## 7 Error Estimation of POD Reduced-Order Extrapolation Galerkin Spectral Method  
It is necessary to introduce a very important lemma before implementing the iterative format of the reduced-order extrapolation Galerkin method for error estimation. **Lemma 1**. : _The known sequences \(\{a_{n}\},\{b_{n}\}\) and \(\{c_{n}\}\) are all three non-negative sequences, if there is a relationship between them: \(a_{0}+b_{0}<c_{0}\), \(a_{n}+b_{m}\leq c_{n}+\alpha\sum_{i=0}^{n-1}a_{i}\;(\alpha>0)\), and the sequence \(\{c_{n}\}\) is a monotone sequence, then, there must be \(a_{n}+b_{n}\leq c_{n}\;exp\;(n\alpha),n\in N\) (positive integer). Next, we will analyze the error of the iterative scheme solution of the POD reduced-order extrapolation Galerkin spectral method._  
Here, \(u_{e}(x,y,t)\) fits formula Equation (35)  
\[\rho c\frac{\partial u_{e}}{\partial t}=\mu(\frac{\partial^{2}u_{e}}{\partial x ^{2}}+\frac{\partial^{2}u_{e}}{\partial y^{2}})+S(x,y,t)+r_{e}(x,y,t) \tag{35}\]  
where, \(r_{e}(x,y,t)\) is the remainder of Formula (35), so there is (36)\[\rho c\frac{\partial d_{e}}{\partial t}=\mu(\frac{\partial^{2}d_{e}}{\partial x^{2} }+\frac{\partial^{2}d_{e}}{\partial y^{2}})-r_{e}(x,y,t) \tag{36}\]  
Then the error function \(d_{e}\) is the solution of the above partial differential Equation (36)  
\[d_{e}^{i}:=u_{H}^{i}-u_{e}^{i} \tag{37}\]  
Combining (30) and (37), the following inequality is obtained. \[\left\|d_{e}^{i}\right\|_{2}=\left\|u_{H}^{i}-u_{e}^{i}\right\|_{2}=\left\|(Z- Z_{e})\varepsilon_{i}\right\|_{2}\leq\sqrt{\lambda_{e+1}}i=1,2,\cdots,L \tag{38}\]  
In particular, if \(i=L,L+1,\cdots,I\), we can replace \(\psi\phi^{i}\) with \(u_{e}^{i}\) in the second equation of equation set (31), then Equation (39) can be obtained:  
\[\rho cu_{e}^{i+1}=\rho cu_{e}^{i-1}+2\Delta t\mu Cu_{e_{x}}^{i}+2\Delta t\mu Du _{e_{y}}^{i}+2\Delta IS^{i} \tag{39}\]  
Then, we made a difference between (19) and (39), and simplified them into  
\[\rho c(u_{H}^{i+1}-u_{e}^{i+1})=\rho c(u_{H}^{i-1}-u_{e}^{i-1})+2\Delta t\mu C (u_{H_{x}}^{i}-u_{e_{x}}^{i})+2\Delta t\mu D(u_{H_{y}}^{i}-u_{e_{y}}^{i}) \tag{40}\]  
\[u_{H}^{i+1}-u_{e}^{i+1}=u_{H}^{i-1}-u_{e}^{i-1}+\frac{2\Delta t\mu C}{\rho c}( u_{H_{x}}^{i}-u_{e_{x}}^{i})+\frac{2\Delta t\mu D}{\rho c}(u_{H_{y}}^{i}-u_{e_{y}}^{i}) \tag{41}\]  
Taking \(u_{d_{x}}=u_{d_{y}}=u_{d}\), \(u_{H_{x}}=u_{H_{y}}=u_{H}\) without affecting the accuracy of the numerical solution, we can obtain the following Equation (42):  
\[u_{H}^{i+1}-u_{e}^{i+1}=u_{H}^{i-1}-u_{e}^{i-1}+\frac{2\Delta t\mu C}{\rho c} (u_{H}^{i}-u_{e}^{i})+\frac{2\Delta t\mu D}{\rho c}(u_{H}^{i}-u_{e}^{i}) \tag{42}\]  
At this time, we substitute \(d_{e}^{i}=u_{H}^{i}-u_{e}^{i}\) into Formula (42), and get:  
\[d_{e}^{i+1}=d_{e}^{i-1} +(\frac{2\Delta t\mu C}{\rho c}+\frac{2\Delta t\mu D}{\rho c})(u_ {H}^{i}-u_{e}^{i})\] \[=d_{e}^{i-1}+(\frac{2\Delta t\mu C}{\rho c}+\frac{2\Delta t\mu D}{ \rho c})d_{e}^{i}\]  
Assume \(\beta=\frac{2\Delta t\mu C}{\rho c}+\frac{2\Delta t\mu D}{\rho c}=\frac{2 \Delta t\mu C+2\Delta t\mu D}{\rho c}\). At this time, \(d_{e}^{i+1}=d_{e}^{i-1}+\  
We will continue to get:  
\[ \begin{split}\left\|d_{e}^{i+1}\right\|_{2}\leq\left\|d_{e}^{L} \right\|_{2}&+\left\|d_{e}^{L-1}\right\|_{2}+\left\|d_{e}^{i} \right\|_{2}+\beta\sum\limits_{k=\text{L}}^{i-1}\left\|d_{e}^{i}\right\|_{2}\\ &\leq 3\sqrt{\lambda_{d+1}}+\beta\sum\limits_{k=L}^{i-1}\left\|d_{e} ^{i}\right\|_{2}\end{split}  \tag{44}\]  
Referring to the conclusion of Lemma 1, it can be known that:  
\[\left\|d_{e}^{i+1}\right\|_{2}\leq 3\sqrt{\lambda_{d+1}}\exp[\beta(i-\text{L}) ]\sum\limits_{k=\text{L}}^{i-1}\left\|d_{e}^{i}\right\|_{2},\;i=L-1,\cdots,I-1\]  
According to the property that the norm is far greater than or equal to the absolute value of the vector element, it is not difficult to know that the numerical solution of the [PERSON] spectral method iterative scheme has the following error estimation results.