## 6 Galerkin Spectral Method for POD Reduced-Order Extrapolation  
According to the essential theory of constructing the POD basis in Section 4 the standard approximate solution of the \(L\) group before the iterative scheme of the central difference Galerkin spectral method can be derived from \(u_{e}^{i}=\psi\psi^{T}u_{H}^{i}:=\left\{\psi\phi^{i}\right\}_{i=1}^{L}\), and \(\phi^{i}=(\phi_{1}^{i},\phi_{2}^{i},\cdots,\phi_{e}^{i})\) is the vector associated with \(i\). Similarly, when \(L+1\leq i\leq I\), the numerical solution of the iterative scheme of the central difference Galerkin spectral method is still approximated by the approximate solution of \(u_{e}^{i}=\psi\phi^{i}\). At this time, \(u_{N}^{i}\) is replaced by \(u_{d}^{i}\). Its essence is that the solution of the first \(L\) step of the reduced-order central difference scheme is obtained by projecting the solution of the initial \(L\) step of the Galerkin spectral method iterative scheme of the central difference of heat conduction onto the POD basis. When the number of iterations is greater than step \(L\), the solution is obtained through finite recursion by completely relying on the POD basis generated in the initial step \(L\) of the central difference Galerkin spectral method iteration scheme and the solution after the reduced-order in step \(L\), which reflects the basic principle of the so-called reduced order extrapolation difference algorithm. The unknown vector \(\phi^{i}=(\phi_{1}^{i},\phi_{2}^{i},\cdots,\phi_{e}^{i})\) represents that the degree of freedom of the iterative scheme of the central difference Galerkin spectral method for the transient nonlinear heat conduction equation is reduced from H to e, and greatly reduces the order. From this, it can be concluded that the iterative format of the reduced-order central difference Galerkin spectral method generated by the least squares operation is (31) (see, e.g., [30]). \[\left\{ \begin{array}{l}u_{e}^{i}=\psi\psi^{\mathrm{T}}u_{H}^{i}:=\psi\phi^{j}(i =1,2,\cdots,L)\\ pc\psi\phi^{i+1}=pc\psi\phi^{i-1}+2\Delta t\mu C\psi\phi_{x}^{i}+2\Delta t\mu D \psi\phi_{y}^{i}+2\Delta tS^{i}\\ (i=L,L+1,\cdots,I-1)\end{array} \right. \tag{31}\]  
According to the orthogonality idea of the matrix, multiply both sides of Formula (31) by \(\psi^{\mathrm{T}}\), and then, obtain the following iterative format of the reduced dimension extrapolation central difference Galerkin spectral method (32):  
\[\left\{ \begin{array}{l}\psi^{\mathrm{T}}u_{H}^{i}=\phi^{j}(i=1,2,\cdots,L)\\ \rho c\phi^{i+1}=\rho c\phi^{i-1}+2\Delta t\mu\psi^{\mathrm{T}}C\psi\phi_{x}^{ i}+2\Delta t\mu\psi^{\mathrm{T}}D\psi\phi_{y}^{i}+2\Delta t\psi^{\mathrm{T}}S^{i}\\ (i=L,L+1,\cdots,I-1)\end{array} \right. \tag{32}\]  
\[u_{e}^{i}=\psi\phi^{j} \tag{33}\]  
Before solving the analytic vector (33) of the iterative format of the reduced-order extrapolation central difference Galerkin spectral method, the iterative format (32) must be used to solve the value of \(\phi^{j}(i=1,2,\cdots,L,L+1,\cdots,I)\). Next, it is easy to obtain the expression of the approximate solution of the iterative scheme of the reduced-order extrapolation central difference Galerkin spectral method for the POD algorithm of the transient nonlinear heat conduction equation at point \((x_{n},y_{m},i\Delta t)\):  
\[u_{e_{k}}^{i}=u_{e}^{i}(x_{n},y_{m}),\;0\leq n,m\leq H \tag{34}\]  
Here, \(m\) and \(n\) still satisfy the inequality: \(1\leq k=n+1+m(H+1)\leq G=(H+1)^{2}\). In addition, the numerical solutions of iterative schemes (32) and (33) can be expressed as \(u_{d}^{i}=(u_{d_{1}}^{i},u_{d_{2}}^{i},\cdots,u_{d_{k}}^{i})^{T}\). **Note 1:** It is obvious that there are \((G=\left(H+1\right)^{2})\) unknown elements involved in each step of the Galerkin spectral method iteration scheme. When the POD reduced-order extrapolation method is used to iterate the format, the unknown elements are reduced from G elements to e elements \((e\leq G)\).