几类发展方程的紧致差分法研究

发布时间：2018-03-04 18:31

本文选题：发展方程　切入点：指数时间差分法　出处：《南昌航空大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文主要研究几类发展方程的紧致差分法,并对设计的相应数值格式进行理论分析,通过一些数值算例来验证数值算法的准确性和有效性。本文共五章,具体的研究工作如下:第二章主要致力于一维Burgers方程的两种高阶数值求解方法的发展和应用,这两种方法在时空方向均有四阶精度。其中,方法一在时间方向使用Crank-Nicolson格式和Richardson外推法,在空间方向用四阶紧致差分法逼近;方法二采取基于padé逼近的时间步方法和空间四阶的紧致差分法。另外,我们运用矩阵分析法分别研究了这两种方法的稳定性。数值实验证实了新算法的合理性和高效性。第三章研究了一维非线性常延迟反应扩散方程的紧致差分法,并运用能量法证明差分解在最大范数意义下具有O(t~2+h~4)的收敛阶。接着,在时间方向运用Richardson外推法,获得了O(t~4+h~4)外推解。然后,将该数值方法推广到其它复杂的延迟问题。最后,数值算例验证了算法的计算精度和有效性。第四章对一维粘性波动方程,构造一个三层紧致差分格式,并运用能量法进行误差分析,证明差分格式在最大范数意义下有O(t~2+h~4)的收敛阶。利用Richardson外推法,得到O(t~4+h~4)的外推解。最后,给出一个数值算例,证实该差分格式的收敛阶和有效性。第五章对全文进行了总结、展望。
[Abstract]:In this paper, the compact difference method for several kinds of evolution equations is studied, and the corresponding numerical schemes are theoretically analyzed. Some numerical examples are used to verify the accuracy and validity of the numerical algorithm. The specific research work is as follows: in chapter 2, we focus on the development and application of two high-order numerical methods for solving one-dimensional Burgers equations, both of which have fourth-order accuracy in the space-time direction. Methods one is to use the Crank-Nicolson scheme and Richardson extrapolation in the time direction, the other is to approximate the space direction by the fourth-order compact difference method, the second method is the time step method based on pad 茅 approximation and the space fourth-order compact difference method. We use matrix analysis method to study the stability of these two methods. Numerical experiments show that the new algorithm is reasonable and efficient. In chapter 3, we study the compact difference method for nonlinear constant delay reaction diffusion equation. The energy method is used to prove the order of convergence of the difference decomposition with 2 h ~ 4) in the sense of maximum norm. Then, using the Richardson extrapolation method in the time direction, the extrapolation solution is obtained. Then, the numerical method is extended to other complex delay problems. Numerical examples verify the accuracy and validity of the algorithm. In Chapter 4th, a three-layer compact difference scheme is constructed for one-dimensional viscous wave equation, and the error is analyzed by energy method. It is proved that the difference scheme has the order of convergence in the sense of the maximum norm. By using the Richardson extrapolation method, the extrapolation solution of OT4) is obtained. Finally, a numerical example is given. The convergence order and validity of the difference scheme are confirmed. Chapter 5th summarizes and prospects the full text.
【学位授予单位】：南昌航空大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

【参考文献】