【作者】 王敏；

【导师】 李德生；

【作者基本信息】 沈阳师范大学 ， 应用数学， 2011， 硕士

【摘要】 众所周知,守恒的差分格式优于非守恒的差分格式。1995年Zhang Fei等人指出非守恒的差分格式容易出现非线性的爆破现象。同年,Li和Vuesquoc也指出“在许多领域,保持原有微分方程的一些固有的属性是判断一种数值模拟成功的标准”。近年来,一些守恒的差分格式分别用来求解广义非线性Sehrodinger方程,正则长波方程,Sine-Gordon方程,Klein-Gordon方程和Zakharov方程,并得到较好的数值结果。人们常常从模拟定解问题的能量守恒律出发构造差分格式,这样的差分格式被称为守恒型格式。本文的目的就是构造一些新的守恒差分格式求解广义Rosenau方程。广义Rosenau方程非线性项包含参数p,当p=1时,就是著名的Rosenau方程。广义Rosenau方程比Rosenau方程更具有一般性,由于目前尚不存在关于该方程精确解的研究报道,所以其数值解的是非常重要的。本文对广义Rosenau方程的初边值问题提出了几类守恒的差分格式,利用Taylor级数展开法建立两个两层的非线性有限差分格式和一个三层的线性有限差分格式,并利用Brouwer不动点定理证明了这些差分格式的存在性,利用离散能量分析方法证明了这些差分格式的收敛性和稳定性。更多还原

【Abstract】 It is well known that conservative difference scheme is better than nonconservation difference scheme. Zhang fei pointed out that non-conservation diffeence scheme can appear nonlinear blasting phenomena easily. Li and Vu-quoc pointed out "in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation."The generalized nonlinear Sehrodinger equation, regularized long wave equation, Sine-Gordon equation, Klein-Gordon equation and Zakharov equation have been solved by some conservative difference scheme, recently, and richer numerical results have been obtained.Difference scheme is constructed by stimulating energy conservation law of set solution, such difference scheme is called conservative scheme.In this paper, some new conservative difference scheme has been prospered to solve generalized Rosenau equation. Parameter p is contained in nonlinear item of generalized Rosenau equation, which is famous Rosenau equation when p=1. The generalized Rosenau equation has more universal than Rosenau equation, however, there is no report about exact solutions of generalized Rosenau equation, so its numerical solution is very important.Several conservative difference schemes of initial boundary value problem of generalized Rosenau equation have been proposed in this paper, and two two-level nonlinear finite difference schemes and a three-level linear finite difference scheme are given by Taylor series expansion, respectively, and the existence of the solution of difference scheme is proved by Brouwer fixed point theorem, the convergence and the stability of these difference schemes are proved by discrete energy analysis.更多还原