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\title{NumWave17}
\author{\raisebox{-.35\height}{\includegraphics[scale=.1]{Pictures/inst_inria.png}}\hspace{1cm}Martin Parisot}
\date{Layerwise Dispersive models\hfill}

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\def\Euler{($E$)\xspace}
\def\SW{($SW$)\xspace}\def\DSW{($SW^\delta$)\xspace}\def\sh{{\sc sh}}
\def\SWL{($SW_L$)\xspace}\def\DSWL{($SW_L^\delta$)\xspace}
\def\NH{($NH$)\xspace}\def\DNH{($NH^\delta$)\xspace}\def\nh{{\sc nh}}
\def\NHL{($NH_L$)\xspace}\def\DNHL{($NH_L^\delta$)\xspace}
\def\GN{($GN$)\xspace}\def\DGN{($GN^\delta$)\xspace}\def\gn{{\sc gn}}
\def\GNL{($GN_L$)\xspace}\def\DGNL{($GN_L^\delta$)\xspace}
\def\Hu{$H_{yp}^u)$\xspace}
\def\Hp{$H_{yp}^p)$\xspace}
\def\Hq{$H_{yp}^w)$\xspace}
\def\xx{{\sc xx}}

\begin{document}
\scriptsize

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		\draw (0,.8\baselineskip) node {\large\sc\color{blanc} A hierarchy of Dispersive Layerwise models}
			(0,-.8\baselineskip) node {\large\sc\color{blanc} for free surface flow}
			(0,-2.3\baselineskip) node {\footnotesize\sc\color{blanc} Derivation and Simulation};			
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		\draw (0,-3.5) node{\underline{Martin Parisot} - {\sc Ange} Inria Paris-Rocquencourt};
		\draw (1,-4) node[left]{in collaboration with}node[right]{Enrique Fernandes-Nieto (Sevile)}
			(1,-4.4)node[right]{Yohan Penel ({\sc Ange})}
			(1,-4.8)node[right]{Jacques Saint-Marie ({\sc Ange})}
				node[left]{and};
	\end{scope} 

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				node[above left]{\includegraphics[height=.25\linewidth]{Pictures/riviere}};
			
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		\draw (0,-7) node {{\bf NumWave} - 11 december 2017};
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			\includegraphics[height=.1\linewidth]{Pictures/logo_inria.jpg}\qquad
			\includegraphics[height=.1\linewidth]{Pictures/logo-LJLL.png}\qquad
			\includegraphics[height=.1\linewidth]{Pictures/logo_cnrs.jpeg}\qquad
			\includegraphics[height=.1\linewidth]{Pictures/logo_cerema.jpg}};
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\section{Introduction}
\subsection{Of waves and jumps}

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\only<2>{\draw(0,3)node[above right]{\includegraphics[width=.49\linewidth,clip,trim=0 0 0 0]{Pictures/plunging}};}
\only<3>{\draw(0,2)node[above right]{\includegraphics[width=.49\linewidth,clip,trim=0 0 0 0]{Pictures/collapsing}};}
\only<4>{\draw(0,0)node[above right]{\includegraphics[width=.49\linewidth,clip,trim=0 30 0 0]{Pictures/surging}};}
\only<5->{\draw(0,0)node[above right]{\includegraphics[width=.49\linewidth,clip,trim=90 50 270 0]{Pictures/Chow}};}
\only<5>{\draw(0,4)node[above left]{\includegraphics[width=.49\linewidth,clip,trim=00 00 0 0]{Pictures/mascaret}};}
\only<5>{\draw(0,0)node[above left]{\includegraphics[width=.49\linewidth,clip,trim=00 00 0 0]{Pictures/rafting}};}
\draw(0,0)node[below]{Vertical profile in breaking wave (left) and hydraulic jump (right) {\color{bleu}[Chow'59]}};
\end{tikzpicture}\end{center}
\end{frame}

\subsection{Brief description of the hierarchy of free surface models}

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\begin{center}\begin{tikzpicture}[xscale=2,yscale=1.5]
	\node (E) at (0,0) {\Euler};
	\node (SW) at (3,-1) {\SW};
	\draw[->] (E.south east)-- (SW.north west) node[midway,above,sloped]{\tiny\Hu and \Hp};
\only<1>{\draw(3.3,-1.5)node[below right]{\includegraphics[width=.4\linewidth,height=2.7cm,clip,trim=30 65 270 0]{Pictures/JumpSW}};}

\only<2->{\node (GN) at (0,-1) {\GN};
	\node (NH) at (2,-1) {\NH};
	\draw[->] (E.south)-- (GN.north) node[midway,above,sloped]{\tiny\Hu};
	\draw[->] (GN.east)-- (NH.west) node[midway,below,sloped]{\tiny\Hq};
	\draw[->] (NH.east)-- (SW.west) node[midway,below,sloped]{\tiny\Hp};}
\only<2>{\draw(3.3,-1.5)node[below right]{\includegraphics[width=.4\linewidth,height=2.7cm,clip,trim=30 65 270 0]{Pictures/JumpGN}};}

\only<3->{\node (SWL) at (3,0) {\SWL};
	\draw[->] (E.east)-- (SWL.west) node[midway,above,sloped]{\tiny\Hp};
	\draw[->] (SWL.south)-- (SW.north) node[midway,above,sloped]{\tiny\Hu};}

\only<4->{	\node (GNL) at (1,.4) {\color{red}\GNL};
	\node (NHL) at (2,.4) {\color{red}\NHL};
	\draw[->,red] (E.north east)-- (GNL.west) node[midway,above,sloped,red]{\tiny regul.};
	\draw[->,red] (GNL.east)-- (NHL.west) node[midway,above,sloped,red]{\tiny\Hq};
	\draw[->,red] (NHL.east)-- (SWL.north west) node[midway,above,sloped,red]{\tiny\Hp};}
	
\only<5>{
	\draw[->,red] (GNL.south)-- (GN.north east) node[midway,above,sloped]{\tiny L=1};
	\draw[->,red] (NHL.south)-- (NH.north) node[midway,above,sloped]{\tiny L=1};}


\begin{scope}[yshift=-1.7cm]
\draw(0,0)node[above right]{\color{bleu}\sc\underline{Models:}};
\draw(.5,-.1)node[left]{{\color{bleu}\Euler}} node[right]{Euler eq. with {\bf monovalued} free surface};
\draw(.5,-.4)node[left]{{\color{bleu}\SW}} node[right]{Shallow water model \only<1-2>{\color{bleu}[Saint-Venant'1871]}};
\only<2->{\draw(.5,-.7)node[left]{{\color{bleu}\NH}} node[right]{Non-hydrostatic model \only<2>{\color{bleu}[Sainte-Marie'11]}};
\draw(.5,-1)node[left]{{\color{bleu}\GN}} node[right]{Green-Naghdi model \only<2>{\color{bleu}[Serre'53]}};}
\only<3->{\draw(2.8,-.4)node[left]{{\color{bleu}\SWL}} node[right]{Layerwise shallow water model {\color{bleu}[ABPS{\sc m}'11]}};}
\only<4->{\draw(2.8,-.7)node[left]{{\color{bleu}\NHL}} node[right]{Layerwise non-hydrostatic model {\color{bleu}[F{\sc n}PPS{\sc m}]}};
\draw(2.8,-1)node[left]{{\color{bleu}\GNL}} node[right]{Layerwise Green-Naghdi model {\color{bleu}[F{\sc n}PPS{\sc m}]}};}

\draw(0,-1.7)node[above right]{\color{bleu}\sc\underline{Assumptions:}};
\draw(.5,-1.8)node[left]{{\color{bleu}\Hp}} node[right]{{\bf Hydrostatic} pressure: \qquad $p\(t,x,z\)=P\(t,x\)+g\(\eta\(t,x\)-z\)$};
\draw(.5,-2.1)node[left]{{\color{bleu}\Hu}} node[right]{{\bf Homogeneity} of the {\bf horizontal} velocity in the column:\qquad$u\(t,x,z\)=\u\(t,x\)$};
\only<2->{\draw(.5,-2.4)node[left]{{\color{bleu}\Hq}} node[right]{{\bf Homogeneity} of the {\bf vertical} velocity in the column:\qquad$w\(t,x,z\)=\w\(t,x\)$};}
\end{scope}

\begin{scope}[xshift=4.1cm,yshift=-1.4cm,xscale=.25,yscale=.4]
\only<1-2>{\def\LM{1}}\only<3->{\def\LM{3}}
	\newcommand\Bottom[1]{(.5-.2*cos(50*#1))+.2*sin(100*#1)-.05*#1)}
	\newcommand\etaM[1]{(5+.1*(1+sin(200*#1)))}
	\newcommand\zetaM[2]{{(\Bottom{#2}+(#1)/\LM*(\etaM{#2}-(\Bottom{#2})))}}
	\newcommand\qb[2]{{#2-14*(\LM-#1)/\LM+30*(\LM-#1)/\LM*(\LM-#1)/\LM-20*(\LM-#1)/\LM*(\LM-#1)/\LM*(\LM-#1)/\LM}}
	\newcommand\qz[1]{\qei+(#1-\ei)*((\qbi-\qei)/(\bi-\ei)+.2*(-1)^(\i)*(#1-\bi))}
\only<1-2>{
	\draw[thick,smooth,samples=100,domain=-3:6] plot(\x,{\etaM{\x}}) node[right]{\footnotesize$\eta$};
	\fill[fill=black!30,smooth,samples=100,domain=-3:6] (-3,-.2)--plot(\x,{\Bottom{\x}})--(6,-.2);
	\draw[thick,smooth,samples=100,domain=-3:6] plot(\x,{\Bottom{\x}}) node[right]{\footnotesize$B$};

	\def\x{2.25}
	\draw[thick,<->] (\x,{\etaM{\x}})--(\x,{\Bottom{\x}}) node[midway,left]{\footnotesize$h$};
	\pgfmathsetmacro{\xi}{\x+1.5}\foreach \i in {1,...,\LM}{
		\pgfmathsetmacro{\bi}{\zetaM{\i-1}{\x}}\pgfmathsetmacro{\ei}{\zetaM{\i}{\x}}
		\pgfmathsetmacro{\qbi}{\qb{(\i-1)}{\x}}\pgfmathsetmacro{\qei}{{\qb{\i}{\x}}}
		\draw[thick,->] ({\xi},{(\bi+\ei)/2})--({\xi+1},{(\bi+\ei)/2}) node[midway,below]{\footnotesize$\u$};
\only<2>{\draw[thick,->] ({\xi},{(\bi+\ei)/2})--({\xi},{(\bi+\ei)/2+1}) node[midway,left]{\footnotesize$\w$};
		\draw[thick,>-<] ({\xi+.25},{(\bi+\ei)/2+.75})--({\xi+.25},{(\bi+\ei)/2+1.25}) node[midway,right]{\footnotesize$\sigma$};
		\draw[thick,dashed] ({\qei},{\ei})--({\qbi},{\bi});
		\draw[thick,domain=\zetaM{\i-1}{\x}:\zetaM{\i}{\x},variable=\t] plot({\qz{\t}},\t);}}
\only<2>{\def\i{1}\pgfmathsetmacro{\im}{1}\pgfmathsetmacro{\qbi}{\qb{0}{\x}}\pgfmathsetmacro{\qei}{\qb{1}{\x}}\pgfmathsetmacro{\bi}{\zetaM{0}{\x}}\pgfmathsetmacro{\ei}{\zetaM{1}{\x}}
		\draw({(\qbi+\qei)/2},{(\bi+\ei)/2}) node[left]{\footnotesize$\q$} ({\qbi},{\bi}) node[xshift=-7]{\footnotesize$q_{B}$};}}

\only<3->{
	\draw[thick,smooth,samples=100,domain=-3:6] plot(\x,{\etaM{\x}}) node[right]{\footnotesize$\eta$};
	\fill[fill=black!30,smooth,samples=100,domain=-3:6] (-3,-.2)--plot(\x,{\Bottom{\x}})--(6,-.2);
	\draw[thick,smooth,samples=100,domain=-3:6] plot(\x,{\Bottom{\x}}) node[right]{\footnotesize$B$};
	\pgfmathsetmacro{\Lm}{\LM-1}
	\foreach \i in {1,...,\Lm}{\pgfmathsetmacro{\ip}{2*\i+1}\draw[very thick,dotted,smooth,samples=100,domain=-3:6] plot(\x,\zetaM{\i}{\x}) node[right,black]{\footnotesize$\zeta_{\sfrac{\pgfmathprintnumber\ip}2}$};}

	\def\x{2.25}
	\draw[thick,<->] (\x,{\etaM{\x}})--(\x,{\Bottom{\x}}) node[midway,left]{\footnotesize$h$};
	\pgfmathsetmacro{\xi}{\x+1.5}\foreach \i in {1,...,\LM}{
		\pgfmathsetmacro{\bi}{\zetaM{\i-1}{\x}}\pgfmathsetmacro{\ei}{\zetaM{\i}{\x}}
		\pgfmathsetmacro{\qbi}{\qb{(\i-1)}{\x}}\pgfmathsetmacro{\qei}{{\qb{\i}{\x}}}
		\draw[thick,->] ({\xi},{(\bi+\ei)/2})--({\xi+1},{(\bi+\ei)/2}) node[midway,below]{\footnotesize$u_\i$};
\only<4->{\draw[thick,->] ({\xi},{(\bi+\ei)/2})--({\xi},{(\bi+\ei)/2+.5}) node[midway,left]{\footnotesize$w_\i$};
		\draw[thick,>-<] ({\xi+.25},{(\bi+\ei)/2+.25})--({\xi+.25},{(\bi+\ei)/2+.75}) node[midway,right]{\footnotesize$\sigma_\i$};
		\draw[thick,dashed] ({\qei},{\ei})--({\qbi},{\bi});
		\draw[thick,domain=\zetaM{\i-1}{\x}:\zetaM{\i}{\x},variable=\t] plot({\qz{\t}},\t);}}
\only<4->{\def\i{1}\pgfmathsetmacro{\im}{2*\i-1}\pgfmathsetmacro{\qbi}{\qb{(\i-1)}{\x}}\pgfmathsetmacro{\qei}{\qb{\i}{\x}}\pgfmathsetmacro{\bi}{\zetaM{(\i-1)}{\x}}\pgfmathsetmacro{\ei}{\zetaM{\i}{\x}}
		\draw({(\qbi+\qei)/2},{(\bi+\ei)/2}) node[left]{\footnotesize$q_\i$} ({\qbi},{\bi}) node[xshift=-7]{\footnotesize$q_{\sfrac{\pgfmathprintnumber\im}2}$};
	\def\i{2}\pgfmathsetmacro{\im}{2*\i-1}\pgfmathsetmacro{\qbi}{\qb{(\i-1)}{\x}}\pgfmathsetmacro{\qei}{\qb{\i}{\x}}\pgfmathsetmacro{\bi}{\zetaM{(\i-1)}{\x}}\pgfmathsetmacro{\ei}{\zetaM{\i}{\x}}
		\draw({(\qbi+\qei)/2},{(\bi+\ei)/2}) node[right]{\footnotesize$q_\i$} ({\qbi},{\bi}) node[xshift=7,yshift=-5]{\footnotesize$q_{\sfrac{\pgfmathprintnumber\im}2}$};
	\def\i{3}\pgfmathsetmacro{\im}{2*\i-1}\pgfmathsetmacro{\qbi}{\qb{(\i-1)}{\x}}\pgfmathsetmacro{\qei}{\qb{\i}{\x}}\pgfmathsetmacro{\bi}{\zetaM{(\i-1)}{\x}}\pgfmathsetmacro{\ei}{\zetaM{\i}{\x}}
		\draw({(\qbi+\qei)/2},{(\bi+\ei)/2}) node[left]{\footnotesize$q_\i$} ({\qbi},{\bi}) node[xshift=-7,yshift=5]{\footnotesize$q_{\sfrac{\pgfmathprintnumber\im}2}$};}}

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\draw(-.2,.7)node{~};
\draw(10,-10)node{~};
\end{tikzpicture}\end{center}
\end{frame}

\section{The hierarchy of layerwise models}
\subsection{Derivation}

\begin{frame}{}{}
\begin{overlayarea}{\linewidth}{\linewidth}
{\color{bleu}\sc\underline{Euler equations \Euler:}}\\
\begin{tabular}{@{\quad}|@{\quad}p{.5\linewidth}@{\quad}p{.45\linewidth}}
Between a {\bf given bottom} $B\(t,x\)$\newline and a {\bf monovalued} free surface $\eta\(t,x\)$:&
(1) $\nabla\cdot u+\partial_z w=0$
\newline
(2) \only<1>{$\partial_tu+u\cdot\nabla u+ w\partial_zu=-\nabla p$}
\only<2->{$\partial_tu+u\cdot\nabla u+ w\partial_zu=-\nabla\(P+g\eta+q\)$}
\newline
(3) \only<1>{$\partial_tw+u\cdot\nabla w+ w\partial_zw=-\partial_z p-g$}
\only<2->{$\partial_tw+u\cdot\nabla w+ w\partial_zw=-\partial_z q$}
\\
with {\bf no-penetration} at bottom is assumed:&
(4) $\partial_tB+u_{|_{z=B}}\cdot\nabla B-w_{|_{z=B}}=0$\\
at free surface, the {\bf pressure is fixed}:&
(5) $p_{|_{z=\eta}}=P\(t,x\)$\\
and the {\bf kinematic condition} is considered:&
(6) $\partial_t\eta+u_{|_{z=\eta}}\cdot\nabla\eta-w_{|_{z=\eta}}=0$\\
and compatible initial condition $\eta^0$, $u^0$ and $w^0$.&$\nabla\cdot u^0+\partial_z w^0=0$\\&$\partial_tB_{|_{t=0}}+u^0_{|_{z=B}}\cdot\nabla B_{|_{t=0}}-w^0_{|_{z=B}}=0$
\end{tabular}\vfill

\vspace{20pt}\only<2>{\myitem We use the {\bf hydrodynamic} pressure:\hfill$q\(t,x,z\)=p\(t,x,z\)-\(P\(t,x\)+g\(\eta\(t,x\)-z\)\)$\\[10pt]
\myitem For $L$ fixed parameter $\(\ell_i\)_{1\le i\le L}$ such that $\sum_i\ell_i=1$, we {\bf discretize} the column by layers of thickness $\ell_ih$ with $h=\eta-B$
and $\L_i=\l[\zeta_{i-\onehalf},\zeta_{i+\onehalf}\r]$ with $\zeta_{i+\onehalf}=\zeta_{i-\onehalf}+\ell_ih$ and $\zeta_{\onehalf}=B$.\\[10pt]
\myitem Assuming u is close enough to its average in $\L_i$: \hfill$u\(t,x,z\in\L_i\)=u_i\(t,x\)+O\(\eps\)$\\}

\only<3->{\vspace{-5pt}\myitem Integrating (1) on $\L_i$:\hfill
$\partial_th+\nabla\cdot\(hu_i\)=\frac1{\ell_i}\l[G\r]_{i-\onehalf}^{i+\onehalf}$\\
\only<3>{with $G$ is the mass exchanged between the layers $G_{i+\onehalf}=\partial_t\zeta_{i+\onehalf}+u_{|_{z=\zeta_{i+\onehalf}}}\cdot\nabla\zeta_{i+\onehalf}-w_{|_{z=\zeta_{i+\onehalf}}}$\\}}
\only<4>{\myitem Integrating (2) on $\L_i$:\hfill
$\partial_t\(h u_i\)+\nabla\cdot\(hu_i\otimes u_i\)=-h\nabla\(P+g\eta\)+\frac1{\ell_i}\l[{\color{red}u}G\r]_{i-\onehalf}^{i+\onehalf}{\color{red}-\frac1{\ell_i}\int_{\L_i}\nabla q\d z}$\\}
\only<5>{\myitem Integrating (2) on $\L_i$:\hfill
$\partial_t\(h u_i\)+\nabla\cdot\(hu_i\otimes u_i\)=-h\nabla\(P+g\eta\)+\frac1{\ell_i}\l[{\color{red}u}G\r]_{i-\onehalf}^{i+\onehalf}{\color{red}-\cancel{\frac1{\ell_i}\int_{\L_i}\nabla q\d z}}$\\
\qquad\myitemb Assuming \Hp, we have $q=0$, and it leads to \SWL.\\
\qquad\myitemb To close the system, we set\hfill$u_{i+\onehalf}G_{i+\onehalf}=\frac{u_i+u_{i+1}}2G_{i+\onehalf}+\gamma\frac{u_{i+1}-u_i}2\l|G_{i+\onehalf}\r|$.\\
}

\only<6->{\myitem Integrating (2) on $\L_i$:\hfill
$\partial_t\(h u_i\)+\nabla\cdot\(hu_i\otimes u_i\)=-h\nabla\(P+g\eta\)+\frac1{\ell_i}\l[uG\r]_{i-\onehalf}^{i+\onehalf}-\nabla\(h{\color{red}q_i}\)+\frac1{\ell_i}\l[\only<-7>{{\color{red}q}}\only<8->{q}\nabla \zeta\r]_{i-\onehalf}^{i+\onehalf}$\\}
\only<7->{\myitem Integrating (3) on $\L_i$:\hfill
$\partial_t\(h w_i\)+\nabla\cdot\(hw_i\ u_i\)=\frac1{\ell_i}\l[wG\r]_{i-\onehalf}^{i+\onehalf}-\frac1{\ell_i}\l[q\r]_{i-\onehalf}^{i+\onehalf}$\\}
\only<8->{\myitem Integrating {\bf twice} (1) on $\L_i$:\hfill
$w_i=\partial_t\zeta_i+u_i\cdot\nabla\zeta_i-G_i$\\}
\only<8>{with $\zeta_i=\frac{\zeta_{i-\onehalf}+\zeta_{i+\onehalf}}2$ and $G_i=\frac{G_{i-\onehalf}+G_{i+\onehalf}}2$}
\only<9>{\qquad\myitemb Assuming \Hq, we have $q_i=\frac{q_{i-\onehalf}+q_{i+\onehalf}}2$, and it leads to \NHL.\\}

\only<10>{\vspace{5pt}\myitem Multiplying (3) by $\(z-\zeta_i\)$ and integrating ({\bf weak derivation}) on $\L_i$:\\\hfill
{\tiny $\partial_t\(h \sigma_i\)+\nabla\cdot\(h\sigma_i\ u_i\)=-\frac1{\ell_i}\(\(\sigma_i-\sqrt{3}\(w_{i+\onehalf}-w_i\)\)G_{i+\onehalf}-\(\sigma_i+\sqrt{3}\(w_{i-\onehalf}-w_i\)\)G_{i-\onehalf}\)+\frac{2\sqrt{3}}{\ell_i}\(q_i-\frac{q_{i-\onehalf}+q_{i+\onehalf}}2\)$}\\}
\only<11->{\myitem \hfill
{\tiny $\partial_t\(h \sigma_i\)+\nabla\cdot\(h\sigma_i\ u_i\)=-\frac1{\ell_i}\(\(\sigma_i-\sqrt{3}\(w_{i+\onehalf}-w_i\)\)G_{i+\onehalf}-\(\sigma_i+\sqrt{3}\(w_{i-\onehalf}-w_i\)\)G_{i-\onehalf}\)+\frac{2\sqrt{3}}{\ell_i}\(q_i-\frac{q_{i-\onehalf}+q_{i+\onehalf}}2\)$}\\
\myitem Looking at the {\bf oriented standard variation} in (1) on $\L_i$:\hfill
$\sigma_i=-\frac{\ell_ih}{2\sqrt{3}}\nabla\cdot u_i$\\}

\only<1>{\begin{block}{\scriptsize Energy conservation of \Euler}
For s.e.s., we have:\hfill
$\partial_t\(\E^{\eta-B}+\K^u+\K^w\)+\nabla\cdot\G=
\(\eta-B\)\partial_tP+p_{|{z=B}}\partial_tB$\\
with $\E^h=h\(gB+P+g\frac h2\)$ and $\K^\psi=\frac12\int_B^\eta\l|\psi\r|^2\d z$.
\end{block}}
\only<5>{\begin{block}{\scriptsize Energy conservation of \SWL}
Assuming $\gamma\ge0$. For s.e.s., we have:\hfill
$\partial_t\(\E^h+\sum_i\K^u_i\)+\nabla\cdot\G\le
h\partial_tP+gh\partial_tB$\\
with $\E^h=h\(gB+P+g\frac h2\)$ and $\K^\psi_i=\frac{\ell_i}2h\l|\psi_i\r|^2$.¡
\end{block}}
\only<9>{\begin{block}{\scriptsize Energy conservation of \NHL}
Assuming $\gamma\ge0$. For s.e.s., we have:\hfill
$\partial_t\(\E^h+\sum_i\(\K^u_i+\K^w_i\)\)+\nabla\cdot\G\le
h\partial_tP+\(gh+q_{\onehalf}\)\partial_tB$\\
with $\E^h=h\(gB+P+g\frac h2\)$ and $\K^\psi_i=\frac{\ell_i}2h\l|\psi_i\r|^2$.
\end{block}}
\only<11>{\vspace{-5pt}\begin{block}{\scriptsize Energy conservation of \GNL}
Assuming $\gamma\ge0$. For s.e.s., we have:\hfill
$\partial_t\(\E^h+\sum_i\(\K^u_i+\K^w_i+\K^\sigma_i\)\)+\nabla\cdot\G\le
h\partial_tP+\(gh+q_{\onehalf}\)\partial_tB$\\
with $\E^h=h\(gB+P+g\frac h2\)$ and $\K^\psi_i=\frac{\ell_i}2h\l|\psi_i\r|^2$.
\end{block}}
\end{overlayarea}
\end{frame}

\subsection{Dispersion relations}

\begin{frame}{}{}
\begin{block}{\scriptsize Dispersion relations of the layerwise models \NHL and \GNL}
The dispersion relation  around the state $\(h,u_i\)=\(H,0\)$ reads:\hfill $\l|c^\xx_L\r|=\sqrt{\l<A_{kH}^{-1}e,\ell\r>}c^\sh$\\
with $c^\sh=\sqrt{gH}$, $\ell=\(\ell_i\)_{1\le i\le L}$, $e=\(1\)_{1\le i\le L}$, $\lambda^\nh=4$, $\lambda^\gn=6$
\\\hfill and $A_{x}=\Id-x^2B$ with $\ds B_{i,j}=\frac{\ell_i^2}{\lambda^\xx}\delta_{i,j}-\ell_j\(\frac{\ell_{\max\(i,j\)}}2+\sum_{l=\max\(i,j\)+1}^L\ell_l\)$
\end{block}
\begin{center}
\only<1>{\includegraphics[height=.49\linewidth]{Pictures/PhaseVelocity}}
\only<2>{
\begin{block}{\scriptsize Theorem:}
For any wave number $k>0$, the phase velocity of the dispersive models converges to the phase velocity of \Euler when the number of layers goes to infinity.
$$c_L^\xx\underset{L\to\infty}{\longrightarrow} c^E=\frac{tanh\(kH\)}{kH}$$
\end{block}
}
\end{center}
\end{frame}

\section{Entropy-satisfying numerical resolution}
\subsection{Numerical scheme}

\begin{frame}{}{}
\begin{overlayarea}{\linewidth}{.4\linewidth}
{\color{bleu}\sc\underline{Layerwise Serre/Green-Naghdi model \GNL:}}
\begin{tabular}{@{\quad}|@{\quad}p{.95\linewidth}}
\only<1,6>{$\partial_tU_i+\nabla\cdot F\(U_i\)=S\(U_i\)+\frac1{\ell_i}\l[\Gamma\r]_{i-\onehalf\oplus}^{i+\onehalf\ominus}+\D_i$}
\only<2>{$\partial_tU_i+\nabla\cdot F\(U_i\)=S\(U_i\){\color{black!20}+\frac1{\ell_i}\l[\Gamma\r]_{i-\onehalf\oplus}^{i+\onehalf\ominus}+\D_i}$}
\only<3>{$\partial_tU_i{\color{black!20}+\nabla\cdot F\(U_i\)}={\color{black!20}S\(U_i\)+}\frac1{\ell_i}\l[\Gamma\r]_{i-\onehalf\oplus}^{i+\onehalf\ominus}{\color{black!20}+\D_i}$}
\only<4-5>{$\partial_tU_i{\color{black!20}+\nabla\cdot F\(U_i\)}={\color{black!20}S\(U_i\)+\frac1{\ell_i}\l[\Gamma\r]_{i-\onehalf\oplus}^{i+\onehalf\ominus}+}\D_i$}
\\\hrule\\
$\mat{@{}r@{~}c@{~}l@{~}c@{~}r@{~}c@{~}l@{}}{
\only<1,6>{\partial_th}
\only<2>{\frac{{\color{red}h_i^{n\star}}-h^n}{\delta_t^n}}
\only<3>{\frac{{\color{red}h^{n+\onehalf}}-h_i^{n\star}}{\delta_t^n}}
\only<4>{\frac{h^{n+1}-h^{n+\onehalf}}{\delta_t^n}}
\only<1,6>{&+&\nabla\cdot\(hu_i\)&=&}
\only<2>{&+&\nabla\cdot\(h^nu_i^n\)&=&}
\only<3-4>{&&&=&}
\only<1,6>{\frac1{\ell_i}\l[G\r]_{i-\onehalf}^{i+\onehalf}}
\only<2>{0}
\only<3>{\frac1{\ell_i}\l[{\color{red}G^{n+\onehalf}}\r]_{i-\onehalf}^{i+\onehalf}}
\only<4>{&&0}
\\
\only<1,6>{\partial_t\(hu_i\)}
\only<2>{\frac{{\color{red}h_i^{n\star}u_i^{n\star}}-h^nu_i^n}{\delta_t^n}}
\only<3>{\frac{{\color{red}h^{n+\onehalf}u_i^{n+\onehalf}}-h_i^{n\star}u_i^{n\star}}{\delta_t^n}}
\only<4>{\frac{h^{n+1}{\color{red}u_i^{n+1}}-h^{n+\onehalf}u_i^{n+\onehalf}}{\delta_t^n}}
\only<5>{\sum_{j=1}^{L}\Bigg(\alpha_{ij}{\color{red}u_{j}^{n+1}}
+\nabla\(\mu_{ij}\cdot{\color{red}u_j^{n+1}}\)
-\nu_{ij}\nabla\cdot{\color{red}u_j^{n+1}}
-\nabla\(\kappa_{ij}\nabla\cdot{\color{red}u_j^{n+1}}\)\Bigg)}
\only<1,6>{&+&\nabla\cdot\(hu_i\otimes u_i\)+h\nabla\(P+g\eta\)&=&}
\only<2>{&+&\nabla\cdot\(h^nu_i^n\otimes u_i^n\)+h^n\nabla\(P+g\eta^n\)&=&}
\only<3-5>{&&&=&}
\only<1,6>{\frac1{\ell_i}\l[uG\r]_{i-\onehalf}^{i+\onehalf}&-&\nabla\(hq_i\)+\frac1{\ell_i}\l[q\nabla \zeta\r]_{i-\onehalf}^{i+\onehalf}}
\only<2>{0}
\only<3>{\frac1{\ell_i}\l[{\color{red}u^{n+\onehalf}G^{n+\onehalf}}\r]_{i-\onehalf}^{i+\onehalf}}
\only<4>{&&-\nabla\(h^{n+1}{\color{red}q_i^{n+1}}\)+\frac1{\ell_i}\l[{\color{red}q^{n+1}}\nabla \zeta^{n+1}\r]_{i-\onehalf}^{i+\onehalf}}
\only<5>{&&\beta_i}
\\
\only<1,6>{\partial_t\(h w_i\)}
\only<2>{\frac{{\color{red}h_i^{n\star}w_i^{n\star}}-h^nw_i^n}{\delta_t^n}}
\only<3>{\frac{{\color{red}h_i^{n+\onehalf}w_i^{n+\onehalf}}-h_i^{n\star}w_i^{n\star}}{\delta_t^n}}
\only<4>{\frac{h_i^{n+1}{\color{red}w_i^{n+1}}-h_i^{n+\onehalf}w_i^{n+\onehalf}}{\delta_t^n}}
\only<1,6>{&+&\nabla\cdot\(hw_i\ u_i\)&=&}
\only<2>{&+&\nabla\cdot\(h^nw^n_i\ u^n_i\)&=&}
\only<3-4>{&&&=&}
\only<1,6>{\frac1{\ell_i}\l[wG\r]_{i-\onehalf}^{i+\onehalf}&-&\frac1{\ell_i}\l[q\r]_{i-\onehalf}^{i+\onehalf}}
\only<2>{0}
\only<3>{\frac1{\ell_i}\l[{\color{red}w^{n+\onehalf}G^{n+\onehalf}}\r]_{i-\onehalf}^{i+\onehalf}}
\only<4>{&&-\frac1{\ell_i}\l[{\color{red}q^{n+1}}\r]_{i-\onehalf}^{i+\onehalf}}
\\
\only<1,6>{\partial_t\(h \sigma_i\)}
\only<2>{\frac{{\color{red}h_i^{n\star}\sigma_i^{n\star}}-h^n\sigma_i^n}{\delta_t^n}}
\only<3>{\frac{{\color{red}h_i^{n+\onehalf}\sigma_i^{n+\onehalf}}-h_i^{n\star}\sigma_i^{n\star}}{\delta_t^n}}
\only<4>{\frac{h_i^{n+1}{\color{red}\sigma_i^{n+1}}-h_i^{n+\onehalf}\sigma_i^{n+\onehalf}}{\delta_t^n}}
\only<1,6>{&+&\nabla\cdot\(h\sigma_i\ u_i\)&=&}
\only<2>{&+&\nabla\cdot\(h^n\sigma_i^n\ u_i^n\)&=&}
\only<3-4>{&&&=&}
\only<1,6>{\frac1{\ell_i}\l[\sigma G\r]_{i-\onehalf\oplus}^{i+\onehalf\ominus}&+&\frac{2\sqrt{3}}{\ell_i}\(q_i-\frac{q_{i-\onehalf}+q_{i+\onehalf}}2\)}
\only<2>{0}
\only<3>{\frac1{\ell_i}\l[{\color{red}\sigma^{n+\onehalf}G^{n+\onehalf}}\r]_{i-\onehalf\oplus}^{i+\onehalf\ominus}}
\only<4>{&&\frac{2\sqrt{3}}{\ell_i}\({\color{red}q_i^{n+1}}-\frac{{\color{red}q_{i-\onehalf}^{n+1}}+{\color{red}q_{i+\onehalf}^{n+1}}}2\)}
\\
\only<1>{w_i&=&\partial_t\zeta_i+u_i\cdot\nabla\zeta_i-G_i&&\sigma_i&=&-\frac{\ell_ih}{2\sqrt{3}}\nabla\cdot u_i}
\only<4-5>{{\color{red}w_{i}^{n+1}}=\partial_t^{n+1}B
+{\color{red}u_{i}^{n+1}}\cdot\nabla\zeta_{i-\sfrac12}^{n+1}
-\frac{\ell_i}{2}h^{n+1}\nabla_k\cdot{\color{red}u_i^{n+1}}
&&&;&&&
{\color{red}\sigma_{i}^{n+1}}=-\frac{\ell_ih^{n+1}}{2\sqrt{3}}\nabla\cdot{\color{red}u_{i}^{n+1}}
}
}$
\end{tabular}\vspace{10pt}
\end{overlayarea}
\begin{overlayarea}{\linewidth}{.5\linewidth}
\only<-5>{Solved using a {\bf splitting} approach:\\
\only<2->{{\color{bleu}\quad1.\ }{\bf Advection and conservative forces}\\
{\color{bleu}\quad\quad1.1\ }Horizontal advection\\}
\only<2>{\quad\quad\qquad using in each layer a {\bf classical shallow water scheme} (Godunov, HLL, kinetic...)}
\only<3->{{\color{bleu}\quad\quad1.2\ }Vertical advection\\}
\only<3>{\quad\quad\qquad implicit up-wind scheme but {\bf solved explicitly}}
\only<4->{{\color{bleu}\quad2.\ }{\bf Dispersion and dissipative forces}\\}
\only<4>{\quad\qquad Linear system but {\bf very big}\hfill 1d: ${\color{red}\(5LN\)^2}$-system with a $3$-point stencil\\}
\only<5->{{\color{bleu}\quad\quad2.1\ }Implicit elliptic scheme on $u_i$\\}
\only<5>{\quad\quad\qquad System smaller but {\bf very large stencil}\hfill 1d: $\(LN\)^2$-system with a {\color{red}$5L$}-point stencil\\}
\only<5->{{\color{bleu}\quad\quad2.1\ }Reconstruction of $w_i$ and $\sigma_i$\\}
\only<5>{\quad\quad\qquad and eventually $q_i$ and $q_{i-\onehalf}$}}
\only<6>{
\begin{block}{\scriptsize Theorem:}
Using a numerical scheme of \SW satisfying the {\bf positivity} and the {\bf entropy-dissipation} under a CFL condition
\hfill $\lambda\(h^n,\u^n\)\delta_t^n\le\delta_k$\hfill\hfill\hfill\\
then under the CFL condition\\
\hfill $\lambda\(h^n,u_i^n\)\delta_t^n\le\delta_k$\hfill\hfill\hfill\\
the proposed scheme satisfies the {\bf positivity} and the {\bf entropy-dissipation} even at {\bf dry front}.
\end{block}}
\end{overlayarea}
\end{frame}

\subsection{Numerical results}

\begin{frame}{}{}
{\color{bleu}\sc\underline{Seawall}}
\only<1>{\includegraphics[width=\linewidth,clip,trim=45 410 65 80]{Pictures/Seawall}}
\only<2>{\includegraphics[width=\linewidth,clip,trim=45 10 65 410]{Pictures/Seawall}}
\end{frame}

\begin{frame}{}{}
{\color{bleu}\sc\underline{Dispersive jump}}
\only<1>{\includegraphics[width=\linewidth,clip,trim=45 10 65 35]{Pictures/DispersifJump}}
\only<2>{\includegraphics[width=\linewidth,clip,trim=45 10 65 35]{Pictures/DispersifJump_charge}}
\end{frame}

\begin{frame}{}{}
{\color{bleu}\sc\underline{Water drop}}
\movie[externalviewer]{\beamergotobutton{video}}{Pictures/Waterdrop.avi}
\only<1>{\includegraphics[width=.95\linewidth,clip,trim=150 75 60 85]{Pictures/Goutte_2d-5}}
\only<2>{\includegraphics[width=\linewidth,clip,trim=80 140 550 70]{Pictures/Goutte_2d-5_6_1d-2}}
\end{frame}

\section*{Realizations}
\subsection*{Perspectives}

\begin{frame}{}{}
{\color{bleu}\sc\underline{Realizations:}}
\begin{tabular}{@{\quad}|@{}p{.95\linewidth}}
\begin{itemize}\vspace{-\baselineskip}
\item[\myitemb] A {\bf dispersive layerwise} hierarchy of models was proposed
\begin{thebibliography}{9}
\bibitem{} [Fernandez-Nieto, Parisot, Penel and Sainte-Marie] {\color{black} hal-01324012}
\end{thebibliography}
Extendable to: viscosity, friction, Corriolis, capillary {\color{bleu}[Marche'07]}.
\item[\myitemb] An {\bf entropy-satisfying} numerical scheme was designed
\begin{thebibliography}{9}
\bibitem{} [Parisot] {\color{black}hal-01242128}.
\end{thebibliography}
\end{itemize}\vspace{-1.7\baselineskip}
\end{tabular}

\vfill

{\color{bleu}\sc\underline{Perspectives:}}
\begin{tabular}{@{\quad}|@{}p{.95\linewidth}}
\begin{itemize}\vspace{-\baselineskip}
\item[\color{vert}\danger] Link with other models
\begin{itemize}\scriptsize
\item[\myitem]the {\bf $\sigma$-transform} {\color{bleu}[Castro]}
\item[\myitem]\GN with vorticity {\color{bleu}[Castro, Lannes'14]}
\end{itemize}
\item[\color{orange}\danger] Boundary condition
\begin{itemize}\scriptsize
\item[\myitem] {\bf transparent boundary} for layerwise model {\color{bleu}[Kazakova, Noble]}
\item[\myitem] imposed {\bf hydraulic head} (for hydraulic jump analysis)
\end{itemize}
\item[\color{orange}\danger] Breaking waves
\begin{itemize}\scriptsize
\item[\myitem] numerical investigation of the {\bf energy dissipation}
\item[\myitem] {\bf adaptative} layer discretization
\end{itemize}
\item[\color{red}\danger]Considering other transformation (for plunging wave {\color{bleu}[Murashige]})
\end{itemize}\vspace{-1.5\baselineskip}
\end{tabular}
\end{frame}

\end{document}

