间断Galerkin方法求解Navier-Stokes和分数阶方程

发布时间：2018-06-22 17:33

本文选题：特征线方法 + 间断Galerkin方法　；参考：《兰州大学》2016年博士论文

【摘要】：本文我们主要考虑两类微分方程:时间依赖的不可压缩Navier-Stokes方程和二维空间Riemann-Liouville分数阶方程.对于Navier-Stokes方程,首先,我们引入一个辅助变量来分离扩散算子使原始的高阶方程转化为一阶系统,从而降低解决高阶方程的难度.其次,通过变分、精心地选择数值流、添加罚项,我们设计了一个稳定对称的局部间断Galerkin(LDG)格式.这样我们完成了方程的空间离散,但是我们并没有离散非线性对流项.我们知道求解Navier-Stokes方程最大的困难有两个:一是如何处理非线性对流项,另一个是如何处理方程中的压强函数.由于特征线方法在求解对流占优问题上有很大的优势,所以我们考虑特征线方法同时离散非线性对流项和时间导数项.从而,我们成功地解决了非线性问题,并且得到了一个稳定的对称的特征线局部间断Galerkin格式(CLDG).在稳定性推导的过程中,我们发现特征线方法使得理论推导简易明了.特别是应用格式的对称性,使得我们消去了一些比较难处理的算子.由于在NavierStokes方程中速度函数和压强函数没有紧密的联系,所以当我们得到速度的误差估计时,不是很容易得到压强的误差估计.为了解决这个困难,我们应用经典的速度和压强的连续上下界条件,建立速度误差和压强误差的联系.实际上,在间断有限元空间我们通常利用速度和压强的离散上下界条件,但是在本文,我们将连续上下界条件和离散空间结合起来得到压强的误差估计,这也是第三章的一个亮点.最后,我们给出四个不同的数值例子来验证理论结果,并且看到数值结果不仅达到了预期的效果而且比预期的更好.对于二维空间Riemann-Liouville分数阶方程,首先,我们引进两个辅助变量来分离Riemann-Liouville分数阶导数.由于Riemann-Liouville分数阶导数本身具有奇异性,Riemann-Liouville分数阶积分没有奇异性,所以一个辅助变量用来代替函数的梯度项,另一个辅助变量用来代替Riemann-Liouville分数阶积分.这样我们成功地分离了Riemann-Liouville分数阶导数并且把高阶导数方程降低为一阶系统.然后,通过变分、精确地选择数值流、添加罚项,我们设计了一个混合的间断Galerkin(HDG)格式,从而完成了方程的半离散.最后,我们给出三种时间离散方法,针对Riemann-Liouville分数阶扩散问题,我们应用一般的差分方法进行时间离散,由于方法的简易性,我们没有给出相应的离散过程.对于Riemann-Liouville分数阶对流扩散问题,我们应用一阶和二阶特征线方法同时离散时间导数项和对流项,并且给出相应的离散格式、稳定性分析、误差估计.在研究二维空间Riemann-Liouville分数阶方程时,我们发现如何找到一种有效的方法进行理论分析是比较困难的,特别是误差估计.为了解决这一困难,我们没有考虑有效的分析工具而是从格式设计出发,设计有效的数值格式.基于这样的考虑我们得到了不同的数值格式,不仅将一维问题推广到二维而且得到较好的数值结果.在第四章最后一节,我们分别应用三个数值例子验证Riemann-Liouville分数阶方程和一阶二阶HDG格式.总之,本篇论文我们成功地将间断Galerkin方法和特征线方法结合起来求解不可压缩的Navier-Stokes方程和二维空间分数阶微分方程.我们可以看到数值实验结果与理论结果是一致的并且是有效的.因此,这些方法可进一步改进且应用到其他问题中.
[Abstract]:In this paper, we mainly consider two kinds of differential equations: time dependent incompressible Navier-Stokes equation and two-dimensional space Riemann-Liouville fractional equation. For the Navier-Stokes equation, first, we introduce a auxiliary variable to separate the diffusion operator and turn the original high order equation into the first order system, thus reducing the high order equation. Secondly, we design a stable symmetric local discontinuous Galerkin (LDG) format by selecting the numerical flow and adding the penalty term by the variation of the variation. We have completed the space discretization of the equation, but we do not have the discrete nonlinear convection term. We know that there are two difficulties in solving the maximum Navier-Stokes equation: one is how The nonlinear convection term is dealt with, and the other is how to deal with the pressure function in the equation. Because the characteristic line method has a great advantage in solving the convection dominated problem, we consider the characteristic line method to discrete the nonlinear convection term and the time derivative. Thus, we successfully solve the nonlinear problem and get a stable problem. In the process of stability derivation, we find that the characteristic line method makes the theoretical derivation simple and clear in the process of stability deduction. In particular, the symmetry of the applied format makes us eliminate some of the more difficult operators. Because the velocity function and pressure function in the NavierStokes equation are not tight, we have not tightened the pressure function in the NavierStokes equation. In order to solve this problem, we apply the continuous upper and lower bounds of the classical velocity and pressure to establish the relation between the velocity error and the pressure error. In fact, we usually use the velocity and pressure in the discontinuous finite element space. In this paper, we combine the continuous upper and lower boundary conditions with the discrete space to get the error estimation of the pressure, which is also a bright spot in the third chapter. Finally, we give four different numerical examples to verify the theoretical results and see that the number results not only achieve the expected effect but also compare with the expected results. Better. For the two-dimensional space Riemann-Liouville fractional equation, first, we introduce two auxiliary variables to separate the Riemann-Liouville fractional derivative. Because the fractional derivative of the Riemann-Liouville is singularity itself, the fractional integral of Riemann-Liouville has no singularity, so a auxiliary variable is used to replace the gradient of the function. The other auxiliary variable is used to replace the Riemann-Liouville fractional integral. So we successfully separate the Riemann-Liouville fractional derivative and reduce the high order derivative equation to the first order system. Then, we choose the numerical flow accurately by the variational, and add the penalty term, and we design a mixed discontinuous Galerkin (HDG) format. And finally, we have completed the semi discrete equation. Finally, we give three time discrete methods. In view of the Riemann-Liouville fractional diffusion problem, we use the general difference method to make time discretization. Because of the simplicity of the method, we do not give the corresponding discrete process. For the Riemann-Liouville fractional convection diffusion problem, we should do it. The first and two order characteristic lines are used to simultaneously discrete time derivatives and convective terms, and the corresponding discrete schemes, stability analysis and error estimation are given. In the study of the two-dimensional space Riemann-Liouville fractional equation, it is found that it is difficult to find an effective method for the theory analysis, especially the error estimation. In order to solve this problem, we do not consider the effective analysis tool but design the effective numerical format from the format design. Based on this, we get different numerical schemes, not only to extend the one-dimensional problem to two-dimensional but also to get better numerical results. In the last section of the fourth chapter, we apply three respectively. The numerical examples verify the Riemann-Liouville fractional equation and the first order two order HDG scheme. In this paper, we successfully combine the discontinuous Galerkin method and the characteristic line method to solve the incompressible Navier-Stokes equation and the two-dimensional fractional order differential equation. We can see that the numerical experiment results are in agreement with the theoretical results. And they are effective. Therefore, these methods can be further improved and applied to other problems.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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