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\proceedings{Finite volumes for complex applications IV}{-}
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\title[Short title of the paper (< 75 mm)]%
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\author{Author1-firstname Author1-name$^\star$ and Author2-firstname Author2-name$^\dagger$}
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\abstract{Your abstract in english}
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\begin{document}
\maketitlepage
\section{Introduction}
Multifluid modeling and numerical simulation have important applications, namely in reactor safety analysis,
where effort is being made to better understand and predict two-phase flow behavior.
\section{Governing equations}
An isothermic two-phase inviscid 2D flow of mixture containing vapor
and liquid is described by the conservation equation for mass and momentum
\begin{equation}
\dfrac{\partial {\bf W}}{\partial t}=
-\dfrac{\partial {\bf F}}{\partial x}
-\dfrac{\partial {\bf G}}{\partial y}+{\bf P},
\label{eq:gov}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cell-vertex finite volume method based on the Lax-Wendroff scheme}
%==========================================================================
\label{sec:2Dscheme}
The numerical method scheme described within this section is originally based
on the method of Ni [NI~81].
\begin{figure}[ht!]
%\centering
\begin{center}
%\vspace*{-0.05\textheight}
%\includegraphics[height=.2\textheight]{fvc.eps}
\caption{Finite volumes $V$ and $V^*$}
\end{center}
\label{fig:volumes}
\end{figure}
\subsection{Some analysis}
%--------------------------------
\noindent Let us consider the scheme described above (without an artificial viscosity term), in the case of
the simple scalar one dimensional advection equation
\begin{theorem}\label{thcasW2p}
We assume
that the nonlinearity satisfies the assumptions.
Let $f\in L^{p'}$ and $g\in
W^{2p}$ and assume that $U \in W^{2,p}$.
Then there exists a constant $C$ depending
on all the problem data ($\|f\|_{L^{p'}}$,
$\|g\|_{W^{2p}}$, $\alpha$ , $\cdots$),
such that
\begin{equation}\label{errorestimate}
\left\{
\right.
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Description of test case - Ransom problem}
The numerical method was tested on a classical test case - Ransom problem.
The computational domain in 2D is a rectangle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Numerical results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\begin{figure}[ht!]
%\includegraphics[width=.3\textwidth]{objem.eps}
%\hspace*{-.08\textwidth}
%\includegraphics[width=.3\textwidth]{fig_channel/iso_alph_2.eps}
%\hspace*{-.08\textwidth}
%\hspace*{.05\textwidth}
%$t=3.25\ ms$ \hspace*{.105\textwidth} $t=3.50\ ms$ \hspace*{.105\textwidth}
%$t=3.75\ ms$ \hspace*{.105\textwidth} $t=4.00\ ms$
%\caption{Isolines of void fraction at denoted time.}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
The numerical method for the solution of two-phase flow equations
is presented.
\begin{thebibliography}{99}
\bibitem[NI~81] {NI81}
{\sc Ni R. H.},
"A~Multiple grid scheme for solving Euler equations",
\textit{ AIAA Journal}, Vol. 20, No. 1.
\bibitem[NI~81] {NI81}
{\sc Ni R. H.},
"A~Multiple grid scheme for solving Euler equations",
\textit{ AIAA Journal}, Vol. 20, No. 1.
\bibitem[NI~81] {NI81}
{\sc Ni R. H.},
"A~Multiple grid scheme for solving Euler equations",
\textit{ AIAA Journal}, Vol. 20, No. 1.
\bibitem[NI~81] {NI81}
{\sc Ni R. H.},
"A~Multiple grid scheme for solving Euler equations",
\textit{ AIAA Journal}, Vol. 20, No. 1.
\end{thebibliography}
%\section{Bibliography}
\section*{Acknowledgements}
This work has been supported by the grant No. \dots
\end{document}