【作者】 邓镇国；

【导师】 马和平；

【作者基本信息】 上海大学 ， 计算数学， 2009， 博士

【摘要】 非局部非线性色散波方程是描述密度分层流体内重力波传播过程的一类模型方程.既然大多数重力内波产生于海水和大气,那么研究这类方程解的性质对于深海石油钻探、水下导航、数值天气预报等都有重要的应用价值,同时对于流体力学、大气学和海洋学等具有重要的理论意义.但是这类方程的色散关系是非局部的,分析方程解的性质不是件容易的事情.因此寻求有效的数值算法是必要的.目前,Fourier变换与非局部算子之间的特殊关系使得Fourier谱或拟谱方法已经成为数值求解这类方程的重要工具.本文主要解决了三个问题:首先,改进了Pelloni和Dougalis对一类非局部非线性色散波方程(包括Benjamin-Ono方程和中等长波方程)的Fourier谱方法给出的L~2误差估计;其次,最近Thomee和Murthy,Pelloni和Dougalis分别在文章中指出,尽管Fourier拟谱方法对Benjamin-Ono方程在数值计算方面是十分有效的,但没有任何的误差分析,本文的工作很好地回答了这个问题;最后,改进了Maday和Quarteroni对Korteweg-de Vries方程的Fourier谱方法给出的L~2误差估计.在第三章,对一类非局部非线性色散波方程的周期边界问题建立了能够显式计算的全离散Fourier谱方法逼近格式,对方程的非线性项显式处理和对线性项隐式处理,改进了Pelloni和Dougalis的L~2误差估计,使之提高到丰满(最优),并且能够放宽对时间步长的限制.在第四章,对最近Thomee和Murthy,Pelloni和Dougalis分别在文章中指出的问题,我们直接对一类非局部非线性色散波方程(包括Benjamin-Ono方程和中等长波方程)建立了能够显式计算的全离散Fourier拟谱逼近格式,利用分数次Sobolev范数度量误差,证明了该格式的稳定性和收敛性,并且具有谱精度.此外,通过一些数值例子表明了本文算法的高精度性和稳定性,并且与其他方法作了比较.在第五章,将第三章中所涉及的方法和证明技巧成功地推广到Korteweg-deVries方程的周期边界问题,改进了Maday和Quarteroni给出的L~2误差估计,使之提高到丰满(最优).此外,通过数值模拟最近受到关注的初始状态重现实验(zabusky和Kruskal),表明本文算法具有很好的计算稳定性.在第六章,讨论了一类非局部非线性色散波方程的修正Fourier拟谱逼近格式,证明了该格式的稳定性和收敛性,并且L~2误差估计是丰满(最优)的.更多还原

【Abstract】 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.更多还原