## 9 Numerical Simulation Example of POD Reduced-Order Extrapolation Galerkin Spectrum Method
(**a**) Temperature reconstruction (y = 0.5 m and \(\mu=20\)\(W/(\mathrm{m\cdot K})\)) at \(\Delta t=0.35\times 10^{-3}\) s; (**b**) Temperature reconstruction (x = 0.5 m and \(\mu=30\)\(W/(\mathrm{m\cdot K})\)) at \(\Delta t=0.35\times 10^{-3}\) s; (**c**) Error graph at \(\Delta t=0.35\times 10^{-3}\) s.  
**Note 3:** It can be observed from Table 1 that since the sixth mode, the contribution of eigenvalues to the POD basis is very small, almost close to 0, and the total energy has reached 100%. It can also be seen from Figure 1 that when the first five singular values are obtained, there is almost no difference between the new matrix and the original matrix, and the abandonment of other singular values after that hardly loses any existing information. This means that only the first five singular values and their corresponding orthogonal matrices U and K are retained, that is, the modal order whose cumulative energy contribution rate reaches 100% is selected as the number of the basis functions. According to the analysis of Figure 1, we reconstruct the temperature field at different time steps, and the results are shown in figure (b) in Figures 2-5. Figure 2 shows a comparison graph between the numerical solutions of POD and the central difference Galerkin spectral method at time step of \(\Delta t=0.125\times 10^{-3}\) s when t = 8 s. We can see from Figure 2c that the temperature reconstruction error is relatively small, and the maximum error cannot exceed \(2.2\times 10^{-4}\). Figure 3 shows the comparison graph between the numerical solutions of POD and the central difference Galerkin spectral method with time step \(\Delta t=0.25\times 10^{-3}\) s at t = 8 s. We can see from Figure 3c that the temperature reconstruction effect is very good, and the maximum error is less than \(1.15\times 10^{-3}\). Table 2 shows the numerical simulation time used for POD extrapolation technology and the central differential Galerkin spectral method at different time steps. From this table, we can see that the time step size in Figure 2 is the smallest, resulting in a longer consumption time for the central differential Galerkin spectral method, but at this point, the speed of POD extrapolation technology increases faster during the simulation process. This reflects that the POD has a higher computational accuracy when the time step is small. Table 2 and Figure 6 show the numerical simulation time of POD extrapolation technique and the central difference Galerkin spectrum method at different time steps. From this table, we can find that the time step in Figure 2 is the smallest, so the central difference Galerkin spectrum method will consume more time, but during this time, the speed of the POD extrapolation technology is faster in the simulation process, which shows that the calculation accuracy of POD is higher when the time step is relatively small. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Order** & **Non-Negative** & **Energy Percentage** & **Accumulated Energy** \\
**Number** & **Eigenvalues** & **of Each Order (\%)** & **Percentage (\%)** \\ \hline
1 & 8.195303 \(\times\)\(10^{8}\) & 98.7348322 & 98.50163 \\
2 & 1.547286 \(\times\)\(10^{4}\) & 2.2651678 & 99.10697 \\
3 & 1.590391 \(\times\)\(10^{2}\) & 1.92054 & 99.83765 \\
4 & 5.102986 \(\times\)\(10^{-1}\) & 0.03461 & 99.94056 \\
5 & 3.171102 \(\times\)\(10^{-6}\) & 0.001913 & 100 \\
6 & 2.806418 \(\times\)\(10^{-9}\) & 0.000006 & 100 \\
7 & 2.950381 \(\times\)\(10^{-13}\) & 0 & 100 \\
8 & 4.539017 \(\times\)\(10^{-16}\) & 0 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Eigenvalues and energy percentage of each order of Eigenmodes (temperature field).