Spectral Methods And Spectral Element Methods For Some Evolutionary Equations

The spectral method,which constitutes a large part of computational mathematics, is one of important numerical methods for solving partial differential equations. The main advantage of the spectral method is so-called "spectral accuracy", which is competitive with the finite difference and the finite element method. However , the spectral method fails to solve the problems defined on complex geometries, so there are limitatiions to its extensive applications. A key to these limitations is the domain decomposition technique which the finite element method is based on—that is the spectral element method.It is very important to analyse errors caused by the spectral (spectral element) discrete scheme for a specific problem. There are a lot of studies about this, but they are not sufficiently subtle and optimal. Especially, for nonlinear problems, there is only a few beautiful work. This dissertation is devoted to spectral methods and spectral element methods for some evolutionary equations. This dissertation employs error analysis of the spectral mehtods and spectral element methods, and gives estimates of stability and convergence results .The main contents of the dissertation are as following:Firstly, a Legendre Petrov Galerkin spectral discrete scheme for fifth-order KdV equation is set up. The method is a generalization of Legendre-Petrov-Galerkin for third-order KdV equation [64]. This method is proved to have the results of stability. It shows that an optimal order of convergence is obtained.Secondly, we discuss a spectral element discretization for convection-diffusion type equations in one space dimensional. Error analysis is presented for the semi-discrete and fully-discrete scheme. And an optimal order of convergence is obtained. In view with the implementation of the approach, parallel program is designed. Efficiency of the approach is verifed by numerical experiments.Thirdly , for linear Schrodinger equation in two space dimensional, we take Crank-Nicolson scheme for time discritization and spectral element scheme for space discretization. Stability of the scheme is examined and optimal order a prior L~2-and H~1- error estimates at each time step are derived. Moreover, we consider alternating direction implicit scheme instead of Crank-Nicolson scheme. Error analysis gives the optimal order a prior L~2- and H~1-error estimates. The implementation of the ADI scheme is discussed. No work like the above has been found in current literature. Next, we establish a Legendre spectral approximation for a class of nonclassical nonlinear parabolic equations in one space dimensional. Stability of the scheme is examined and optimal order of convergence rate is obtained. Then Chebyshev pseudospectral method and Chebyshev-Legendre pseudospectral method is established for the equations, Numerical test tells efficiency of the methods. In all tentative study of the numerical solution to the problems, it is verified that the application of spectral method by numerical experiments and theoretical analysis are successful.Finally, We present a stabilized Hermite spectral method for the generalized Burgers equations in unbounded domain. We try to use Hermite polynomials directly as approximation solution of problem in the method. We obtain stability and convergence of the method. Error estimate shows an optimal order of convergence rate of the method. Numerical experiments verify the theoretical results. In addition, we discuss the pseudo-spectral method and composite Hermite spectral method.In the dissertation conventional energy estimate method is used to establish stability and convergence results. And the stability for nonlinear problems is in the sense of generalized stability proposed by Guo Ben-yu.