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\tikzset{zigzag/.style={decorate, decoration=zigzag}}
\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}
    In the isotropic coordinate for Schwarzschild black hole, the spical metric
    \begin{equation*}
      \dd{l^2} = \left( 1+\frac{M}{2r} \right)^4 (\dd{r^2} + r^2 \dd{\Omega^2})
    \end{equation*}
    is conformally flat, with the conformal factor $\psi = 1 + \frac{M}{2r}$.

    General $N$ black hole puncture initial data:
    \begin{equation*}
      \psi = \sum_{i=1}^N \frac{M_i}{2|\vb*{r}-\vb*{r}_i|} + (\text{regular part}),
    \end{equation*}
    again, if we factor out the conformal factor $\psi$, the remaining conformal metric is regular everywhere.
  \end{frame}

  \begin{frame}{Methods on evolving black holes: moving puncture}

    In the moving puncture mothod, we won't cut out the black hole singularity, but choose a gauge condition that makes the singularity invisible to the numerical evolution.
    \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}}
    There are two main numerical methods in numerical relativity:
    \begin{itemize}
      \item Finite Difference / Finite Volume
      \begin{itemize}
        \item Easy to implement, and mature codes available
      \end{itemize}
      \item Spectral / Discontinuous Galerkin
      \begin{itemize}
        \item High accuracy for smooth solutions
      \end{itemize}
    \end{itemize}
  \end{frame}

  \begin{frame}{Discontinuous Galerkin method and \texttt{nmesh}}
    Core idea of DG method: if we have a first-order PDE system:
    \begin{equation*}
      \pdvt{u} + A^i \pdv{u}{x^i} = S,
    \end{equation*}
    then for some test functions $\{v_a\}$, we have
    \begin{equation*}
      \left(v_a, \pdvt{u}\right) + \left(v_a, A^i \pdv{u}{x^i}\right) = (v_a, S),
    \end{equation*}
    which reduce to a linear system of $\{ u_a = (v_a, u)\}$ after intergrating by parts. Boundary terms are replaced by numerical fluxes.

    We use Lagrange polynomials over Gauss-Legendre points on each element.
    \begin{center}
      \begin{tikzpicture}[scale=4]
        \draw[thick, blue] (0,0) -- (1,0);
        \draw[thick, red] (1,0) -- (2,0);

        \pgfmathsetmacro{\a}{0.5 - 0.5*sqrt(3/7)}
        \pgfmathsetmacro{\b}{0.5 + 0.5*sqrt(3/7)}
        \pgfmathsetmacro{\c}{1.5 - 0.5*sqrt(3/7)}
        \pgfmathsetmacro{\d}{1.5 + 0.5*sqrt(3/7)}

        \foreach \x in {0,\a,0.5,\b,1}
          \fill[blue] (\x,0) circle (0.018);

        \foreach \x in {1,\c,1.5,\d,2}
          \fill[red] (\x,0) circle (0.018);

        \fill[blue] (1,0) circle (0.022);
        \fill[red]  (1,0) circle (0.013);
      \end{tikzpicture}
    \end{center}
    Information exchange only happens at the boundaries of elements through numerical fluxes, which makes it easy to parallelize.
  \end{frame}

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

\end{document}