## 9 Numerical Simulation Example of POD Reduced-Order Extrapolation Galerkin Spectrum Method  
In Equation (1), we take \(\ S(x,y,t)=0\), the initial conditions  
\[S_{1}(x,y)=sin(\tau x)cos(\tau y),\ \ S_{2}(x,y)=0\]  
Here, we choose \(H=100\) as the number of spectral approximation basis functions. We agree that the stability conditions are \(\Delta t\leq 4.8\ \mathrm{H^{-3}s}\), \(\rho=40\ \mathrm{kg/m^{3}}\), \(c=80\ \mathrm{J/(kg\cdot K)}\), \(\mu=20\ \mathrm{W/(m\cdot K)}\) and \(\mu=30\ \mathrm{W/(m\cdot K)}\). First, according to the iterative scheme (19) of the central difference Galerkin spectral method, the numerical solution of the classical collocation point spectral method is calculated while \(t=8\ \mathrm{s}\), and the time steps \(\Delta t\) are taken as \(0.125\times 10^{-3}\ \mathrm{s}\), \(0.25\times 10^{-3}\ \mathrm{s}\), \(0.3\times 10^{-3}\ \mathrm{s}\) and \(0.35\times 10^{-3}\ \mathrm{s}\), respectively, then we make figures, which are shown in Figures 2-5, respectively. Secondly, the numerical solution of the first L = 5 steps at time position node \(\left\{\ t_{k}\right\}_{k=1}^{5}\) is solved by using the spectral method iteration format (19). We can construct sample matrix \(Z=\left(U_{H}^{1},U_{H}^{2},\cdots,U_{H}^{5}\right)\). At this time, we can know that the eigenvector of matrix \(ZZ^{T}\) is \(\psi_{k}(k=1,2,\cdots,r)\) through expression \(\psi_{k}=\frac{1}{\sqrt{\lambda_{k}}}Z\varphi_{k}\), \(r\) stands for \(rank(Z)\). After calculation, we know that \(\sqrt{\lambda_{5}}\leq 9\times 10^{-4}\), which is shown in Figure 1; only the first five POD bases can meet the actual accuracy requirements. It is known that it is not necessary to reselect the POD base, and at this time, we can calculate the numerical approximate solution of the reduced dimension spectrum method in the time steps of \(\Delta t=0.125\times 10^{-3}\ \mathrm{s}\), \(\Delta t=0.25\times 10^{-3}\ \mathrm{s}\), \(\Delta t=0.3\times 10^{-3}\ \mathrm{s}\) and \(\Delta t=0.35\times 10^{-3}\ \mathrm{s}\), then, we can draw their figures, which are shown in Figures 2b, 3b, 4b and 5b, respectively. Finally, we can use the MATLAB program language to compile the error analysis program of the central difference Galerkin spectrum method and the reduced-order extrapolation central difference Galerkin spectrum method and draw pictures to show. Figure 1: Modal Matrix. Figure 2: (**a**) Central difference spectral method; (**b**) POD reduced-order extrapolation method; (**c**) Error graph at \(\Delta t=0.125\times 10^{-3}\) s; (**d**) Energy contribution rate at \(\Delta t=0.125\times 10^{-3}\) s.  
Figure 4: The central differential temperature and POD temperature values of each node at y = 0.25 m and \(\mu=20\,\mathrm{W}/(\mathrm{m}\cdot\mathrm{K})\). From the figure, it can be seen that the consistency between POD and the central differential Galerkin spectral method is very good, with the maximum error not exceeding, thus demonstrating the feasibility of POD algorithm technology. The maximum error cannot exceed \(7\times 10^{-3}\), thus demonstrating the feasibility of POD algorithm technology. **(a)** Temperature reconstruction at \(\Delta t=0.3\times 10^{-3}\) s; **(b)** Error graph at \(\Delta t=0.3\times 10^{-3}\) s.  
Figure 3: (**a**) Central difference spectral method; (**b**) POD reduced-order extrapolation method; (**c**) Error graph at \(\Delta t=0.25\times 10^{-3}\) s; **(d)** Energy contribution rate at \(\Delta t=0.25\times 10^{-3}\) s.  
Figure 5: (**a–c**), respectively, show the effectiveness of the POD and central difference Galerkin spectroscopy method under different thermal conductivity coefficients and non spatial directions. It can be seen that only a few points have slight fluctuations, while the matching effect at other points is quite ideal, and its error does not exceed \(2.75\times 10^{-2}\).