Study Of Chebyshev Collocation Spectral Method Based On Fast Transformation

Spectral methods have the properties of high-order accuracy and exponential conver-gence.With the development of computer technology,the numerical simulations of real prob-lems using spectral methods have become popular.Generally speaking,spectral methods can be executed by two ways,the first is through matrix multiplication transformation,and the second is through fast transformation.Spectral methods based on matrix multiplication transformation are faster than fast transformation if the grid number is smaller.However,the computational efficiency of fast transformation is better than matrix multiplication transformation if the grid number is larger,thus,it is best to choose the fact transformation for large scale complex problems.In this study,the Chebyshev spectral method based on fast transformation is used to solve the 1D and 2D equations.In the computation,the Chebyshev-Gauss-Lobatto collocation point is used for spatial approximation,and the finite difference scheme is used for time ap-proximation.The main work of present dissertation includes the following three aspects:1.The Chebyshev collocation spectral method based on fast transformation is used to solve the strongly nonlinear hear transfer of a fin with variable thermal conductivity,convec-tion heat transfer coefficient and internal heat generation.The feasibility and accuracy are validated by comparing the present solutions with the solutions which are based on matrix multiplication transformation,and then the difference of processing of nonlinear equation is also considered.The results show that the fast transformation can provide high accuracy and solve the nonlinear equation in 1D more directly.2.The Chebyshev collocation spectral method based on fast transformation is used to solve the 2D Poisson equation in Cartesian system.The present solutions are compared with exact solutions.The results show that the present method has good accuracy and efficiency for solving 2D problems.3.The Chebyshev collocation spectral method based on fast transformation and matrix multiplication transformation are used to solve the 1D and 2D Poisson equations with transi-ent terms respectively.Then the computation time in the same condition with different grid number is compared between them.The results show that when the grid number is less than 200,the matrix multiplication transformation is faster than fast transformation and when the grid number is more than 200,the fast transformation is more efficient than matrix multiplica-tion transformation.This will provide some references for using spectral methods in the fu-ture.