修正局部Crank-Nicolson格式对Rosenau-Burgers方程的应用

发布时间：2018-04-01 00:17

本文选题：Rosenau-Burgers　切入点：方程　出处：《新疆大学》2017年硕士论文

【摘要】：Rosenau-Burgers方程是自然界中的一类很重要的动力学模型,它广泛出现在爆炸和水波的传播问题,离散动力学问题,波动力学等众多领域.因为非线性项的处理困难,给数值求解该问题带来一定的困难.因此构造有效数值计算方法的研究具有重要的理论和现实意义.本文首先Rosenau-Burgers方程的非线性项滞后一个时间步长来进行线性化.线性的Rosenau-Burgers方程空间变量进行中心差分离散,把所研究的方程转化为常微分方程组,其次利用指数函数的Trotter Product公式来近似该常微分方程组的系数矩阵,然后五对角稀疏矩阵分别按行和按元素分离成一些简单矩阵的和形式,再利用Crank-Nicolson方法提出两种求解Rosenau-Burgers方程定解问题的修正局部Crank-Nicolson格式,该文提出的两种数值格式对空间和时间具有二阶精度的绝对稳定的显式差分格式.数值格式通过Taylor级数展开进行了局部截断误差分析,相容性给出了证明.能量不等式方法证明了稳定性,并且收敛性给出了证明.为了验证两种数值格式的有效性,举例两种数值算例进行数值实验,本文中两种数值格式的实验结果与参考文献中几个差分格式的数值结果进行比较,发现本文两种差分格式比其它格式的数值结果具有明显的优越性.两种差分格式而言,按元素分裂的修正局部Crank-Nicolson 格式的数值结果比按行分裂的修正局部 Crank-Nicolson 格式的数值结果好一些.本文按行和按元素分裂的修正局部Crank-Nicolson格式不仅求解Rosenau-Burgers方程,而且数值求解非线性偏微分方程提供了参考。
[Abstract]:Rosenau-Burgers equation is a very important dynamic model in nature. It is widely used in explosion and water wave propagation problem, discrete dynamics problem, wave mechanics and so on. It is difficult to solve the problem numerically. Therefore, the study of constructing effective numerical method is of great theoretical and practical significance. In this paper, the nonlinear term of Rosenau-Burgers equation is linearized by one time step behind the nonlinear term. The spatial variables of linear Rosenau-Burgers equation are discretized by central difference. The studied equation is transformed into ordinary differential equation system, then the coefficient matrix of the ordinary differential equation system is approximated by the Trotter Product formula of exponential function, then the five-diagonal sparse matrix is separated into some simple matrix sum form by row and element, respectively. Then two modified local Crank-Nicolson schemes for solving the definite solution of Rosenau-Burgers equation are proposed by using the Crank-Nicolson method. In this paper, two explicit difference schemes with second order accuracy for space and time are presented. The local truncation error is analyzed by Taylor series expansion. In order to verify the validity of the two numerical schemes, two numerical examples are given for numerical experiments. By comparing the experimental results of the two schemes with the numerical results of several difference schemes in the reference, it is found that the two schemes have obvious advantages over the numerical results of the other schemes. The numerical results of the modified local Crank-Nicolson scheme by element splitting are better than those of the modified local Crank-Nicolson scheme by line splitting. In this paper, the modified local Crank-Nicolson scheme by line and by element is not only used to solve the Rosenau-Burgers equation. Moreover, the numerical solution of nonlinear partial differential equations provides a reference.
【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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