Fully Diagonalized Chebyshev Spectral Methods

As an effective numerical method for solving differential equations,spectral method has gained rapid development in recent decades.Compared with other methods,the main advantage of this method is its high order accuracy,which makes it become one of the most important tools for numerical solutions of various practical problems in science and engineering.Spectral method has also been applied to the numerical simulations of various problems such as fluid mechanics,quantum mechanics and material science,etc.The mathematical models in many practical problems are based on the boundary value problems of differential equations.The research on the numerical methods of second or fourth order elliptic boundary value problems is a very important and meaningful work both in theory and in practice.At present,many effective computational methods have been developed for such problems,and the relevant references are numerous.In this thesis,we study the fully diagonalized Chebyshev spectral method for solving the second and fourth order elliptic boundary value problems.They are based on appropriate base functions for the Galerkin formulations which are complete and biorthogonal with respect to certain Sobolev inner product.The suggested base functions lead to diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series.Numerical results demonstrate the effectiveness and the spectral accuracy.This thesis is divided into four chapters:The first chapter is an introduction.We first introduce the history of Chebyshev spectral method.Then we point out the importance to choose the appropriate base functions,and introduce the related spectral methods as well as the existing base functions.Finally,we briefly describe the research contents and innovations of this thesis.In the second chapter,some notations and basic properties of Chebyshev polynomials are introduced.In the third chapter,we establish new Chebyshev basis functions for second-order problems,which are biorthogonal with respect to certain Sobolev inner product.Accordingly,we construct the diagonalized Chebyshev spectral method for solving the second-order elliptic boundary value problem with Direchlet boundary conditions.Some numerical results are presented to demonstrate its high accuracy.In the fourth chapter,we establish new Chebyshev basis functions for fourth-order problems,which are Sobolev biorthogonal.Accordingly,we construct the diagonalized Chebyshev spectral method for solving the fourth-order elliptic boundary value problem with Direchlet boundary conditions.Numerical results also show the accuracy and efficiency of the proposed method.