In order to meet the requirements of high accuracy and fast algorithm for numerical heat transfer simulation, a POD reduced-order extrapolation algorithm based on the iterative format of the classical central difference Gakerkin spectral method is proposed for solving two-dimensional transient heat conduction problems. Using the calculation results of classical central difference Gakerkin spectral method as sample data constructs POD reduced-order spectral method model; A study on numerical algorithm characteristics of flow and heat transfer using partial differential equation as mathematical model, and gives the error estimate; Finally, taking different time steps as parameters to conduct experimental simulation in turn. The results show that: the spectral method based on POD reduced-order extrapolation method is at least 253 times faster than the central difference Gakerkin spectral method, and gradually decreases with the interval of variable parameters. The calculation speed of POD reduced-order model and the calculation speed of central difference are both reduced. However, the average relative error between the temperature field reconstructed by POD reduced dimension model and the central difference result is smaller and the calculation accuracy is higher, the maximum value is not more than 0.11%.The average computational efficiency is 458 times higher than that of the traditional central difference Gakerkin spectral method.

This paper generalizes a class of controllability problems based on the scale structure population system model. Based on the comparison principle of linear systems, the solution of the nonlinear system model is obtained by referring to the fixed point theorem. The non-negative, boundedness, existence, and uniqueness of the solution of the system model are established. The optimality condition is described in detail by means of a normal cone and conjugate system under the condition of proving the continuous dependence of the state environment on the solution to control variables.

In order to meet the requirements of high accuracy and fast algorithm for numerical heat transfer simulation, an iterative scheme of Proper Orthogonal Decomposition (for short, POD) dimension reduction based on the classical central difference Galerkin spectral method is proposed for solving two-dimensional transient heat conduction problems. The POD dimension reduction spectral method model is constructed by taking the calculation results of classical central difference Galerkin spectral method as sample data. The numerical algorithm characteristics of flow and heat transfer are studied by using a partial differential equation as a mathematical model, and the error estimation is given. Finally, different time intervals are used as parameters to simulate experiments. The results show that the POD method is applicable to transient nonlinear heat conduction problems, and the maximum average relative error of the reconstructed temperature field is 0.89675%. Moreover, the POD method not only has a high calculation accuracy, but also has an average calculation speed as high as 310.25 times that of the central difference Galerkin algorithm. It can be seen that under the condition that the error between the solution of POD dimension reduction extrapolation algorithm and the solution of classical central difference Galerkin spectrum method is small enough, the POD method can greatly reduce the calculation amount, shorten the running time, and ensure a high accuracy of the calculation results, thus verifying the effectiveness and feasibility of the algorithm.