Fourier Spectral Methods For A Class Of Nonlocal, Nonlinear Dispersive Wave Equations

The nonlocal,nonlinear dispersive wave equations are a class of model equations for describing the propagation of gravitational waves in the density stratified fluid.Since most of the gravitational waves arise in the seawater and the atmosphere,studying the properties of the solution to the equations not only has the important application in deepsea oil drilling,underwater navigation and numerical weather prediction,but also has the theoretical significance for fluid mechanics,atmospheric sciences and oceanic sciences. But the dispersive relations of the equations are nonlocal and thus it is not an easy thing to analyze the properties of the solution to the equations.Hence it is necessary to find an efficient numerical method.At present,the Fourier spectral/pseudo-spectral method provides a powerful technique for the numerical solutions of such problems due to the special relations between Fourier transform and nonlocal operator.This paper mainly solves three problems.First,we improve the error estimates in L~2- norm of the Fourier spectral method for a class of nonlocal,nonlinear dispersive wave equations including the Benjamin-Ono equation and intermediate long wave equation by Pelloni and Dougalis;Second,recently Thomee and Murthy,Pelloni and Dougalis,point out in their papers respectively that Fourier pseudo-spectral method solves the Benjamin-Ono cquaiton well but no error analyses are given,and our works answer the problem well; Third,we improve the error estimates in L~2- norm of the Fourier spectral method for the Korteweg-de Vries equation.In Chapter 3,we establish the fully discrete spectral method for the explicitly numerical solution to periodic boundary-value problem for two nonlocal,nonlinear dispersive wave equations,the Benjamin-Ono and the Intermediate Long Wave equations.We treat the linear terms in the equation implicitly and the nonlinear terms cxplicitly.We improve the error estimates in L~2-norm by Pelloni and Dougalis and make them optimal.In addition,we relax the restriction on the time-step.In Chapter 4,for the recent problem pointed out in the papers by Thomnee and Murthy,Pelloni and Dougalis,we directly present the fully discrete spectral method for the explicitly numerical solution to periodic boundary-value problem for two nonlocal, nonlinear dispersive wave equations,the Benjamin-Ono and the Intermediate Long Wave equations.Using the fractional order Sobolev norm for measuring the error,we prove the stability and spectral accuracy of the method.In addition,some numerical examples are given to show the high order and stability of our method and our method is compared with other methods.In Chapter 5,we successfully generalize the methods and proving skills involved in Chapter 3 to the boundary-value problem for the Korteweg-de Vrics equation.We improve the error estimates in L~2- norm by Maday and Quarteroni and make them optimal.In addition,numerically modeling the recent attractive experiment for the recurrence of initial states by Zabusky and Kruskal,shows that our method has good computational stability.In Chapter 6,we discuss the modified Fourier pseudo-spectral method for a class of nonlocal,nonlinear dispersive wave equations.We prove the stability and convergence of the method and obtain the optimal error estimates in L~2-norm.