间断Galerkin 方法求解Navier-Stokes 和分数阶方程

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	间断Galerkin 方法求解Navier-Stokes 和分数阶方程
Alternative Title	The discontinuous Galerkin methods for solving Navier-Stokes and fractional equations
	王淑琴
Thesis Advisor	伍渝江 ; Jinyun Yuan
	2016-05-29
Degree Grantor	兰州大学
Place of Conferral	兰州
Degree Name	博士
Keyword	特征线方法间断Galerkin 方法 Navier- Stokes 方程空间分数阶微分方程
Abstract	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. 由于在~Navier-Stokes 方程中速度函数~$\bm u$ 和压强函数~$p$ 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. 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. 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. 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.
Other Abstract	For Navier-Stokes equations, firstly, we introduce an auxiliary variable to split the diffusion operator and make the high-order equation into one order system, then we reduce the difficulty of the high-order problem. Secondly, making the variational formulas, carefully choosing numerical fluxes and adding penalty terms, we propose a stable and symmetric local discontinuous Galerkin (LDG) scheme. Because the velocity function $\bm u$ and the pressure function $p$ do not have the closed relation in the Navier-Stokes equations, then after getting the error estimate of velocity, it is not easy to get the estimate of pressure. In order to tackle this difficulty, we use the continuous inf-sup condition for velocity and pressure to make a connection for velocity and pressure. Finally, we give four different examples to verify theoretical results, and find that the numerical results reach the desired results and even much better. For the space-fractional Riemann-Liouville equations in 2D, firstly, we introduce two auxiliary variables to split the Riemann-Liouville derivative. Since the Riemann-Liouville derivative has the singularity, Riemann-Liouville integral does not have the sigularity, then we use one auxiliary variable to substitute the grad term, another one to substitute the Riemann-Liouville integral. Secondly, making the variational formulas, carefully choosing numerical fluxes and adding the penalty terms, we obtain a hybridized discontinuous Galerkin (HDG) scheme, then we accomplish the semi-discrete process. Finally we give three ways to perform the time discretization, i.e., for the Riemann-Liouville diffusion problems, we use the general finite difference method to discretize the time derivative. In the last section of the chapter 4, we use three numerical examples to verify the Riemann-Liouville equations and the one order and the two order HDG scheme, respectively.
URL	查看原文
Language	中文
Document Type	学位论文
Identifier	https://ir.lzu.edu.cn/handle/262010/225023
Collection	数学与统计学院
Recommended Citation GB/T 7714	王淑琴. 间断Galerkin 方法求解Navier-Stokes 和分数阶方程[D]. 兰州. 兰州大学,2016.

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