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\title{NUMWAVE 2017}
\author{\raisebox{-.35\height}{\includegraphics[scale=.1]{Pictures/inst_inria.png}}\hspace{1cm}Emmanuel AUDUSSE}
\date{Layerwise Discretized model\hfill}

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\begin{document}
\scriptsize

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		\draw (0,.8\baselineskip) node {\large\sc\color{blanc} Hyperbolicity of the}
			(0,-.8\baselineskip) node {\large\sc\color{blanc}  Layerwise Discretized Shallow Water equations}
			(0,-2.3\baselineskip) node {\footnotesize\sc\color{blanc}  The bilayer case};			
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	\begin{scope}
		\draw (0,-3.5) node{\underline{Emmanuel Audusse} - P13};
		\draw (1,-4) node[left]{in collaboration with} node[right]{Nina Aguillon (P6)}
			(1,-4.4)node[right]{Edwige Godlewski (P6 - {\sc Ange})}
			(1,-4.8)node[right]{Martin Parisot (INRIA - {\sc Ange})}
			(1,-4.8)node[left]{and};
		\draw (0,-7) node {{\bf NUMWAVE} - 11 d\'ecembre 2017};
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			\includegraphics[height=.1\linewidth]{Pictures/logo_cnrs.jpeg}\qquad
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\section{Introduction}

\subsection{Layerwise Discretized Shallow Water model}

\begin{frame}{}{}
{\color{bleu}\sc\underline{Layerwise Discretized Shallow Water model:}}
\begin{thebibliography}{9}
\bibitem{Audusse11} [Audusse, Bristeau, Perthame and Sainte-Marie'11]
\end{thebibliography}
\begin{tabular}{@{\qquad}p{.9\linewidth}}
\only<1>{\vspace{-20pt}
\begin{equation*}
\mat{@{}ll@{}}{&\textrm{for }i\in\l[\!\!\l[1,L\r]\!\!\r]\\
\(SW_L\)&\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{
\ds\partial_th&+&\ds\partial_x\(hu_i\)&+&\ds\partial_y\(hv_i\)&=&\ell_i^{-1}\l[G\r]_{i-\sfrac12}^{i+\sfrac12}
\\\ds
\partial_t\(hu_i\)&+&\partial_x\(hu_i^2+\frac g2 h^2\)&+&\ds\partial_y\(hu_i v_i\)&=&-gh\partial_xB+\ell_i^{-1}\l[uG\r]_{i-\sfrac12}^{i+\sfrac12}
\\\ds
\partial_t\(hv_i\)&+&\ds\partial_x\(hv_i u_i\)&+&\partial_y\(hv_i^2+ \frac g2 h^2\)&=&-gh\partial_yB+\ell_i^{-1}\l[vG\r]_{i-\sfrac12}^{i+\sfrac12}
}\r.}\vspace{-10pt}\end{equation*}
with 
$h_i=\ell_ih$ and 
$\ds\sum_{i}\ell_i=1$.
\vspace{-10pt}}
\only<2>{
\begin{equation*}
\mat{@{}ll@{}}{
\(SW_1\)&\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{
\ds\partial_th_1&+&\ds\partial_x\(h_1u_1\)&=&0
\\\ds
\partial_t\(h_1u_1\)&+&\partial_x\(h_1u_1^2+\frac g2 h_1^2\)&=&0
\\\ds
\partial_t\(h_1v_1\)&+&\ds\partial_x\(h_1v_1 u_1\)&=&0
}\r.}\vspace{-10pt}\end{equation*}}

\only<3>{\vspace{-20pt}
\begin{changemargin}{-1cm}{1cm}
\begin{equation*}
\mat{@{}l@{}ll@{}}{
\mat{@{}c@{}}{\(SW_2\)\\ h_1=h_2\\
\textcolor{white}{
u_{\sfrac32}=\frac{u_1+u_2}2}
\\
\textcolor{white}{
v_{\sfrac32}=\frac{v_1+v_2}2
}
}
&\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{
\ds\partial_th&+&\ds\partial_x\(hu_1\)&=&-2G_{\sfrac32}
\\\ds
\ds\partial_th&+&\ds\partial_x\(hu_2\)&=&2G_{\sfrac32}
\\\ds
\partial_t\(hu_1\)&+&\partial_x\(hu_1^2+\frac g2 h^2\)&=&-2u_{\sfrac32}G_{\sfrac32}
\\\ds
\partial_t\(hu_2\)&+&\partial_x\(hu_2^2+\frac g2 h^2\)&=&2u_{\sfrac32}G_{\sfrac32}
\\\ds
\partial_t\(hv_1\)&+&\ds\partial_x\(hv_1 u_1\)&=&-2v_{\sfrac32}G_{\sfrac32}
\\\ds
\partial_t\(hv_2\)&+&\ds\partial_x\(hv_2 u_2\)&=&2v_{\sfrac32}G_{\sfrac32}
}\r.
\textcolor{white}{\Leftrightarrow
\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{\ds
\partial_th&+&\ds\partial_x\(h\u\) &=&0
\\\ds
\partial_t\(h\u\) &+&\partial_x\( h\( \u^2+\du^2\) +\frac g2h^2\)&=&\ds0
\\\ds
\partial_t\du&+&\partial_x\( \du\,\u\)&=&\ds0
\\\ds
\partial_t\( h\v\) &+&\ds\partial_x\( h\( \u\,\v+\du\,\dv\) \) &=&\ds0
\\\ds
\partial_t\dv&+&\du\partial_x\v+\u\partial_x\dv &=&\ds0
\\
\u=\frac{u_1+u_2}2&&\du=\frac{u_2-u_1}2
\\
\v=\frac{v_1+v_2}2&&\dv=\frac{v_2-v_1}2
}\r.}
}\vspace{-10pt}
\end{equation*}
\end{changemargin}
}

\only<4>{\vspace{-20pt}
\begin{changemargin}{-1cm}{1cm}
\begin{equation*}
\mat{@{}l@{}ll@{}}{
\mat{@{}c@{}}{\(SW_2\)\\ h_1=h_2\\u_{\sfrac32}=\frac{u_1+u_2}2\\v_{\sfrac32}=\frac{v_1+v_2}2}
&\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{
\ds\partial_th&+&\ds\partial_x\(hu_1\)&=&-2G_{\sfrac32}
\\\ds
\ds\partial_th&+&\ds\partial_x\(hu_2\)&=&2G_{\sfrac32}
\\\ds
\partial_t\(hu_1\)&+&\partial_x\(hu_1^2+\frac g2 h^2\)&=&-2u_{\sfrac32}G_{\sfrac32}
\\\ds
\partial_t\(hu_2\)&+&\partial_x\(hu_2^2+\frac g2 h^2\)&=&2u_{\sfrac32}G_{\sfrac32}
\\\ds
\partial_t\(hv_1\)&+&\ds\partial_x\(hv_1 u_1\)&=&-2v_{\sfrac32}G_{\sfrac32}
\\\ds
\partial_t\(hv_2\)&+&\ds\partial_x\(hv_2 u_2\)&=&2v_{\sfrac32}G_{\sfrac32}
}\r.
%\ds\partial_th_1&+&\ds\partial_x\(h_1u_1\)&=&-G_{\sfrac32}
%\\\ds
%\ds\partial_th_2&+&\ds\partial_x\(h_2u_2\)&=&G_{\sfrac32}
%\\\ds
%\partial_t\(h_1u_1\)&+&\partial_x\(h_1u_1^2+\frac g2 h_1h\)&=&-u_{\sfrac32}G_{\sfrac32}
%\\\ds
%\partial_t\(h_2u_2\)&+&\partial_x\(h_2u_2^2+\frac g2 h_2h\)&=&u_{\sfrac32}G_{\sfrac32}
%\\\ds
%\partial_t\(h_1v_1\)&+&\ds\partial_x\(h_1v_1 u_1\)&=&-v_{\sfrac32}G_{\sfrac32}
%\\\ds
%\partial_t\(h_2v_2\)&+&\ds\partial_x\(h_2v_2 u_2\)&=&v_{\sfrac32}G_{\sfrac32}
%}\r.
\Leftrightarrow
\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{\ds
\partial_th&+&\ds\partial_x\(h\u\) &=&0
\\\ds
\partial_t\(h\u\) &+&\partial_x\( h\( \u^2+\du^2\) +\frac g2h^2\)&=&\ds0
\\\ds
\partial_t\du&+&\partial_x\( \du\,\u\)&=&\ds0
\\\ds
\partial_t\( h\v\) &+&\ds\partial_x\( h\( \u\,\v+\du\,\dv\) \) &=&\ds0
\\\ds
\partial_t\dv&+&\du\partial_x\v+\u\partial_x\dv &=&\ds0
\\
\u=\frac{u_1+u_2}2&&\du=\frac{u_2-u_1}2
\\
\v=\frac{v_1+v_2}2&&\dv=\frac{v_2-v_1}2
}\r.
}\vspace{-10pt}\end{equation*}
\end{changemargin}}

\end{tabular}
    \begin{center}
      \begin{tikzpicture}[xscale=.8, yscale=.5]\def\pi{4*atan(1)}      
	\draw[thick,->] (-7,3)--(-7,2) node[midway,right]{$g$};
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	\only<2->{\def\B{(.5)}}
%	\def\vit{(\y/(\L+1))^(.4)}
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%	\def\vit{-.05*(\B-\ym)^2*(3/5*(\B+\h)-\ym)*(11/10*(\B+\h)-\ym)}
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	\draw[thick,<->] (\x,{\B})--(\x,{\h+\B}) node[midway,left]{$h$};
	\def\ym{(\B+.55*\h)}\draw[thick,->] (\x,{\ym})--({\x+\vit},{\ym}) node[above]{\color{black}$\u$};
	\draw[thick,>-<] ({\x+.75*\vit},{\ym-.1*\h})--({\x+1.25*\vit},{\ym-.1*\h}) node[below, midway]{\color{black}$\du$};}
      \end{tikzpicture}
    \end{center}
\only<1>{
\begin{thebibliography}{9}
\bibitem{Audusse11b} [Audusse, Bristeau, Pelanti and Sainte-Marie'11] {\color{black} with variable density.}
\movie[externalviewer]{\beamergotobutton{Gibraltar}}{Pictures/gibraltar.avi}
\bibitem{Cindy16} [Bristeau, Guichard, Di Martino and Sainte-Marie'16] {\color{black} with viscous terms.}
\bibitem{Fernandez} [Fernandez-Nieto, Parisot, Penel and Sainte-Marie] {\color{black} with non-hydrostatic terms.}
\end{thebibliography}}
\only<2>{
\begin{minipage}{.6\linewidth}
\begin{itemize}
\item[\myitemb] {\bf Strictly hyperbolic} equations% $u-\sqrt{gh}$, $u$ and $u+\sqrt{gh}$.
\item[\myitemb] Admissible shock define to ensure the {\bf mechanical energy dissipation} : $E=\frac{h_1}2\(u_1^2+v_1^2\)+\frac g2 h_1^2$\\
$$\partial_tE+\partial_x\(\(\frac{u_1^2+v_1^2}2+gh_1\)h_1u_1\)\le0$$
\end{itemize}
\end{minipage}
\begin{minipage}{.35\linewidth}\vspace{-10pt}
\begin{center}\begin{tikzpicture}[xscale=.5]\def\ub{.5}\def\hu{2.5}\def\ya{.75}\def\yb{.8}
\draw[thick,->] (-3,0)--(3,0) node[below]{\small$x$};
\fill[color=black!20] ({\ub-1.2*\hu},1) node[above,black]{$u_1-\sqrt{gh_1}$}--(0,0)--({\ub-.9*\hu},1);
\draw (0,0)--({\ub+\hu},1) node[above]{$u_1+\sqrt{gh_1}$};
\draw[dashed](0,0)--({\ub},1) node[above]{$u_1$};
\end{tikzpicture}
\vspace{-10pt}
\movie[externalviewer]{\beamergotobutton{Dambreak $SW_1$}}{Pictures/dambreak.avi}
\end{center}
\end{minipage}}
\only<3>{
\begin{thebibliography}{9}
%\bibitem[]
\textcolor{white}{
[Teshukov'07, Richard and Gavrilyuk'12] {\color{white} shear model.}
}
%\bibitem[]
\textcolor{white}{
[Castro and Lannes'14] {\color{white} with non-hydrostatic terms.}
}
\end{thebibliography}}
\only<4>{
\begin{thebibliography}{9}
\bibitem{Gavrilyk} [Teshukov'07, Richard and Gavrilyuk'12] {\color{black} shear model.}
\bibitem{Gavrilyk} [Castro and Lannes'14] {\color{black} with non-hydrostatic terms.}
\end{thebibliography}}
\end{frame}

\begin{frame}{}{}
\begin{minipage}{.45\linewidth}
{\color{bleu}\sc\underline{Bi-layer model:}}
\begin{equation*}
\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{\ds
\partial_th&+&\ds\partial_x\(h\u\) &=&0,
\\\ds
\partial_t\(h\u\) &+&\ds\partial_x\( h \u^2 +h\(\du^2+\frac g2h\)\)&=&\ds0,
\\\ds
\partial_t\du&+&\partial_x\( \du\,\u\)&=&\ds0,
\\\ds\only<1>{\color{black!20}}
\partial_t\( h\v\) &\only<1>{\color{black!20}}+&\only<1>{\color{black!20}}\ds\partial_x\( h\( \u\,\v+\du\,\dv\) \) &\only<1>{\color{black!20}}=&\only<1>{\color{black!20}}\ds0,
\\\ds\only<1>{\color{black!20}}
\partial_t\dv&\only<1>{\color{black!20}}+&\only<1>{\color{black!20}}\du\partial_x\v+\u\partial_x\dv &\only<1>{\color{black!20}}=&\only<1>{\color{black!20}}\ds0
}\r.
\end{equation*}
\begin{itemize}
\item[\myitemb] Admissible shock define to ensure the {\bf mechanical energy dissipation} : $E=\frac{h}2 \(\u^2+\du^2{\only<1>{\color{black!20}}+\v^2+\dv^2}\)+\frac g2 h^2$\\
\end{itemize}
$$\partial_tE+\partial_x\(
%\(\frac {\u^2+\du^2{\only<1>{\color{black!20}} + \v^2 + \dv^2}}2+\frac{gh}2\) h
\(E
+h\(\du^2+\frac g2h\)\)\u {\only<1>{\color{black!20}} +h\v\dv\du} \) \le0$$
\end{minipage}\hspace{.7cm}
\begin{minipage}{.45\linewidth}
{\color{bleu}\sc\underline{Full-Euler model:}}
\begin{equation*}
\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{\ds
\partial_t\rho&+&\ds\partial_x\(\rho u\) &=&0,
\\\ds
\partial_t\(\rho u\) &+&\ds\partial_x\( \rho u^2+p\)&=&\ds0,
\\\ds
\partial_t\(\rho e\) &+&\ds\partial_x\( \(\rho e+p\)u\)&=&\ds0,
\\\ds\only<1>{\color{black!20}}
\partial_t\( \rho v\) &\only<1>{\color{black!20}}+&\only<1>{\color{black!20}}\ds\partial_x\( \rho u\,v \) &\only<1>{\color{black!20}}=&\only<1>{\color{black!20}}\ds0,
}\r.
\end{equation*}
\begin{itemize}
\item[\myitemb] Admissible shock define to ensure the {\bf entropy dissipation} :\\
\end{itemize}
$$\partial_t\eta+\partial_x\(\eta u \) \le0$$
\vspace{10pt}
\end{minipage}
\vfill
\begin{itemize}
\item[\myitemb] 1D analogous to the full-Euler equations {\bf except at the shock}.
\onslide<2>{\item[\myitemb] 2D {\color{red}NOT} analogous : {\bf non-conservative products, coalescence, resonance.}}
\end{itemize}
\end{frame}

\section{1D bilayer model}

\subsection{Hyperbolicity and waves patterns}
\begin{frame}{}{}
{\color{bleu}\sc\underline{1D Bi-layer model:}}
$$\l\{\mat{@{~}l@{~}l@{~}l@{~}l@{~}l@{}l@{~}l@{}}{\ds
\partial_th&+&\ds\partial_x\(h\u\) &=&0,
\\\ds
\partial_t\(h\u\) &+&\ds\partial_x\( h \u^2 +h\(\du^2+\frac g2h\)\)&=&\ds0,
\\\ds
\partial_t\du&+&\partial_x\( \du\,\u\)&=&\ds0,
}\r.
\qquad\textrm{We set }U=\(\mat{c}{h\\h\u\\\du}\)
$$
\begin{block}{Proposition : hyperbolicity of $\(1D-SW_2\)$}
For physical solution, i.e. $h>0$, the 1D bilayer model~$\(SW_2\)$ is {\bf strictly hyperbolic}.\\
More precisely, the eigenvalues are given by
\begin{equation*}
\mat{@{}c@{}}{\ds
\lambda_L=\u-\sqrt{gh+3\du^2}
\quad<\quad
\lambda_*=\u
\quad<\quad
\lambda_R=\u+\sqrt{gh+3\du^2}.
}\end{equation*}
In addition, the $\lambda_L$-wave and the $\lambda_R$-wave are {\bf genuinely nonlinear},\\
whereas the $\lambda_*$-wave is {\bf linearly degenerate}.
\end{block}
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\draw[thick,->] (-3,0)--(3,0) node[below]{\small$x$};
\fill[color=black!20] ({\ub-1.2*\hu},1) node[above,black]{$\lambda_{L}\( U_L\) $\quad~}--(0,0)--({\ub-.9*\hu},1) node[above,black]{\quad$\lambda_{L}\( U_{L*}\) $};
\draw (0,0)--({\ub+\hu},1) node[above]{$\sigma_R$};
\draw[dashed](0,0)--({\ub},1) node[above]{$\u_*$};
\draw ({\ya*(\ub-1.7*\hu)},\ya)node{$U_L$}
	({\ya*(\ub+7/4*\hu)},\ya)node{$U_R$}
	({\ya*(\ub-1/2*\hu)},\ya)node{$U_{L*}$}
	({\ya*(\ub+1/2*\hu)},\ya)node{$U_{R*}$};
\end{tikzpicture}\end{center}
\end{frame}

\subsection{The 1D Riemann problem}
\begin{frame}{}{}
\begin{block}{Proposition : admissible shock of $\(1D-SW_2\)$}
We denote by $\sigma_k$ the speed of the $\lambda_k$-shock.\\
Assuming that the water depth $h$ is positive, the following properties are equivalent:
\begin{enumerate}[i)]
\item\label{entropy_jump} The mechanical energy $E=\frac g2h^2+\frac h2\(\u^2+\du^2\)$ is {\bf decreasing} through a shock, i.e.
\begin{equation*}
-\sigma_k\l[E\r]+\l[
%\( \frac{\u^2+3\du^2}2+gh\) 
\(E +h\(\du^2+\frac g2h\)\)\u
\r]<0.
\end{equation*}
\item\label{shockCompressible} the shock is {\bf compressive}, i.e. we have
\begin{equation*}
%\mat{l@{\qquad}l}{&\ds
-\sigma_k\l[h^2\r]+\l[h^2\u\r]>0
\qquad\textrm{or}\quad
%\\\textrm{or}&\ds
-\sigma_k\l[h\u^2\r]+\l[h\u^3\r]<0
\qquad\textrm{or}\quad
%\\\textrm{or}&\ds
-\sigma_k\l[h\du^2\r]+\l[h\du^2\u\r]>0.
%}
\end{equation*}
\item\label{LaxCondition} the {\bf Lax entropy condition} is satisfied
\begin{equation*}
\lambda_L\( U_{L*}\)<\sigma_L<\lambda_L\( U_{L}\)
\qquad\textrm{and}\qquad
\lambda_R\( U_{R}\)<\sigma_R<\lambda_R\( U_{R*}\).
\end{equation*}
\end{enumerate}
\end{block}
\underline{\bf Remark :} It is {\color{red}NOT} a corollary of the classical theorem [Godlewsli, Raviart'96] since the mechanical energy $E$ (acting as a mathematical entropy) is {\bf not a convex function} of the conserved variable $U$.
\end{frame}

\begin{frame}{}{}
\begin{block}{Theorem : Riemann problem of $\(1D-SW_2\)$}
Consider the initial condition
%\begin{equation*} \label{1D_Riemann_data}
$U\( 0,x\) = 
\begin{cases}
 U_L = \( h_L,\u_L,\du_L \) ^t \in \R_+^* \times \R^2& \ds \textrm{if} \ x<0, \\
 U_R = \( h_R,\u_R,\du_R \) ^t \in \R_+^* \times \R^2& \ds \textrm{if} \ x \geq 0.
\end{cases}
$\\
%\end{equation*}
If the following condition is fulfilled :
$\u_R-\u_L < \mu^r\(U_L\) +  \mu^r\(U_R\)$ with
\begin{equation*}
 \mu^r(U) =  \sqrt{g h + 3 \du^2} + \frac{g h}{\sqrt{3}\du}  \log\( \sqrt{1+ \frac{3 \du^2}{g h}} +    \du\sqrt{\frac{3}{gh}} \, \)
\end{equation*}
then there exists a {\bf unique selfsimilar} solution $U\in\( L^\infty\( \R_+^*\times\R\) \) ^3$ to the 1D Riemann problem $\(SW_2\)$ satisfying the {\bf mechanical energy dissipation}.\\
In addition the water depth $h$ is {\bf strictly positive} for all $\( t,x\) \in\R_+\times\R$.
\end{block}
\begin{minipage}{.55\linewidth}
\underline{\bf Sketch of proof :}
\begin{enumerate}
\item Determine the rarefaction and shock curves.
\item Conclude by monotony.
\end{enumerate}
\end{minipage}\hspace{-20pt}
\begin{minipage}{.35\linewidth}
\begin{center}\begin{tikzpicture}[scale=2.5]\def\ub{.25}\def\hu{1}
%\begin{scope}[xshift=50]
%\draw[thick,->] (0,-.1)--(0,1.1) node[left]{\small$\u$};
%\draw[thick,->] (-.1,.1)--(1.5,.1) node[below]{\small$p$};
%\draw[domain=0:1.45] plot (\x,{.75*sqrt(\x)});
%\draw[domain=0:1.45] plot (\x,{1/(1+\x)});
%\def\x{1.1}\draw (\x,{1/(1+\x)}) node{$\bullet$}node[above right]{\small$\( p_L,\u_L\) $};
%\def\x{.2}\draw (\x,{.75*sqrt(\x)}) node{$\bullet$}node[below right]{\small$\( p_R,\u_R\) $};
%\def\x{.65}\draw (\x,{.75*sqrt(\x)}) node{$\bullet$}node[above]{$\small\( p_{*},\u_*\) $};
%\end{scope}
\draw[thick,->] (0,-.1)--(0,1.1) node[left]{\small$h_{L*}$};
\draw[thick,->] (-.1,0)--(1.5,0) node[below]{\small$h_{R*}$};
\draw[domain=0:1.4,samples=200] plot (\x,{.75*sqrt(\x)}) node[right]{\small$p_{L*}=p_{R*}$};
\draw[domain=0:1.4,samples=200] plot (\x,{1/(1+\x)}) node[right]{\small$\u_{L*}=\u_{R*}$};
\def\x{.65}\draw (\x,{.75*sqrt(\x)}) node{$\bullet$};%node[right]{\small\quad$\( h_{R*},h_{L*}\) $};
\draw[thick,dotted] (.5,0) node[below]{\small$h_{R}$} --(.5,1.1);
\draw[thick,dotted] (0,.75) node[left]{\small$h_{L}$} --(1.45,.75);
\begin{scope}[scale=.1,xshift=70,yshift=50]
	\draw (-1.5,0)--(1.5,0);
	\fill[color=black!20] ({\ub-1.3*\hu},1)--(0,0)--({\ub-.9*\hu},1);
	\fill[color=black!20] ({\ub+1.3*\hu},1)--(0,0)--({\ub+.9*\hu},1);
	\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[scale=.1,xshift=275,yshift=50]
	\draw (-1.5,0)--(1.5,0);
	\fill[color=black!20] ({\ub-1.3*\hu},1)--(0,0)--({\ub-.9*\hu},1);
	\draw (0,0)--({\ub+\hu},1);
	\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[scale=.1,xshift=70,yshift=250]
	\draw (-1.5,0)--(1.5,0);
	\fill[color=black!20] ({\ub+1.3*\hu},1)--(0,0)--({\ub+.9*\hu},1);
	\draw (0,0)--({\ub-\hu},1);
	\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[scale=.1,xshift=275,yshift=250]
	\draw (-1.5,0)--(1.5,0);
	\draw (0,0)--({\ub-\hu},1);
	\draw (0,0)--({\ub+\hu},1);
	\draw[dashed](0,0)--({\ub},1);
\end{scope}
\end{tikzpicture}\end{center}
\end{minipage}
\end{frame}



\section{2D bilayer model}
\subsection{Hyperbolicity and waves patterns}

\begin{frame}{}{}
{\color{bleu}\sc\underline{2D Bi-layer model:}}
$$\l\{\mat{@{}l@{~}c@{~}l@{~}c@{~}l@{}}{\ds
\only<2->{\color{vert}}\partial_th&\only<2->{\color{vert}}+&\ds\only<2->{\color{vert}}\partial_x\( h\u\) &\only<2->{\color{vert}}=&\only<2->{\color{vert}}0,
\\\ds
\only<2->{\color{vert}}\partial_t\( h\u\) &\only<2->{\color{vert}}+&\only<2->{\color{vert}}\ds\partial_x\( h\( \u^2+\du^2\) +\frac g2h^2\) &\only<2->{\color{vert}}=&\only<2->{\color{vert}}\ds0,
\\\ds
\only<2->{\color{vert}}\partial_t\du&\only<2->{\color{vert}}+&\only<2->{\color{vert}}\partial_x\( \du\,\u\) &\only<2->{\color{vert}}=&\only<2->{\color{vert}}\ds0,
\\\ds
\partial_t\( h\v\) &+&\ds\partial_x\( h\( \u\,\v+\du\,\dv\) \) &=&\ds0,
\\\ds
\partial_t\dv&+&{\only<3->{\color{red}}\du\partial_x\v+\u\partial_x\dv}&=&\ds0,
}\r.
\qquad\textrm{We set }V=\(\mat{c}{\only<2->{\color{vert}}h\\\only<2->{\color{vert}}h\u\\\only<2->{\color{vert}}\du\\h\v\\\dv}\)
\onslide<2->{\mat{c}{\l\}\mat{c}{~\\{\only<2->{\color{vert}}U}\qquad(1D-SW_2)\\~}\r.\\~\\~}}
$$
\begin{block}{Proposition : hyperbolicity of $\(2D-SW_2\)$}
For physical solution, i.e. $h>0$, the 2D bilayer model~$\(SW_2\)$ is {\bf hyperbolic}.\\
More precisely, the eigenvalues are given by\vspace{-5pt}
$$\mat{@{}c@{}}{\ds
{\only<2->{\color{vert}}\lambda_L=\u-\sqrt{gh+3\du^2}}
\quad{\only<4->{\color{red}}<}\quad
{\only<2->{\color{vert}}\gamma_L=\u-\l|\du\r|}
\quad{\only<3->{\color{red}}\le}\quad
{\only<2->{\color{vert}}\lambda_*=\u}
\quad{\only<3->{\color{red}}\le}\quad
{\only<2->{\color{vert}}\gamma_R=\u+\l|\du\r|}
\quad{\only<4->{\color{red}}<}\quad
{\only<2->{\color{vert}}\lambda_R=\u+\sqrt{gh+3\du^2}}.
\vspace{-5pt}
}$$
In addition, the $\lambda_L$-wave and the $\lambda_R$-wave are {\bf genuinely nonlinear},\\
whereas the $\gamma_L$-wave, $\lambda_*$-wave and $\gamma_R$-wave are {\bf linearly degenerate}.
\end{block}
\begin{center}\begin{tikzpicture}[xscale=1.5,yscale=2]\def\ub{.5}\def\hu{2.5}\def\ya{.5}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1) node[above]{$\gamma_L$};
\onslide<1-2>{\draw[dashed](0,0)--({\ub+\hu/2},1) node[above]{$\gamma_R$};}
\onslide<1-2,4>{\draw[dashed](0,0)--({\ub},1) node[above]{$\lambda_*$};}
\onslide<3>{\draw[dashed,double](0,0)--({\ub},1) node[above]{\color{red}$\lambda_*=\gamma_R\(U_{R*}\)$};}
\onslide<1-3>{\draw (0,0)--({\ub+\hu},1) node[above]{$\sigma_R$};}
\onslide<4>{
\draw[dashed,xshift=-1.5pt] (0,0)--({\ub+\hu},1) node[above]{\color{red}$\sigma_R<\gamma_R\(U_{R*}\)$};
\draw (0,0)--({\ub+\hu},1) ;}
\draw[thick,->] (-3,0)--(3,0) node[below]{\small$x$};
\fill[color=black!20] ({\ub-1.2*\hu},1) node[above,black]{$\lambda_{L}\( V_L\) $}--(0,0)--({\ub-.9*\hu},1) node[above,black]{$\lambda_{L}\( V_{L*}\) $};
\draw
%	({\ya*(\ub-1.7*\hu)},\ya)node{$W_L^{1D}$}
%	({\ya*(\ub+7/4*\hu)},\ya)node{$W_R^{1D}$}
%	({\ya*(\ub-1/2*\hu)},\ya)node{$W_{L*}^{1D}$}
%	({\ya*(\ub+1/2*\hu)},\ya)node{$W_{R*}^{1D}$}
	({\yb*(\ub-1.5*\hu)},\yb)node{$V_L$}
	({\yb*(\ub+1.3*\hu)},\yb)node{$V_R$}
	({\yb*(\ub-11/16*\hu)},\yb)node{$V_{L1}$}
	({\yb*(\ub-1/4*\hu)},\yb)node{$V_{L2}$};
\onslide<1-2,4>{\draw({\yb*(\ub+1/4*\hu)},\yb)node{$V_{R2}$};}
\onslide<1-3>{\draw({\yb*(\ub+3/4*\hu)},\yb)node{$V_{R1}$};}
\only<2->{\draw({\ya*(\ub-1/2*\hu)},\ya)node[fill=white]{\color{vert}$U_{L*}$}
	({\ya*(\ub+1/2*\hu)},\ya)node[fill=white]{\color{vert}$U_{R*}$};}
\end{tikzpicture}\end{center}
\end{frame}

\begin{frame}{}{}
\begin{block}{Proposition : coalescence}
Let us define the following polynomial function for $X>h$\vspace{-10pt}
\begin{equation*}
P\( U\ ;X\) :=\frac12\(1+\frac Xh\) gh+\( 1+ \frac{X}{h}+\( \frac{X}{h}\) ^2-\( \frac{X}{h}\) ^3\) \du^2.\vspace{-10pt}
\end{equation*}
If $\du \neq 0$, let $\eta\( U\)$ be the {\bf unique real root} larger than $h$.\\
{\bf The coalescence occurs if and only if $h_{k*}\ge\eta_k:=\eta\( U_k\) $}.
\end{block}\vspace{-15pt}
\begin{center}\begin{tikzpicture}[scale=4]\def\ub{.25}\def\hu{1}\def\hL{.4}\def\hR{.5}\def\etaL{.85}\def\etaR{1.5}
\begin{scope}[xscale=.05,yscale=.1,xshift=140,yshift=75]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\fill[color=black!20] ({\ub-1.5*\hu},1)--(0,0)--({\ub-1*\hu},1);
\fill[color=black!20] ({\ub+1.5*\hu},1)--(0,0)--({\ub+1*\hu},1);
\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=550,yshift=25]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\fill[color=black!20] ({\ub-1.5*\hu},1)--(0,0)--({\ub-1*\hu},1);
\draw (0,0)--({\ub+\hu},1);
\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=140,yshift=200]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\draw (0,0)--({\ub-\hu},1);
\fill[color=black!20] ({\ub+1.5*\hu},1)--(0,0)--({\ub+1*\hu},1);
\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=550,yshift=200]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\draw (0,0)--({\ub-\hu},1);
\draw (0,0)--({\ub+\hu},1);
\draw[dashed](0,0)--({\ub},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=1000,yshift=75]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw (-3,0)--(3,0);
\fill[color=black!20] ({\ub-1.5*\hu},1)--(0,0)--({\ub-1*\hu},1);
\draw[dashed](0,0)--({\ub},1);
\draw[dashed,xshift=-3pt] (0,0)--({\ub+\hu},1);\draw[xshift=3pt] (0,0)--({\ub+\hu},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=1000,yshift=200]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw (-3,0)--(3,0);
\draw (0,0)--({\ub-\hu},1);
\draw[dashed](0,0)--({\ub},1);
\draw[dashed,xshift=-3pt] (0,0)--({\ub+\hu},1);\draw[xshift=3pt] (0,0)--({\ub+\hu},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=1000,yshift=260]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw (-3,0)--(3,0);
\draw[xshift=-2pt] (0,0)--({\ub-\hu},1);\draw[dashed,xshift=2pt] (0,0)--({\ub-\hu},1);
\draw[dashed](0,0)--({\ub},1);
\draw[dashed,xshift=-3pt] (0,0)--({\ub+\hu},1);\draw[xshift=3pt] (0,0)--({\ub+\hu},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=550,yshift=260]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\draw[xshift=-2pt] (0,0)--({\ub-\hu},1);\draw[dashed,xshift=2pt] (0,0)--({\ub-\hu},1);
\draw[dashed](0,0)--({\ub},1);
\draw (0,0)--({\ub+\hu},1);
\end{scope}
\begin{scope}[xscale=.05,yscale=.1,xshift=140,yshift=260]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\draw[xshift=-2pt] (0,0)--({\ub-\hu},1);\draw[dashed,xshift=2pt] (0,0)--({\ub-\hu},1);
\draw[dashed](0,0)--({\ub},1);
\fill[color=black!20] ({\ub+1.5*\hu},1)--(0,0)--({\ub+1*\hu},1);
\end{scope}
\draw[domain=0:1.95,samples=200] plot (\x,{.1*\x+.3*sqrt(\x)}) node[right]{\small$p_{L*}=p_{R*}$};
\draw[domain=0:1.95,samples=200] plot (\x,{1/(1+\x)-.25}) node[right]{\small$\u_{L*}=\u_{R*}$};
\def\x{.727}\draw (\x,{.1*\x+.3*sqrt(\x)}) node{$\bullet$};
\draw[thick,->] (0,-.1)--(0,1.1) node[left]{\small$h_{L*}$};
\draw[thick,->] (-.1,0)--(2,0) node[below]{\small$h_{R*}$};
\draw[thick,dotted] (\hR,0) node[below]{\small$h_{R}$} --(\hR,1.05);
\draw[thick,dotted] (0,\hL) node[left]{\small$h_{L}$} --(1.95,\hL);
\draw[thick,dotted] (\etaR,0) node[below]{\small$\eta_R$} --(\etaR,1.05);
\draw[thick,dotted] (0,\etaL) node[left]{\small$\eta_L$} --(1.95,\etaL);
\end{tikzpicture}\end{center}\vspace{-10pt}
{\bf In practice}, it is enough to test $P\(U_k;h_{k*}\)\l\{\mat{ll}{>0&\textrm{no-coalescence}\\<0&\textrm{coalescence.}}\r.$
\end{frame}

\subsection{Definition of the non-conservative products}

\begin{frame}{}{}
\begin{thebibliography}{9}
\bibitem{Lagoutiere08} [Lagouti\`ere, Seguin, Takahashi'08] {\color{black}{\bf Path-independent} non-conservative product definition.}
\end{thebibliography}
\begin{block}{Definition : non-conservative products}
Through a $\lambda_k$-shock, the following {\bf jump conditions} hold:\vspace{-10pt}
\begin{equation*}
\l\{\mat{@{}l@{~}c@{~}l@{~}c@{~}l@{}}{\ds
\partial_th&+&\ds\partial_x\( h\u\) &=&0,
\\\ds
\partial_t\( h\u\) &+&\ds\partial_x\( h\( \u^2+\du^2\) +\frac g2h^2\) &=&\ds0,
\\\ds
\partial_t\du&+&\partial_x\( \du\,\u\) &=&\ds0,
\\\ds
\partial_t\( h\v\) &+&\ds\partial_x\( h\( \u\,\v+\du\,\dv\) \) &=&\ds0,
\\\ds
\partial_t\dv&+&\du\partial_x\v+\u\partial_x\dv&=&\ds0,
}\r.\Leftrightarrow
\l\{\mat{r@{~}l}{
\sigma_k\l[h\r]&=\l[h \u \r] \\
\sigma_k\l[ h \u \r]&=\l[h \( \u^2 + \du^2 \) + \frac{g}{2} h^2 \r] \\
\sigma_k\l[\du \r]&=\l[ \du \u \r] \\
 \sigma_k\l[h\v\r]&=\l[h\( \u\,\v+\du\dv \) \r]\\
 \l[h\sqrt{\(\u-\sigma_k \)^2-\du^2}\ \dv\r]&=0
\qquad\textrm{if {\bf no-coalescence}}.
}\r.
\end{equation*}
\end{block}
\begin{minipage}{.65\linewidth}
\underline{\bf Arguments :} $z=x-\sigma_kt$, $\w=\u-\sigma_k$ and $\dw=\du$
\begin{enumerate}
\onslide<2->{\item Regularization : $h_\eps\(z\)\in C^1[-\sfrac{\eps}2,-\sfrac{\eps}2]$ monotonic.}
\onslide<3->{\item $\(h_\eps\w_\eps\)'=0\Rightarrow \w_\eps\(z\)=\frac{h_L\w_L}{h_\eps\(z\)}\in C^1[-\sfrac{\eps}2,-\sfrac{\eps}2]$\\}
\onslide<4->{$\(\dw_\eps\w_\eps\)'=0\Rightarrow \dw_\eps\(z\)=\frac{h_\eps\(z\)\w_L}{h_L}\in C^1[-\sfrac{\eps}2,-\sfrac{\eps}2]$}
\onslide<5->{\item $\l\{\mat{@{}l@{}}{\(h_\eps\(\w\v+\dw\dv\)\)'=0\\\dw\v'+\w\dv'=0}\r.
\Rightarrow\(1-\(\frac{\dw_\eps}{\w_\eps}\)^2\)\dv_\eps'-\frac{\dw_\eps}{\w_\eps}\(\frac{\dw_\eps}{\w_\eps}\)'\dv_\eps=0$}
\onslide<6->{$\Rightarrow\dv_\eps\(z\)= C \frac{h_L}{h_\eps(z)} \sqrt{\l|\frac{\w_L^2-\dw_L^2}{\w_\eps(z)^2-\dw_\eps(z)^2}\r|}
\qquad\Rightarrow
\v_\eps\(z\).
$}
\end{enumerate}
\end{minipage}\hfill
\begin{minipage}{.35\linewidth}
\begin{center}\begin{tikzpicture}
\def\hl{1}\def\hls{2.62}
\def\wl{3.5}\pgfmathsetmacro\wls{\hl*\wl/\hls}
\def\dwl{.4}\pgfmathsetmacro\dwls{\hls*\dwl/\hl}
\def\dvl{2.05}\pgfmathsetmacro\dvls{\dvl*\hl/\hls*sqrt((\wl^2-\dwl^2)/(\wls^2-\dwls^2)}
\def\vl{2.45}\pgfmathsetmacro\vls{(\hl/\hls*(\wl*\vl+\dwl*\dvl)-\dwls*\dvls)/(\wls)}
\def\ep{1}\def\H{(((\x+\ep)+.1*sin(500*(\x+\ep)))/((2*\ep)+.1*sin(1000*\ep)))}
\draw[->] (-2,0)--(2,0)node[below]{$z$};
\onslide<1>{\draw (-2,\hl)node[above]{$h_{L}$}--(0,\hl)--(0,\hls)--(2,\hls)node[above]{$h_{L*}$}
					(-2,\wl)node[above]{$\w_{L}$}--(0,\wl)--(0,\wls)--(2,\wls)node[above]{$\w_{L*}$}
					(-2,\dwl)node[above]{$\dw_{L}$}--(0,\dwl)--(0,\dwls)--(2,\dwls)node[above]{$\dw_{L*}$};
\draw[red] (-2,\vl)node[above]{$\v_{L}$}--(0,\vl)
		(-2,\dvl)node[above]{$\dv_{L}$}--(0,\dvl);}
\def\y{3.2}\draw[dotted,thick] (-2,\y)node[below]{$\eta_L$}--(2,\y);
\onslide<2->{\draw (-2,\hl)node[above]{$h_{L}$}--(-\ep,\hl) (\ep,\hls)--(2,\hls)node[above]{$h_{L*}$}
				(-2,\wl)node[above]{$\w_{L}$}--(-\ep,\wl) (\ep,\wls)--(2,\wls)node[above]{$\w_{L*}$}
				(-2,\dwl)node[above]{$\dw_{L}$}--(-\ep,\dwl) (\ep,\dwls)--(2,\dwls)node[above]{$\dw_{L*}$};
\draw[red](-2,\vl)node[above]{$\v_{L}$}--(-\ep,\vl)
		(-2,\dvl)node[above]{$\dv_{L}$}--(-\ep,\dvl);
\draw[smooth,domain=-\ep:\ep] plot[smooth,domain=-\ep:\ep](\x,{\hl+(\hls-\hl)*\H});
\draw[dashed,thick] (-\ep,0)node[below]{$-\sfrac\eps2$}--(-\ep,4) (\ep,0)node[below]{$\sfrac\eps2$}--(\ep,4);}
\onslide<3->{\draw[smooth,domain=-\ep:\ep] plot[smooth,domain=-\ep:\ep](\x,{\wl+(\wls-\wl)*(\H)});}
\onslide<4->{\draw[smooth,domain=-\ep:\ep] plot[smooth,domain=-\ep:\ep](\x,{\dwl+(\dwls-\dwl)*(\H)});}
\onslide<6>{\draw[smooth,domain=-\ep:\ep,red] plot[smooth,domain=-\ep:\ep](\x,{\dvl*\hl/(\hl+(\hls-\hl)*\H)*sqrt(abs((\wl^2-\dwl^2)/((\wl+(\wls-\wl)*(\H))^2-(\dwl+(\dwls-\dwl)*(\H))^2)))})
								plot[smooth,domain=-\ep:\ep](\x,{(\hl/(\hl+(\hls-\hl)*\H)*(\wl*\vl+\dwl*\dvl)-(\dwl+(\dwls-\dwl)*(\H))*(\dvl*\hl/(\hl+(\hls-\hl)*\H)*sqrt(abs((\wl^2-\dwl^2)/((\wl+(\wls-\wl)*(\H))^2-(\dwl+(\dwls-\dwl)*(\H))^2)))))/(\wl+(\wls-\wl)*(\H))});
\draw[red] (\ep,\dvls)--(2,\dvls)node[above]{$\dv_{L*}$}
		(\ep,\vls)--(2,\vls)node[above]{$\v_{L*}$};}
\end{tikzpicture}\end{center}
\end{minipage}
\end{frame}

\begin{frame}{}{}
\begin{thebibliography}{9}
\bibitem{Lagoutiere08} [Lagouti\`ere, Seguin, Takahashi'08] {\color{black}{\bf Path-independent} non-conservative product definition.}
\end{thebibliography}
\begin{block}{Definition : non-conservative products}
Through a $\lambda_k$-shock, the following {\bf jump conditions} hold:\vspace{-10pt}
\begin{equation*}
\l\{\mat{@{}l@{~}c@{~}l@{~}c@{~}l@{}}{\ds
\partial_th&+&\ds\partial_x\( h\u\) &=&0,
\\\ds
\partial_t\( h\u\) &+&\ds\partial_x\( h\( \u^2+\du^2\) +\frac g2h^2\) &=&\ds0,
\\\ds
\partial_t\du&+&\partial_x\( \du\,\u\) &=&\ds0,
\\\ds
\partial_t\( h\v\) &+&\ds\partial_x\( h\( \u\,\v+\du\,\dv\) \) &=&\ds0,
\\\ds
\partial_t\dv&+&\du\partial_x\v+\u\partial_x\dv&=&\ds0,
}\r.\Leftrightarrow
\l\{\mat{r@{~}l}{
\sigma_k\l[h\r]&=\l[h \u \r] \\
\sigma_k\l[ h \u \r]&=\l[h \( \u^2 + \du^2 \) + \frac{g}{2} h^2 \r] \\
\sigma_k\l[\du \r]&=\l[ \du \u \r] \\
 \sigma_k\l[h\v\r]&=\l[h\( \u\,\v+\du\dv \) \r]\\
 \l[h\sqrt{\(\u-\sigma_k \)^2-\du^2}\ \dv\r]&=0
\qquad\textrm{if {\bf no-coalescence}}.
}\r.
\end{equation*}
\end{block}
\begin{minipage}{.65\linewidth}
\underline{\bf Arguments :} $z=x-\sigma_kt$, $\w=\u-\sigma_k$ and $\dw=\du$
\begin{enumerate}
\item Regularization : $h_\eps\(z\)\in C^1[-\sfrac{\eps}2,-\sfrac{\eps}2]$ monotonic.
\item $\(h_\eps\w_\eps\)'=0\Rightarrow \w_\eps\(z\)=\frac{h_L\w_L}{h_\eps\(z\)}\in C^1[-\sfrac{\eps}2,-\sfrac{\eps}2]$\\
$\(\dw_\eps\w_\eps\)'=0\Rightarrow \dw_\eps\(z\)=\frac{h_\eps\(z\)\w_L}{h_L}\in C^1[-\sfrac{\eps}2,-\sfrac{\eps}2]$
\item $\l\{\mat{@{}l@{}}{\(h_\eps\(\w\v+\dw\dv\)\)'=0\\\dw\v'+\w\dv'=0}\r.
\Rightarrow\(1-\(\frac{\dw_\eps}{\w_\eps}\)^2\)\dv_\eps'-\frac{\dw_\eps}{\w_\eps}\(\frac{\dw_\eps}{\w_\eps}\)'\dv_\eps=0$
$\Rightarrow\dv_\eps\(z\)= C \frac{h_L}{h_\eps(z)} \sqrt{\l|\frac{\w_L^2-\dw_L^2}{\w_\eps(z)^2-\dw_\eps(z)^2}\r|}
\qquad\Rightarrow
\v_\eps\(z\).$
\end{enumerate}
\end{minipage}\hfill
\begin{minipage}{.35\linewidth}
\begin{center}\begin{tikzpicture}
\def\hl{1}\def\hls{3.5}
\def\wl{3.5}\pgfmathsetmacro\wls{\hl*\wl/\hls}
\def\dwl{.4}\pgfmathsetmacro\dwls{\hls*\dwl/\hl}
\def\dvl{2.05}\def\dvls{1.8}
\def\vl{2.45}\pgfmathsetmacro\vls{(\hl/\hls*(\wl*\vl+\dwl*\dvl)-\dwls*\dvls)/(\wls)}

\def\ep{1}\def\H{(((\x+\ep)+.1*sin(500*(\x+\ep)))/((2*\ep)+.1*sin(1000*\ep)))}
\draw[->] (-2,0)--(2,0)node[below]{$z$};
\draw (-2,\hl)node[above]{$h_{L}$}--(-\ep,\hl) (\ep,\hls)--(2,\hls)node[above]{$h_{L*}$}
				(-2,\wl)node[above]{$\w_{L}$}--(-\ep,\wl) (\ep,\wls)--(2,\wls)node[above]{$\w_{L*}$}
				(-2,\dwl)node[above]{$\dw_{L}$}--(-\ep,\dwl) (\ep,\dwls)--(2,\dwls)node[above]{$\dw_{L*}$};
\draw[red](-2,\vl)node[above]{$\v_{L}$}--(-\ep,\vl)
		(-2,\dvl)node[above]{$\dv_{L}$}--(-\ep,\dvl);
\draw[smooth,domain=-\ep:\ep] plot[smooth,domain=-\ep:\ep](\x,{\hl+(\hls-\hl)*\H});
\draw[dashed,thick] (-\ep,0)node[below]{$-\sfrac\eps2$}--(-\ep,4) (\ep,0)node[below]{$\sfrac\eps2$}--(\ep,4);
\def\xb{.59}\def\y{3.2}\draw[dotted,thick] (\xb,0)--(\xb,4) (-2,\y)--(2,\y)node[below]{$\eta_L$};
\only<2->{\draw[smooth,domain=-\ep:\ep] plot[smooth,domain=-\ep:\ep](\x,{\wl+(\wls-\wl)*(\H)});
\draw[smooth,domain=-\ep:\ep] plot[smooth,domain=-\ep:\ep](\x,{\dwl+(\dwls-\dwl)*(\H)});}
\only<3->{\draw[smooth,red] plot[domain=-\ep:\xb-.05](\x,{\dvl*\hl/(\hl+(\hls-\hl)*\H)*sqrt((\wl^2-\dwl^2)/((\wl+(\wls-\wl)*(\H))^2-(\dwl+(\dwls-\dwl)*(\H))^2))})
					plot[domain=-\ep:\xb-.107](\x,{(\hl/(\hl+(\hls-\hl)*\H)*(\wl*\vl+\dwl*\dvl)-(\dwl+(\dwls-\dwl)*(\H))*(\dvl*\hl/(\hl+(\hls-\hl)*\H)*sqrt((\wl^2-\dwl^2)/((\wl+(\wls-\wl)*(\H))^2-(\dwl+(\dwls-\dwl)*(\H))^2))))/(\wl+(\wls-\wl)*(\H))});
\draw[smooth,red] plot[domain=\xb+.043:\ep](\x,{\dvls*\hls/(\hl+(\hls-\hl)*\H)*sqrt((\wls^2-\dwls^2)/((\wl+(\wls-\wl)*(\H))^2-(\dwl+(\dwls-\dwl)*(\H))^2))})
			plot[domain=\xb+.38:\ep](\x,{(\hl/(\hl+(\hls-\hl)*\H)*(\wl*\vl+\dwl*\dvl)-(\dwl+(\dwls-\dwl)*(\H))*(\dvls*\hls/(\hl+(\hls-\hl)*\H)*sqrt((\wls^2-\dwls^2)/((\wl+(\wls-\wl)*(\H))^2-(\dwl+(\dwls-\dwl)*(\H))^2))))/(\wl+(\wls-\wl)*(\H))});
\draw[red] (\ep,\dvls)--(2,\dvls)node[above]{$\dv_{L*}$}
		(\ep,\vls)--(2,\vls)node[above]{$\v_{L*}$};}
\end{tikzpicture}\end{center}
\end{minipage}
\end{frame}

\subsection{The 2D Riemann problem}
\begin{frame}{}{}
\begin{block}{Proposition : admissible shock of $\(2D-SW_2\)$}
We denote by $\sigma_k$ the speed of the $\lambda_k$-shock.\\
Assuming that the water depth $h$ is positive, the following properties are equivalent:
\begin{enumerate}[i)]
\item\label{entropy_jump_2D} The mechanical energy $E=\frac g2h^2+\frac h2\(\u^2+\du^2+\v^2+\dv^2\)$ is {\bf decreasing} through a shock, i.e.
\begin{equation*}
%-\sigma_k\l[E\r]+\l[\( \frac{\u^2+3\du^2+\v^2+\dv^2}2+gh\) h\u+h\v\dv\du\r]<0.
-\sigma_k\l[E\r]+\l[\(E +h\(\du^2+\frac g2h\)\)\u+h\v\dv\du\r]<0.
\end{equation*}
\item\label{LaxCondition_2D} the {\bf Lax entropy condition} is satisfied
\begin{equation*}
\lambda_L\( U_{L*}\)<\sigma_L<\lambda_L\( U_{L}\)
\qquad\textrm{and}\qquad
\lambda_R\( U_{R}\)<\sigma_R<\lambda_R\( U_{R*}\).
\end{equation*}
\end{enumerate}
\end{block}
\underline{\bf Remark :}
The transverse kinetic energy is {\bf preserved if there is no-coalescence}\\and {\bf dissipated if there is coalescence}.
More precisely we have
\begin{equation*}
\l[\frac{\v^2+\dv^2}2 h\w+h\v\dv\dw\r]
=
\frac 1{2Q}\l[h^2\( \w^2-\dw^2\)\dv^2\r].
\end{equation*}
\end{frame}

\begin{frame}{}{}
\begin{block}{Theorem : Riemann problem of $(2D-SW_2)$}
Consider the initial condition
%\begin{equation*} \label{1D_Riemann_data}
$V\( 0,x\) = 
\begin{cases}
 V_L = \( h_L,\u_L,\du_L,\v_L,\dv_L \) ^t \in \R_+^* \times \R^4& \ds \textrm{if} \ x<0, \\
 V_R = \( h_R,\u_R,\du_R,\v_R,\dv_R \) ^t \in \R_+^* \times \R^4& \ds \textrm{if} \ x \geq 0.
\end{cases}
$\\
%\end{equation*}
If the following condition is fulfilled : $\u_R-\u_L < \mu^r\(V_L\) +  \mu^r\(V_R\)$\\
then there exists a {\bf unique selfsimilar} solution $V\in\( L^\infty\( \R_+^*\times\R\) \) ^5$ to the 2D Riemann problem $\(SW_2\)$ satisfying the {\bf mechanical energy dissipation}.\\
In addition the water depth $h$ is {\bf strictly positive} for all $\( t,x\) \in\R_+\times\R$.
\end{block}
\begin{center}\begin{tikzpicture}[xscale=1.5,yscale=2]\def\ub{.5}\def\hu{2.5}\def\ya{.5}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1) node[above]{$\gamma_L$};
\onslide<1>{\draw[dashed](0,0)--({\ub+\hu/2},1) node[above]{$\gamma_R$};}
\draw[dashed](0,0)--({\ub},1) node[above]{$\lambda_*$};
\onslide<1>{\draw (0,0)--({\ub+\hu},1) node[above]{$\sigma_R$};}
\onslide<2>{
\draw[dashed,xshift=-1.5pt] (0,0)--({\ub+\hu},1) node[above]{\color{red}$\sigma_R<\gamma_R\(U_{R*}\)$};
\draw (0,0)--({\ub+\hu},1) ;}
\draw[thick,->] (-3,0)--(3,0) node[below]{\small$x$};
\fill[color=black!20] ({\ub-1.2*\hu},1) node[above,black]{$\lambda_{L}\( V_L\) $}--(0,0)--({\ub-.9*\hu},1) node[above,black]{$\lambda_{L}\( V_{L*}\) $};
\draw
%	({\ya*(\ub-1.7*\hu)},\ya)node{$W_L^{1D}$}
%	({\ya*(\ub+7/4*\hu)},\ya)node{$W_R^{1D}$}
%	({\ya*(\ub-1/2*\hu)},\ya)node{$W_{L*}^{1D}$}
%	({\ya*(\ub+1/2*\hu)},\ya)node{$W_{R*}^{1D}$}
	({\yb*(\ub-1.5*\hu)},\yb)node{$V_L$}
	({\yb*(\ub+1.3*\hu)},\yb)node{$V_R$}
	({\yb*(\ub-11/16*\hu)},\yb)node{$V_{L1}$}
	({\yb*(\ub-1/4*\hu)},\yb)node{$V_{L2}$};
\draw({\yb*(\ub+1/4*\hu)},\yb)node{$V_{R2}$};
\onslide<1>{\draw({\yb*(\ub+3/4*\hu)},\yb)node{$V_{R1}$};}
\draw[red,double,->] ({1.1*\ya*(\ub-10/8*\hu)},1.1*\ya)--({1.1*\ya*(\ub-7/8*\hu)},1.1*\ya);
\draw[red,double,->] ({.9*\ya*(\ub-10/8*\hu)},.9*\ya)--({.9*\ya*(\ub-7/8*\hu)},.9*\ya);
\draw[red,double,->] ({\ya*(\ub-5/8*\hu)},\ya)--({\ya*(\ub-3/8*\hu)},\ya);
\draw[red,double,<->] ({\ya*(\ub-1/8*\hu)},\ya)--({\ya*(\ub+1/8*\hu)},\ya);
\onslide<1>{\draw[red,double,->] ({1.1*\ya*(\ub+9/8*\hu)},1.1*\ya)--({1.1*\ya*(\ub+7/8*\hu)},1.1*\ya);
\draw[red,double,->] ({.9*\ya*(\ub+9/8*\hu)},.9*\ya)--({.9*\ya*(\ub+7/8*\hu)},.9*\ya);
\draw[red,double,->] ({\ya*(\ub+5/8*\hu)},\ya)--({\ya*(\ub+3/8*\hu)},\ya);}
\onslide<2>{\draw[red,double,->] ({\ya*(\ub+9/8*\hu)},\ya)--({\ya*(\ub+7/8*\hu)},\ya);}
\end{tikzpicture}\end{center}
\end{frame}

\subsection{Some analytical solutions}
\begin{frame}{}{}
\begin{center}
\begin{tabular}{cc}

\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\fill[color=black!20] ({\ub-1.5*\hu},1)--(0,0)--({\ub-1*\hu},1);
\draw (0,0)--({\ub+\hu},1);
\draw[dashed](0,0)--({\ub},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{general}}{Pictures/General.avi}
\end{tabular}

\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{0}\def\hu{2}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\draw[dashed](0,0)--({\ub},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{circulation}}{Pictures/Circulation.avi}
\end{tabular}

\\~\\~\\~\\
\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{1}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub+\hu/2},1);
\draw (-3,0)--(3,0);
\fill[color=black!20] ({\ub-1.5*\hu},1)--(0,0)--({\ub-1*\hu},1);
\draw (0,0)--({\ub+\hu},1);
\draw[double, dashed](0,0)--({\ub},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{simple resonance}}{Pictures/Resonance1.avi}
\end{tabular}
\hfill

\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{1.2}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw[dashed](0,0)--({\ub-\hu/2},1);
\draw (-3,0)--(3,0);
\draw (0,0)--({\ub-\hu},1);
\draw[dashed](0,0)--({\ub},1);
\draw[xshift=-2pt,dashed] (0,0)--({\ub+\hu},1);
\draw[xshift=2pt] (0,0)--({\ub+\hu},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{coalescence}}{Pictures/Coalescence1.avi}
\end{tabular}

\\~\\~\\~\\
\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{1.31}\def\hu{2}\def\ya{.4}\def\yb{.75}
\draw (-3,0)--(3,0);
\fill[color=black!20] ({\ub-1.5*\hu},1)--(0,0)--({\ub-1*\hu},1);
\draw (0,0)--({\ub+\hu},1);
\draw[dashed,xshift=-3pt](0,0)--({\ub},1);
\draw[dashed](0,0)--({\ub},1);
\draw[dashed,xshift=3pt](0,0)--({\ub},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{double resonance}}{Pictures/Resonance2.avi}
\end{tabular}

\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{1.2}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw (-3,0)--(3,0);
\draw (0,0)--({\ub-\hu},1);
\draw[double, dashed](0,0)--({\ub},1);
\draw[xshift=-2pt,dashed] (0,0)--({\ub+\hu},1);
\draw[xshift=2pt] (0,0)--({\ub+\hu},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{coalescence + resonance}}{Pictures/CoalescenceResonance.avi}
\end{tabular}

\begin{tabular}{c}\tikz{
\begin{scope}[xscale=.5,yscale=1.5]
\def\ub{1.2}\def\hu{2.5}\def\ya{.4}\def\yb{.75}
\draw (-3,0)--(3,0);
\draw[xshift=-2pt] (0,0)--({\ub-\hu},1);
\draw[xshift=2pt,dashed] (0,0)--({\ub-\hu},1);
\draw[dashed](0,0)--({\ub},1);
\draw[xshift=-2pt,dashed] (0,0)--({\ub+\hu},1);
\draw[xshift=2pt] (0,0)--({\ub+\hu},1);
\end{scope}}\\\movie[externalviewer]{\beamergotobutton{2 coalescences}}{Pictures/Coalescences2.avi}
\end{tabular}

\end{tabular}\end{center}
\end{frame}

\section{Conclusions}

\begin{frame}{}{}
{\color{bleu}\sc\underline{Realizations:}}
\begin{tabular}{@{\quad}|@{}p{.95\linewidth}}
\begin{itemize}\vspace{-\baselineskip}
\item[\myitemb] well-possedness of the bi-layerswise discretized model $(2D-SW)$.
\item[\myitemb] analysis of resonance phenomena, coalescence phenomena, non-conservative products
\end{itemize}\vspace{-1.7\baselineskip}
\end{tabular}
\begin{thebibliography}{9}
\bibitem{Aguillon17} [Aguillon, Audusse, Godlewski, Parisot] {\color{black}to appear...}
\end{thebibliography}\vfill


{\color{bleu}\sc\underline{Perspectives for analysis:}}
\begin{tabular}{@{\quad}|@{}p{.95\linewidth}}
\begin{itemize}\vspace{-\baselineskip}
%\item[\hspace{.25em}\tikz{\fill[color=red](0,0)circle(3.5pt);\fill[color=white](-.06,-.02)rectangle(.06,.02);}\hspace{.15em}]
%\item[\color{orange}\danger]with 3 layers (Skewness), 4 layers (Kurtosis)...
\item[\color{orange}\danger]
with an arbitrary number of layers %\hfill{\bf out of reach}
\item[\color{orange}\danger]in an asymptotic regime : small shear $\frac{\l|u_{i+1}-u_i\r|}{\sqrt{gh}}\ll 1$%\hfill{\bf oceanography}
\item[\color{orange}\danger]with active tracers%\hfill{\bf oceanography}
\item[\color{orange}\danger]with dispersive terms
\end{itemize}\vspace{-1.5\baselineskip}
\end{tabular}

\vfill

{\color{bleu}\sc\underline{Perspectives for numeric:}}
\begin{tabular}{@{\quad}|@{}p{.95\linewidth}}
\begin{itemize}\vspace{-\baselineskip}
\item[\myitemb] preservation of the steady state with circulation
\item[\myitemb] numerical analysis in case of coalescence
\item[\color{orange}\danger]explanation of the uncoupled numerical scheme \hfill CFL {\bf not enough} restrictive
\end{itemize}\vspace{-1.5\baselineskip}
\end{tabular}

\end{frame}

\end{document}


