Burgers方程的高阶人工边界条件法

发布时间：2018-06-20 11:01

本文选题：Burgers方程 + 高阶人工边界条件　；参考：《北方工业大学》2017年硕士论文

【摘要】：本文主要研究了无界域的Burgers方程的高阶人工边界方法。第一章首先介绍了 Burgers方程的研究价值,解决无界域问题的常用方法,以及人工边界条件法的应用。第二章应用积分型人工边界条件来求无界域的一维Burgers方程的数值解。使用Hopf-Cole变换将原问题转换为无界域中的热传导方程,再引入两个积分型的人工边界条件,将得到的热传导方程简化为有界的计算域中的等价方程,之后用降阶法为这个等价方程构建有限差分格式,求解线性方程组即可得到所得热传导方程的数值解,进而得到原Burgers方程的数值解。该方法被证明是唯一可解、无条件稳定的,且具有空间上的2阶收敛和时间的3/2阶收敛,并用算例加以验证。第三章应用高阶人工边界条件来求无界域的Burgers方程的求解。首先,同样通过Hopf-Cole变换将原来的Burgers方程(非线性)转化为无界域中的热传导方程(线性),即克服了Burgers方程本身的非线性问题。然后,通过使用Pade逼近、Laplace变换及其逆变换给出高阶人工边界条件将所得的热传导方程限制在有限的计算域上。之后,我们证明了所得的热传导方程与Burgers方程解的稳定性。之后,将应用高阶人工边界条件而得到有限计算域的热传导方程,只在空间方向上通过Taylor展开进行离散。对于所得空间方向上离散的热传导方程,在理论上证明了求得的半离散解的稳定性及收敛性。最后,我们在这个有界的计算域上建立了 Burgers方程的有限差分格式,并用两个数值例子说明了该方法的稳定性、有效性,且在空间方向上二阶收敛,在时间方向上约为二阶收敛。
[Abstract]:In this paper, the higher order artificial boundary method for unbounded Burgers equation is studied. In the first chapter, the research value of Burgers equation, the common methods to solve the unbounded domain problem and the application of artificial boundary condition method are introduced. In chapter 2, the integral artificial boundary condition is used to obtain the numerical solution of one-dimensional Burgers equation in unbounded domain. The original problem is transformed into the heat conduction equation in the unbounded domain by Hopf-Cole transform. Two artificial boundary conditions of integral type are introduced, and the obtained heat conduction equation is simplified to the equivalent equation in the bounded computational domain. Then the finite difference scheme is constructed for the equivalent equation by using the reduced order method, and the numerical solution of the heat conduction equation is obtained by solving the linear equation system, and the numerical solution of the original Burgers equation is obtained. It is proved that the method is solvable, unconditionally stable, and has the convergence of order 2 in space and the convergence of order 3 / 2 in time, which is verified by an example. In chapter 3, the higher order artificial boundary condition is used to solve the Burgers equation in unbounded domain. Firstly, the original Burgers equation (nonlinear) is also transformed into the heat conduction equation in the unbounded domain by Hopf-Cole transformation (linear equation), which overcomes the nonlinear problem of Burgers equation itself. Then, by using the Pade approximation Laplace transform and its inverse transformation, the higher order artificial boundary conditions are given to limit the heat conduction equation to the finite computational domain. Then we prove the stability of the solutions of the heat conduction equation and the Burgers equation. After that, the heat conduction equations in finite computational domain are obtained by using higher order artificial boundary conditions, and are discretized only in the spatial direction by Taylor expansion. The stability and convergence of the obtained semi-discrete solutions are theoretically proved for the discrete heat conduction equations in the direction of the obtained space. Finally, we establish the finite difference scheme of Burgers equation in this bounded computational domain. Two numerical examples are given to illustrate the stability and validity of the method, and the second order convergence in the space direction. In the time direction, the convergence is about second order.
【学位授予单位】：北方工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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