commit 962b6aae46913eb88f7f4d2cd790ea544f27ccd0
parent 34433435cdf448e98fc98f08f68aa0a889f62fc8
Author: miksa <milutin@popovic.xyz>
Date: Thu, 30 Jun 2022 16:35:48 +0200
done app pde
Diffstat:
10 files changed, 627 insertions(+), 32 deletions(-)
diff --git a/app_pde/build/main.bbl b/app_pde/build/main.bbl
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diff --git a/app_pde/build/main.pdf b/app_pde/build/main.pdf
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diff --git a/app_pde/chap1.tex b/app_pde/chap1.tex
@@ -199,8 +199,8 @@ Thereby the equations become
For now we have separated two simplifications, that define an
\textbf{idealized/perfect fluid}
\begin{enumerate}
- \item \textbf{incompressible} $\qquad \mu=0$
- \item \textbf{inviscid} $\quad \rho = \text{const.},\ \nabla \mathbf{u}=
+ \item \textbf{inviscid} $\qquad \mu=0$
+ \item \textbf{incompressible} $\quad \rho = \text{const.},\ \nabla \mathbf{u}=
0$
\end{enumerate}
\subsection{Vorticity and irrotational Flow}
diff --git a/app_pde/pres/build/main.bbl b/app_pde/pres/build/main.bbl
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+\headcommand {\beamer@framepages {20}{20}}
diff --git a/app_pde/pres/build/main.pdf b/app_pde/pres/build/main.pdf
Binary files differ.
diff --git a/app_pde/pres/build/main.run.xml b/app_pde/pres/build/main.run.xml
@@ -0,0 +1,85 @@
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diff --git a/app_pde/pres/build/main.snm b/app_pde/pres/build/main.snm
diff --git a/app_pde/pres/main.tex b/app_pde/pres/main.tex
@@ -49,13 +49,20 @@
\title
{Mathematical Modeling of Water-Wave Problems}
\subtitle{Applied PDE Seminar}
-\author[Popović Milutin]
-{Popović Milutin}
-\date{17. March 2021}
+%\author[Popović Milutin]
+%{Popović Milutin \inst{1}\\[1ex] {\small supervisor\inst{1,2}}
+
+\author[Popović Milutin]{Popović Milutin\\[10mm]{\small Supervisor: Sabine Hittmeir}}
+\date{29. June 2022}
\begin{document}
\begin{frame}
\titlepage
+ \nocite{johnson_1997}
+ \nocite{vallis_2017}
+ \nocite{constantin_tsunami}
+ \nocite{rupert_2009}
+ \nocite{mathe-physik}
\end{frame}
@@ -70,7 +77,6 @@
\item[$\circ$] Fluid density\\ $\rho(\vec{x}, t)$
\item[$\circ$] Velocity Field\\ $\vec{u}(\vec{x}, t) = (u, v,
w)$
- \item[$\circ$] Pressure \\$P(\vec{x}, t)$
\end{itemize}
\column{0.57\textwidth}
@@ -122,7 +128,7 @@
conservation}
\begin{align}
\hspace{0.3\linewidth} \frac{\partial \rho}{\partial t}
- \nabla \cdot (\rho \vec{u}) =
+ +\nabla \cdot (\rho \vec{u}) =
0 \nonumber
\end{align}
\end{itemize}
@@ -192,6 +198,16 @@
\end{frame}
\begin{frame}
+ \frametitle{Perfect Fluid}
+ \begin{itemize}
+ \item \textbf{inviscid} $\mu = 0$
+ \item \textbf{incompressible} $\rho = \text{const}.$, then
+ $\nabla \vec{u} = 0$
+ \end{itemize}
+
+ \end{frame}
+
+ \begin{frame}
\frametitle{Boundary Conditions for Water Waves}
\begin{center}
\begin{tabular}{@{}l@{}}
@@ -211,10 +227,10 @@
\begin{tabular}{@{}l@{}}
\tabitem $h_0$ for the typical water depth\\
\tabitem $\lambda$ for the typical wavelength\\
- \tabitem $\frac{\lambda}{\sqrt{g h_0}}$ time scale
- of wave propagation\\
\tabitem $\sqrt{g h_0}$ velocity scale of waves in
$(x, y)$\\
+ \tabitem $\frac{\lambda}{\sqrt{g h_0}}$ time scale
+ of wave propagation\\
\tabitem $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in $z$
\end{tabular}
\end{center}
@@ -223,7 +239,7 @@
\frac{h_0}{\lambda}$
\\
\centering
- $\rightarrow$ \textbf{Amplitude Parameter}
+ $\rightarrow$ \textbf{Amplitude parameter}
$\varepsilon=\frac{a}{h_0}$
\end{frame}
@@ -302,28 +318,27 @@
\begin{frame}
\frametitle{History of the Soliton}
\begin{itemize}
+ \item[$\circ$] John Scott Russell discovered the solitary wave in 1834,
+ firstly calling it the \textbf{wave of translation}
\item[$\circ$] a \textbf{soliton} is a solitary wave that resists
dispersion, maintaining its shape while it propagates at constant
velocity\\
- \item[$\circ$] John Scott Russell discovered the solitary wave in 1834,
- firstly calling it the \textbf{wave of translation}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Korteweg-de Vries equation (KdV)}
- Korteweg-de Vries equation: nonlinear, dispersive PDE
+ Korteweg-de Vries equation: nonlinear PDE
\begin{ceqn}
\begin{align}
- 2\eta_t + 3 \eta \eta_\xi + \frac{K}{3} \eta_{\xi\xi\xi} =
- 0\qquad \left(\xi = x-ct, \tau = \varepsilon t\right)\nonumber
+ \eta_t + 6K \eta \eta_{x} + \eta_{x x x} = 0\nonumber
\end{align}
\end{ceqn}
With Solution
\begin{ceqn}
\begin{align}
- \eta(\xi, \tau) = 2c^2 \text{sech}^2\left( \sqrt{\frac{3}{2K}}
- \left(\xi- \tau\right) \right)\nonumber
+ \eta(x, t) = 2c^2 \text{sech}^2\Big( c
+ \left(x - 4c^2t\right) \Big)\nonumber
\end{align}
\end{ceqn}
\end{frame}
@@ -347,13 +362,8 @@
\centering
\includegraphics[width=0.35\textwidth]{./pics/water-surface.png}
\caption{Earthquake generating a tsunami with $\lambda = 100\
- \text{km},\;\; a = 1\ \text{m}$}
+ \text{km},\;\; a = 1\ \text{m}$ (found in \cite{graph_meter})}
\end{figure}
- \begin{ceqn}
- \begin{align}
- \end{align}
- \end{ceqn}
-
\end{frame}
\begin{frame}
@@ -361,17 +371,17 @@
\begin{itemize}
\item[$\circ$] $\varepsilon = \frac{a}{h_0}$ and $\delta =
- \frac{h_0}{\lambda}$ need to ender the regime
+ \frac{h_0}{\lambda}$ need to enter the regime
$\varepsilon=O(\delta^2)$ for the KdV equation to become
relevant
\item[$\circ$] But also the geophysical scales need to be $\xi = O(1)$ and $\tau =
- O(1)$ for the KdV-balance to become relevant, that is
+ O(1)$ for the KdV dynamics to become relevant, that is
\begin{ceqn}
\begin{align}
x = O\left(\varepsilon^{-1} \lambda \right)
\end{align}
\end{ceqn}
- \item[$\circ$] KdV balance is when the waves are ordered with the
+ \item[$\circ$] KdV dynamics is when the waves are ordered with the
highest in front following an oscillatory tale
\item[$\circ$] This happens because wave amplitude is proportional to
wave speed
@@ -383,7 +393,7 @@
\frametitle{2004 Tsunami: Regime of Validity}
\begin{ceqn}
\begin{align}
- \lambda = 10\ \text{km}\qquad a = 1\ \text{m}\nonumber
+ \lambda = 100\ \text{km}\qquad a = 1\ \text{m}\nonumber
\end{align}
\end{ceqn}
\begin{columns}
@@ -440,11 +450,6 @@
\end{frame}
\begin{frame}{Bibliography}
- \nocite{johnson_1997}
- \nocite{vallis_2017}
- \nocite{constantin_tsunami}
- \nocite{rupert_2009}
- \nocite{mathe-physik}
\printbibliography
\end{frame}
\end{document}
