网易 
新闻 LOFTER 邮箱 相册 阅读 图书 有道 摄影 企业邮箱 优惠券 云笔记 闪电邮 邮箱大师 印像派 考拉海购
 
更多
博客 
手机博客 博客搬家 博客VIP服务 发现 小组 风格
 
LiveWriter写博 word写博 邮件写博 短信写博
 
群博客 博客油菜地 博客话题 博客热点 博客圈子 找朋友
 
注册 登录  
 加关注
   显示下一条  |  关闭
温馨提示！由于新浪微博认证机制调整，您的新浪微博帐号绑定已过期，请重新绑定！立即重新绑定新浪微博》  |  关闭

阿弥陀佛

街树飘影未见尘 潭月潜水了无声 般若观照心空静...

导航

 
 
 
 
 

日志

 
 
关于我
险峰

一直从事气象预报、服务建模实践应用。 注重气象物理场、实况场、地理信息、本体知识库、分布式气象内容管理系统建立。 对Barnes客观分析, 小波,计算神经网络、信任传播、贝叶斯推理、专家系统、网络本体语言有一定体会。 一直使用Java、Delphi、Prolog、SQL编程。

文章分类
网易考拉推荐

有限体积法背景  

2016-09-30 14:42:52|  分类： Spark |  标签： |举报 |字号大中小 订阅

  下载LOFTER 我的照片书  |

Some Background on Finite Volume Methods

We are generally interested in solving PDE's of the form

$\displaystyle \frac{\partial \textbf u}{\partial t} + \frac{\partial \textbf f(\textbf u)}{\partial \textbf x} = 0 $

For the moment, let's focus our attention even further, on one of the simplest PDE's of that form, known as Burger's equation (inviscid form).

$\displaystyle \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0 $
Though this class of equations may appear relatively straightforward, it turns out that many such PDE's lead to the formation of discontinuities, or shocks, even from smooth initial data. In order to solve PDE's whose solutions contain shocks, we must turn to methods which do not assume the continuity of the solution, unlike the more intuitive finite difference methods. One way in which we can make sense of such solutions is to think of the average value of the solution over a given cell, rather than the value at specific grid points. Let the cell $C_{i}^{n+\frac{1}{2}}$ describe the spatial region between $x_{i-\frac{1}{2}}$ and $x_{i+\frac{1}{2}}$ and the temporal region between $t_i$ and $t_{i+1}$. The average value of the solution over this cell is
\begin{displaymath}\textbf u_i^n = \frac{1}{\triangle x} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} u(x,t_n)\ \,dx\ \end{displaymath}

and let the numerical flux F be given by


\begin{displaymath}\textbf F^{n+\frac{1}{2}}_i = \frac{1}{\triangle t} \int_{t_n}^{t_{n+1}} f(u(x_i,t))\ \,dt\ \end{displaymath}

Integrating the conservation form of Burger's eqn. and dividing by total volume gives,


\begin{displaymath}\frac{1}{\triangle x \triangle t}\int_{t_n}^{t_{n+1}} \int_{x... ...\frac{1}{2}}} \frac{\partial f(u)}{\partial x} \,dx\\ \,dt\ = 0\end{displaymath}


\begin{displaymath}\frac{1}{\triangle x \triangle t} \int_{x_{i-\frac{1}{2}}}^{x... ...}} [f(u_{i + \frac{1}{2}}) - f(u_{i - \frac{1}{2}})]\ \,dt\ = 0\end{displaymath}


\begin{displaymath}\frac{\textbf u_i^{n+1} - \textbf u_i^n}{\triangle t} + \fr... ...} - \textbf F^{n+\frac{1}{2}}_{i-\frac{1}{2}}}{\triangle x} = 0\end{displaymath}

If we can find a way to estimate the numerical fluxes $\textbf F^{n+\frac{1}{2}}_{i+\frac{1}{2}}$ and $\textbf F^{n+\frac{1}{2}}_{i-\frac{1}{2}}$ using information from the current time step, we will have a method for advancing the solution in time. Specifically


\begin{displaymath}\textbf u_i^{n+1} = \textbf u_i^n - \frac{\triangle t}{\tri... ...}}_{i+\frac{1}{2}} - \textbf F^{n+\frac{1}{2}}_{i-\frac{1}{2}})\end{displaymath}

To see how this can be implemented in practice, go to the next section

Roe Solver

One technique for estimating these numerical fluxes, developed by Roe, involves linearizing the system by evaluating the Jacobian matrix $\textbf A = \frac{\partial \textbf f}{\partial \textbf u}$ at $(x_i, t_n)$ and approximating $\textbf f(\textbf u) = \textbf A \textbf u$. The original equation then becomes $\frac{\partial \textbf u}{\partial t} + \textbf A \frac{\partial \textbf u}{\partial x} = 0$. If the basis of eigenvectors $r_\alpha$ of A is chosen, the system is uncoupled and is reduced to a series of 1D equations of the form


\begin{displaymath}\frac {\partial u_{(\alpha)}}{\partial t} + \lambda_\alpha \frac {\partial u_{(\alpha)}}{\partial x} = 0\end{displaymath}

where $\lambda_\alpha$ is the eigenvalue corresponding to the eigenvector $r_\alpha$. Given initial data


\begin{displaymath}u_{(\alpha)}(x, 0) = \left\{ \begin{array}{lr} u_L & \quad ... ...2}}\\ u_R & \quad x > x_{i+\frac{1}{2}} \end{array} \right. \end{displaymath}

the value at $x_{i+\frac{1}{2}}$ at subsequent times is $q_L$ if $\lambda_\alpha > 0$ and $q_R$ if $\lambda_\alpha < 0$. The $\alpha$ component of the numerical flux at $x_{i+\frac{1}{2}}$ is then given by


\begin{displaymath}F_{i + \frac{1}{2}, \alpha} = \left\{ \begin{array}{lr} u_L... ...f_\alpha (u_R) & \quad \lambda_\alpha < 0 \end{array} \right. \end{displaymath}

The two results can be combined in the following expression


\begin{displaymath}F_{i + \frac{1}{2}, \alpha} = \frac{1}{2}(f_\alpha (u_L) + f_\alpha (u_R) - \vert\lambda_\alpha\vert \omega_\alpha)\end{displaymath}

Summing over all $\alpha$, the final result is


\begin{displaymath}\textbf F_{i+\frac{1}{2}} = \frac{1}{2}(\textbf f_\alpha (\te... ... \sum \vert\lambda_\alpha\vert \omega_\alpha \textbf r_\alpha) \end{displaymath}

  评论这张
 
阅读(49)| 评论(0)
推荐 转载

历史上的今天

在LOFTER的更多文章

评论

<#--最新日志，群博日志--> <#--推荐日志--> <#--引用记录--> <#--博主推荐--> <#--随机阅读--> <#--首页推荐--> <#--历史上的今天--> <#--被推荐日志--> <#--上一篇，下一篇--> <#-- 热度 --> <#-- 网易新闻广告 --> <#--右边模块结构--> <#--评论模块结构--> <#--引用模块结构--> <#--博主发起的投票-->
 
 
 
 
 
 
 
 
 
 
 
 
 
 

页脚

我的照片书 - 博客风格 - 手机博客 - 下载LOFTER APP - 订阅此博客

网易公司版权所有 ©1997-2018