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\def \L {2.}


\title{Towards Moving Picture Simulations in a Discontinuous Galerkin Framework}
\author{Yingjie Wang}
\institute{FAU}

\addbibresource{ref.bib}

\begin{document}
	\maketitle

  \begin{frame}{Contents}
		\tableofcontents
	\end{frame}

  \begin{frame}{Outline of the Project}
    \section{Introduction}
    \begin{enumerate}
      \item A first order evolution system, to be named FOZ4c
      \item An implementation in our discontinuous Galerkin code \texttt{nmesh}
      \item Simulations of binary black holes with large mass ratio with the moving puncture gauge
    \end{enumerate}
  \end{frame}

  \begin{frame}{Final Goal: binary black holes with large mass ratio}
    \subsection{binary black holes with large mass ratio}

    Why binary black holes?

    \begin{itemize}
      \item Main sources of gravitational waves
      \item Simplest two-body problem in general relativity
      \item Connecting strong field GR with post-Newtonian approximations
    \end{itemize}

    Why large mass ratio?
    \begin{itemize}
      \item LISA is able to detect extreme mass ratio inspirals (EMRIs)
      \item Compare with perturbation theory
    \end{itemize}
  \end{frame}

  \begin{frame}{Methods on evolving black holes}
    \subsection{Methods on evolving black holes}
    There are two main methods to evolve black holes in numerical relativity:
    \begin{itemize}
      \item Excision method: excise the black hole interior from the computational domain, and impose boundary conditions on the excision surface
      \item Moving puncture method: evolve the black hole as a puncture, and use a suitable gauge condition to avoid the singularity.
    \end{itemize}
  \end{frame}

  \begin{frame}{Methods on evolving black holes: excision}
    \begin{figure}
      \centering
      \includegraphics[width=0.7\textwidth]{imgs/black_hole_excision_mesh.png}
      \caption{An example mesh using excision method from \cite{Hemberger_2013}.}
    \end{figure}
  \end{frame}

  \begin{frame}{Methods on evolving black holes}
    \begin{tikzpicture}[>=Latex, line cap=round, line join=round]
      % causal diamond
        \draw[thick,red,zigzag] (-\L,\L) coordinate(stl) -- (\L,\L) coordinate (str);
        \draw[thick,black] (\L,-\L) coordinate (sbr)
          -- (0,0) coordinate (bif) -- (stl);
        \draw[thick,black,fill=blue, fill opacity=0.2,text opacity=1]
          (bif) -- (str) -- (2*\L,0) node[right] (io) {$i^0$} -- (sbr);

        % null labels
        \draw[black] (1.4*\L,0.7*\L) node[right]  (scrip) {$\mathcal{I}^+$}
                    (1.5*\L,-0.6*\L) node[right] (scrip) {$\mathcal{I}^-$}
                    (0.2*\L,-0.6*\L) node[right] (scrip) {$\mathcal{H}^-$}
                    (0.5*\L,0.85*\L) node[right] (scrip) {$\mathcal{H}^+$};

        % singularity label
        \draw[thick,red,<-] (0,1.05*\L) 
          -- (0,1.2*\L) node[above] {\color{red} singularity};
        % % Scwharzschild surface
        % \draw[thick,blue] (bif) .. controls (1.*\L,-0.35*\L) .. (2*\L,0);
        % \draw[thick,blue,<-] (1.75*\L,-0.1*\L)  -- (1.9*\L,-0.5*\L)
        %   -- (2*\L,-0.5*\L) node[right,align=left]
        %   {$t=$ constant\\in Schwarzschild\\coordinates};
        % excision surface
        \draw[thick,dashed,red] (-0.3*\L,0.3*\L) -- (0.4*\L,\L);
        \draw[thick,red,<-] (-0.33*\L,0.3*\L) 
          -- (-0.5*\L,0.26*\L) node[left,align=right] {excision\\surface};
        % Kerr-Schild surface
        \draw[green,thick] (0.325*\L,0.325*\L) .. controls (\L,0) .. (2*\L,0);
        \draw[green,dashed,thick] (0.325*\L,0.325*\L) -- (-0.051*\L,0.5*\L);
        % Kerr-Schild label
        \draw[green,thick,<-] (0.95*\L,0.15*\L) -- (1.2*\L,0.5*\L)
          -- (2*\L,0.5*\L) node[right,align=left]
          {time slice};
      \end{tikzpicture}
  \end{frame}

  \begin{frame}{Methods on evolving black holes: moving puncture}
    \begin{figure}
      \centering
      \includegraphics[width=0.42\textwidth]{imgs/moving_puncture_init.png}
      \includegraphics[width=0.45\textwidth]{imgs/moving_puncture_stable.png}
      \caption{The embedding of the initial slice (left) and the final slice (right) in the moving puncture method, from \cite{Hannam_2008}.}
    \end{figure}
  \end{frame}

  \begin{frame}{Methods on evolving black holes: moving puncture}
    \begin{figure}
      \centering
      \includegraphics[width=0.7\textwidth]{imgs/moving_puncture_penrose.png}
      \caption{The numerical time slices in the moving puncture method in the Penrose diagram, from \cite{Hannam_2008}.}
    \end{figure}
  \end{frame}

  \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
    \subsection{Discontinuous Galerkin method and \texttt{nmesh}}
  \end{frame}

  \begin{frame}{References}
    \printbibliography
  \end{frame}

\end{document}