\beta}u_{i}^{j}\right)\\ &+f_{i}^{j+\frac{1}{2}}.\end{split}  \tag{4}\]  
We define the diagonal matrix as  
\[D=\text{diag}(d_{1},d_{2},\ldots,d_{N})\]  
and the vectors as  
\[u^{j}=\left[u_{1}^{j},u_{2}^{j},\ldots,u_{N}^{j}\right]^{\mathrm{T}},f^{j+ \frac{1}{2}}=\left[f_{1}^{j+\frac{1}{2}},f_{2}^{j+\frac{1}{2}},\ldots,f_{N}^{ j+\frac{1}{2}}\right]^{\mathrm{T}}.\]  
For \(j=0,1,\ldots,M\) and a given \(u^{0}\), the differential scheme (4) admits a matrix representation expressed as  
\[(I+DG)u^{j+1}=(I-DG)u^{j}+\Delta tf^{j+\frac{1}{2}}, \tag{5}\]  
where I is the identity matrix, and G is an SPD Toeplitz matrix, defined as \(G=G_{\beta}+G_{\beta}^{\mathrm{T}}\), where \(G_{\beta}^{\mathrm{T}}\) represents the transpose of \(G_{\beta}\). \(G_{\beta}\) can be represented as  
\[G_{\beta}=-\frac{\Delta t}{2h^{\beta}}