1D RKDG to shallow water equations
RKDG to shallow water equations
1.Governing Equations
\]
F = \begin{bmatrix} q \cr gh^2/2 + q^2/h \end{bmatrix}\]
2.Discrete with DGM
\]
\int_{\Omega} l_i \frac{\partial l_j}{\partial x} F(U_j) dx= 0 \end{equation}\]
\int_{\Omega} l_i \frac{\partial l_j}{\partial x} F(U_j) dx+
\oint_{\partial \Omega} l_i l_j (F^* - F)\cdot \vec{n} ds = 0 \end{equation}\]
\]
ODE:
\]
\]
It is important to point out that at dry cells no flux is flow inside the elemnt. Therefor, for dry cells
\]
3.Numerical Flux
3.1.HLL flux function
Formulations are given as
F^- \cr
\frac{S_R F^- - S_L F^+ + S_L S_R(U^+ - U^-)}{S_R S_L} \cr
F^+ \end{matrix} \right.
\begin{matrix}
S_L \geq 0 \cr
S_L < 0 < S_R \cr
S_R \leq 0
\end{matrix}\]
Wave Speed is suggested by Fraccarollo and Toro (1995)
\]
\]
\(u^*\) and \(c^*\) is defined by
\]
\]
for wet-dry interface, the wave speed is giving as
- left-hand dry bed
S_L = u^+ - 2\sqrt{g h^+} \quad S_R = u^+ + \sqrt{g h^+}
\end{equation}\]
- right-hand dry bed
S_L = u^- - \sqrt{g h^-} \quad S_R = u^- + 2\sqrt{g h^-}
\end{equation}\]
- both sides are dry
S_L = 0 \quad S_R = 0
\end{equation}\]
Noticing. 1
For flux terms, the discharge \(q^2\) is divided by water depth \(h\)
\]
so a threadhold of water depth \(h_{flux}\) ( \(10^{-3}\)m ) is add into flux function SWEFlux.m
. When \(h\) is less than \(h_{flux}\), the \(q^2/h\) is approximated to 0 as there is no flow at this node.
Noticing. 2
When defining the dry beds, another threadhold of water depth \(h_{dry}\) is used. It is convenient to deine \(h_{dry}\) equals to \(h_{flux}\).
3.2.Rotational invariance
0 & n_x\end{bmatrix} \quad
T^{-1} = \begin{bmatrix} 1 & 0 \cr
0 & n_x\end{bmatrix}\]
\]
defining \(Q = TU\), the numerical flux \(\hat{\mathbf{F}}\) can be obtained through the evaluation of numerical flux \(\mathbf{F}\) by
\]
4.Limiter
Note: discontinuity detector from Krivodonova (2003) is not working
For better numerical stability, minmod limiter is used in limiting the discharge and elevation.
Check testing/Limiter1D/doc
for more details about minmod limiter.
5. Positive preserving limiter
For the thin layer approach, a small depth ( \(h_{positive} = 10^{-3} m\)) and zeros velocity are prescribed for dry nodes.
The first step is to define wet elements. After each time step, the whole domain is calculated; If the any depth of nodes in \(\Omega_i\) is greater than \(h_{positive}\), then the element is defined as wet element, otherwise the water height of all nodes are remain unchanged.
The second step is to modify wet cells; If the depth of any nodes is less than \(h_{positive}\), then the flow rate is reset to zero and the new water depth is constructed as
\mathrm{M}\Pi_h h_i(x) = \theta_1 \left( h_i(x) - \bar{h}_i \right) + \bar{h}_i
\end{equation}\]
where
\theta_1 = min \left\{ \frac{\bar{h}_i - \xi }{\bar{h}_i - h_{min}}, 1 \right\}, \quad h_{min} = min\{ h_i (x_i) \}
\end{equation}\]
It is necessary to fulfill the restriction that the mean depth \(\bar{h}_i\) is greater than \(\xi\), i.e. \(10^{-4}\)m. In the function PositiveOperator
, if the mean depth of element is less than \(\xi\), all nodes will add a small depth \(\xi - \bar{h}_i\) to re-fulfill the restriction.
At last, all values of water height at nodes with negative \(h_i(x_j) <0\) will be modified to zero and the discharge of dry nodes ( \(h_i \le h_{positive}\) ) will be reseted to zero.
6. Wet/Dry reconstruction
No special treatment is introduced in the model at the moment.
5.Numerical Test
5.1.Wet dam break
Model Setting | value |
---|---|
channel length | 1000m |
dam position | 500m |
upstream depth | 10m |
downstream depth | 2m |
element num | 400 |
Final Time | 20s |
5.2.Dry dam break
Model Setting | value |
---|---|
channel length | 1000m |
dam position | 500m |
upstream depth | 10m |
downstream depth | 0m |
element num | 400 |
Final Time | 20s |
5.3.Parabolic bowl
Model Setting | value |
---|---|
channel length | 2000m |
\(h_0\) | 10m |
\(a\) | 600m |
\(B\) | 5m/s |
\(T\) | 269s |
Exact solution
Z(x,t) = \frac{-B^2 \mathrm{cos}(2wt) - B^2 - 4Bw \mathrm{cos}(wt)x}{4g}
\end{equation}\]
\(t = T/2\)
\(t = 3T/4\)
\(t = T\)
1D RKDG to shallow water equations的更多相关文章
- 论文翻译:2018_Source localization using deep neural networks in a shallow water environment
论文地址:https://asa.scitation.org/doi/abs/10.1121/1.5036725 深度神经网络在浅水环境中的源定位 摘要: 深度神经网络(DNNs)在表征复杂的非线性关 ...
- BJ2 斜率限制器
BJ2 斜率限制器 本文介绍斜率限制器取自于 Anastasiou 与 Chan (1997)[1]研究,其所利用的斜率限制器也是 Barth 与 Jespersen 限制器的一种修正形式,并且包含一 ...
- 体积与边精确积分DGM方法
Triangular DGM 1. Basis functions decomposing the domain \(\Omega\) into \(N_e\) conforming non-over ...
- TRANSFORM YOUR HABITS
TRANSFORM YOUR HABITS3rd EditionNote from James Clear:I wrote Transform Your Habits to create a free ...
- TPO 02 - The Origins of Cetaceans
TPO 02 - The Origins of Cetaceans It should be obvious that cetaceans[n. 鲸目动物]-whales, porpoises [n. ...
- 洛谷P3070 [USACO13JAN]岛游记Island Travels
P3070 [USACO13JAN]岛游记Island Travels 题目描述 Farmer John has taken the cows to a vacation out on the oce ...
- BZOJ_3049_[Usaco2013 Jan]Island Travels _状压DP+BFS
BZOJ_3049_[Usaco2013 Jan]Island Travels _状压DP+BFS Description Farmer John has taken the cows to a va ...
- viva correction statements
* List of amendments| No. | Location | Amendments ...
- [Usaco2013 Jan]Island Travels
Description Farmer John has taken the cows to a vacation out on the ocean! The cows are living on N ...
随机推荐
- 【Java虚拟机10】ClassLoader.getSystemClassLoader()流程简析
前言 学习类加载必然离开不了sun.misc.Launcher这个类和Class.forName()这个方法. 分析ClassLoader.getSystemClassLoader()这个流程可以明白 ...
- VS2019 及 Visual Assist X 安装配置
Visual Studio 2019 安装 下载 https://visualstudio.microsoft.com/zh-hans/downloads/ 安装 设置 扩大 Solution Con ...
- 记一次 .NET 某资讯论坛 CPU爆高分析
大概有11天没发文了,真的不是因为懒,本想前几天抽空写,不知道为啥最近求助的朋友比较多,一天都能拿到2-3个求助dump,晚上回来就是一顿分析,有点意思的是大多朋友自己都分析了几遍或者公司多年的牛皮藓 ...
- Alpha-功能规格说明书
项目 内容 这个作业属于哪个课程 2021春季软件工程(罗杰 任健) 这个作业的要求在哪里 团队项目-计划-功能规格说明书 一.引言 1. 项目简介 项目团队:删库跑路对不队 项目名称:题士 项目内容 ...
- 【二食堂】Beta - Scrum Meeting 9
Scrum Meeting 9 例会时间:5.24 20:00~20:20 进度情况 组员 当前进度 今日任务 李健 1. 文本导入.保存部分未完成issue 2. 知识图谱导出的前端issue3. ...
- Alpha阶段发布声明
发布声明 Alpha 1.Alpha版本功能说明 功能列表和详情图 模块 功能 展示 首页 查看首页博文,搜索博文,可供未登录用户使用 动态 查看推荐动态给未登录用户使用,登录用户可以查看关注动态.我 ...
- BUAA2020软工作业(一)——谈谈我和计算机的缘分
项目 内容 这个作业属于哪个课程 2020春季计算机学院软件工程(罗杰 任健) 这个作业的要求在哪里 第一次作业-热身! 我在这个课程的目标是 进一步提高自己的编码能力,工程能力 这个作业在哪个具体方 ...
- zip和flatMap没有生效
在Reactor 中flatMap和zip等没有生效 1.一个简单的示例代码如下: 2.示例运行结果 3.得到结论 最近在项目中使用了 Project Reactor ,但发现代码在写着写着有些地方没 ...
- Python中的括号()、[]、{}
长时间不用容易混淆,仅记! 在Python语言中最常见的括号有三种,分别是:小括号().中括号[].花括号{} . Python中的小括号(): 代表tuple元祖数据类型,元祖是一种不可变序列.大多 ...
- 暴力尝试安卓gesture.key
import hashlib import os import itertools f = open(r'D:\KEY\gesture.key','r') psd = f.readline() f.c ...