## 10 Conclusions  
This paper studies the transient nonlinear convection and heat transfer model based on the spectral method and the POD reduced dimension extrapolation method. The purpose is to analyze the feasibility of the Galerkin spectral method based on POD dimension reduction technology for solving flow and heat transfer problems. We have carried out a series of experimental verifications and analyses on the POD dimension reduction center and the Galerkin spectral method in terms of their iterative format construction, error estimation, calculation efficiency, principal component contribution rate, etc. The following conclusions are drawn:  
1. The temperature field reconstructed by the POD dimension reduction is in good agreement with the results calculated by the classical central difference Galerkin spectral method, and the POD reduced-order extrapolation method is superior to the classical central difference Galerkin spectral method, namely: it reflects the advantages of the POD algorithm, such as a short running time, a high accuracy of calculation results, and a relatively small load of the algorithm.  
Figure 6: Error values of POD reduced-order extrapolation technology and center difference Galerkin spectral method at different time steps.  
2. This article uses the POD basis to derive a simplified spectral method format for transient two-dimensional nonlinear heat transfer problems. Moreover, the theoretical analysis of the relationship between the number of instantaneous images and the overall solution are an improvement and innovation of existing methods. We also validated the correctness of our theoretical method with numerical examples. In the effective computational space, the error will decrease with the decrease in the time step. However, along with a decrease in the time step, in the POD reduced-order and extrapolation environment, the computational load of the computer will also increase slightly in a short period of time, resulting in a slight delay in the corresponding calculation time. This means that selecting a reasonable time step in the POD reduced-order and extrapolation environment will obtain a more ideal calculation accuracy and energy contribution rate. This method can greatly save computation and memory capacity and improve computational efficiency. Although only a linear heat conduction problem is discussed in this article, our method can be extended to the calculation of more complex nonlinear problems. Our research work is to apply this reduced-order extrapolation technique to efficient numerical simulations of mid to deep geothermal energy, as well as more complex engineering calculation problems in the future. Furthermore, we will design more effective calculation methods that can better serve the HVAC field.  
Z.B.: Methodology; Software; Visualization; Writing--original draft; Y.W.: Supervision, Writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
The work is supported by the National Natural Science Foundation of China (51476073, 51266004), Gansu Province Natural Science Foundation (21R7 RA304).  
Not applicable.  
The original contributions presented in this study are included in the article and further inquiries can be directed to the corresponding authors.  
The authors declare no competing interests.