[Commits] [svn:einsteintoolkit] Workshop_Spring_2012/numerical_relativity/ (Rev. 29)
diener at cct.lsu.edu
diener at cct.lsu.edu
Tue Apr 3 16:52:38 CDT 2012
User: diener
Date: 2012/04/03 04:52 PM
Modified:
/numerical_relativity/
numerical_relativity.tex
Log:
Add some slides on finite differencing.
File Changes:
Directory: /numerical_relativity/
=================================
File [modified]: numerical_relativity.tex
Delta lines: +89 -27
===================================================================
--- numerical_relativity/numerical_relativity.tex 2012-04-03 18:30:44 UTC (rev 28)
+++ numerical_relativity/numerical_relativity.tex 2012-04-03 21:52:37 UTC (rev 29)
@@ -15,7 +15,7 @@
% \item 3+1 spacetime decomposition.
\item The BSSN formulation.
\item Puncture data.
-% \item Finite Differencing.
+ \item Finite Differencing.
% \item The method of lines.
\end{itemize}
}
@@ -31,33 +31,33 @@
\begin{align}
{\cal H} & \equiv R + K^2 - K_{ij} K^{ij} = 0, \\
{\cal M}^i & \equiv D_j (K^{ij} - \gamma^{ij} K) = 0.
-\end{align}
-These equations were used a lot in the early years of numerical relativity.
+\end{align}\pause
+These equations were used a lot in the early years of numerical relativity.\pause
However, it was later discovered that the ADM evolution equations are only
-weakly hyperbolic.
+weakly hyperbolic.\pause
In the mid to late 90's a new formulation was introduced that proved to be
much more robust and stable: The BSSN formulation.
}
\frame{\frametitle{The BSSN formulation (continued)}
-\setlength{\vs}{-1.5ex}
+\setlength{\vs}{-0.5ex}
Introduce a conformal rescaling of the three metric\vspace{\vs}
\begin{equation}
\gamma_{ij} = \psi^4 \tg_{ij}.\vspace{\vs}
-\end{equation}
-We choose $\psi = \gamma^{1/12}$ such that the determinant of $\tg_{ij}$ is 1\vspace{\vs}
+\end{equation}\pause
+We choose $\psi = \gamma^{1/12}$ such that the determinant of $\tg_{ij}$ is 1%\vspace{\vs}
\begin{equation}
\mathrm{det}(\tg_{ij}) = \mathrm{det}(\psi^{-4}\gamma_{ij}) =
\mathrm{det}(\gamma^{-1/3}\gamma_{ij}) =
-\gamma^{-1} \mathrm{det}(\gamma_{ij}) = 1. \vspace{\vs}
-\end{equation}
+\gamma^{-1} \mathrm{det}(\gamma_{ij}) = 1.\vspace{\vs}
+\end{equation}\pause
In addition we introduce a trace decomposition of the extrinsic curvature.\vspace{\vs}
\begin{align}
K & = \gamma^{ij} K_{ij}, \\
A_{ij} & = K_{ij} - \frac{1}{3} \gamma_{ij} K.\vspace{\vs}
-\end{align}
+\end{align}\pause
We then promote the following variables to evolution variables\vspace{\vs}
\begin{align}
\phi &= \ln \psi = \frac{1}{12} \ln \gamma,
@@ -75,16 +75,17 @@
\begin{equation}
\tG^i = \tg^{jk} \tG^i{}_{jk} = - \partial_j \tg^{ij},
\end{equation}
-to evolved variables as well.
+to evolved variables as well.\pause
+
The final set of evolution variables are $\phi$, $K$, $\tg_{ij}$, $\tA_{ij}$
-and $\tG^i$.
+and $\tG^i$.\pause
The BSSN evolution equations can be derived from the ADM equations. As an
example take the equation for $\phi$.
\begin{equation}
\dt \phi = \partial_0\phi = \partial_0 \left (\frac{1}{12}\ln\gamma\right)
= \frac{1}{12}\frac{1}{\gamma}\partial_0\gamma.
-\end{equation}
+\end{equation}\pause
Using the expression for the derivative of the determinant of the metric
in terms of the derivatives of the metric ($\partial_0\gamma = \gamma
\gamma^{ij} \partial_0\gamma_{ij}$) we find
@@ -114,13 +115,11 @@
& - 2 \tilde{A}^{ij} \partial_j\alpha
+ 2 \alpha ( \tilde{\Gamma}^i{}_{jk} \tilde{A}^{jk} + 6 \tilde{A}^{ij}
\partial_j \phi - \frac{2}{3} \tg^{ij} \partial_j K).
-\end{align}%\vspace{\vs}
+\end{align}
}
\frame{\frametitle{The BSSN formulation (continued)}
-%\raisebox{-1.2\vs}[0pt][0pt]{
Here $R_{ij} = \tilde{R}_{ij} + R^{\phi}_{ij}$, where\vspace{1.0\vs}
-%}\vspace{\vs}
\begin{align}
R^{\phi}_{ij} = {} & - 2 \tilde{D}_i
\tilde{D}_j \phi - 2 \tilde{\gamma}_{ij}
@@ -142,18 +141,18 @@
\tilde{{\cal G}} & \equiv \tg-1 = 0, \\
\tilde{{\cal A}} & \equiv \tg^{ij}\tA_{ij} = 0, \\
\tilde{{\cal L}}^{i} & \equiv \tG^{i}+\partial_j\tg^{ij} = 0.
-\end{align}
+\end{align}\pause
The constraints $\tilde{{\cal G}}$ and $\tilde{{\cal A}}$ are enforced actively
-at each timestep.
+at each timestep.\pause
The other constraints ($\tilde{{\cal H}}$, $\tilde{{\cal M}}^i$ and
-$\tilde{{\cal L}}^{i}$) are not enforced.
+$\tilde{{\cal L}}^{i}$) are not enforced.\pause
To improve stability and to help maintain $\tilde{{\cal L}}^{i}$ at a low level
the following rule is employed in an implementation
\begin{itemize}
\item Where derivatives of $\tG^{i}$ are needed the evolved $\tG^{i}$ are used
- directly.
+ directly.\pause
\item Where $\tG^{i}$ are needed without taking derivatives
$\tg^{jk}\tG^{i}{}_{jk}$ are used instead.
\end{itemize}
@@ -161,7 +160,7 @@
\frame{\frametitle{The BSSN formulation (continued)}
In order to evolve a spacetime with the BSSN equations, you have to specify
-the gauges $\alpha$ and $\beta^{i}$.
+the gauges $\alpha$ and $\beta^{i}$.\pause
Most codes use the ``moving puncture'' gauges
\begin{align}
@@ -179,12 +178,12 @@
\gamma^{\mathrm{ph}}_{ij} = \psi^4\gamma_{ab}, \mbox{\hspace{2em}} K^{\mathrm{ph}}_{ij} = \psi^{-2} K_{ij},
\end{equation}
where $\gamma_{ij}$ is chosen to be the flat metric and $K_{ij}$ is assumed
-to be tracefree. \\
+to be tracefree.\pause \\
The constraint equations become\vspace{\vs}
\begin{align}
0 & = \Delta\psi +\frac{1}{8}K^{ij}K_{ij}\psi^{-7} \\
0 & = D_j K^{ij}.
-\end{align}
+\end{align}\pause
The momentum constraint has an analytic solution\vspace{\vs}
\begin{align}
K^{ij}_{\mathrm{BY}} = {} & \frac{3}{2r^2}\left ( P^i n^j + P^j n^i -
@@ -197,21 +196,84 @@
\frame{\frametitle{Puncture data (continued)}
\setlength{\vs}{-0.8ex}
-For $K_{ij} = 0$ the Hamiltonian constraint has a simple solution for $N$ black holes.
+For $K_{ij} = 0$ the Hamiltonian constraint has a simple solution for $N$
+black holes.
\begin{equation}
\psi = 1+\sum_{i=1}^N\frac{m_i}{2|\vec{r}-\vec{r}_i|}.
-\end{equation}
+\end{equation}\pause
Inspired by this, for $K_{ij}\ne 0$ we make the ansatz
\begin{equation}
\psi=\frac{1}{\alpha}+u, \mbox{\hspace{2em}}
\frac{1}{\alpha} = \sum_{i=1}^N\frac{m_i}{2|\vec{r}-\vec{r}_i|},
-\end{equation}
+\end{equation}\pause
In which case the Hamiltonian constraint becomes an equation for $u$
\begin{equation}
\Delta u+\frac{1}{8}\alpha^7 K^{ij}K_{ij}(1+\alpha u)^{-7} = 0.
-\end{equation}
+\end{equation}\pause
It can be shown that $u$ will be $C^2$ at the location of the `punctures' and
that $\psi$ has a unique solution.
}
+\frame{\frametitle{Finite Differencing}
+With finite differencing we discretize a function by sampling it at a
+collection of grid points.\pause
+
+The grid points are usually (but not necessarily) equally spaced. \pause
+
+We can then approximate derivatives of a function at a grid point by a
+weigthed sum of function values at grid points in the neighbourhood (the
+stencil) of the grid point. \pause
+
+As an example consider a stencil containing the grid point ($f_i$) and it's two
+nearest neighbors ($f_{i-1}$ and $f_{i+1}$) with
+$\Delta x=x_{i+1}-x_i=x_i-x_{i-1}$.
+\begin{equation}
+ \left. \frac{df}{dx} \right |_{x_i}\approx \frac{1}{\Delta x}\sum_{j=-1}^{j=1} a_j f_{i+j}.
+\end{equation}
+}
+\frame{\frametitle{Finite Differencing (continued)}
+The coefficients $a_j$ can be found be expanding $f$ in a Taylor series around
+$x_i$ for the grid points in the stencil
+\begin{eqnarray}
+f(x_{i-1}) & = & f(x_i) - \left. \frac{df}{dx}\right |_{x_i} \Delta x
+ + \frac{1}{2} \left. \frac{d^2 f}{dx^2}\right |_{x_i} \Delta x^2
+ + O((\Delta x)^3) \nonumber \\
+f(x_i) & = & f(x_i) \nonumber \\
+f(x_{i+1}) & = & f(x_i) + \left. \frac{df}{dx}\right |_{x_i} \Delta x
+ + \frac{1}{2} \left. \frac{d^2 f}{dx^2}\right |_{x_i} \Delta x^2
+ + O((\Delta x)^3) \nonumber
+\end{eqnarray} \pause
+Requiring that the weighted sum approximates the derivative yields
+the following equations for $a_{-1}$, $a_0$ and $a_1$
+\begin{eqnarray}
+0 & = & a_{-1}+a_0+a_1 \nonumber \\
+1 & = & -a_{-1}+a_1 \nonumber \\
+0 & = & a_{-1}+a_1 \nonumber
+\end{eqnarray}
+with the solution $a_{-1}=-1/2, a_0=0, a_1=1/2$.
+}
+
+\frame{\frametitle{Finite Differencing (continued)}
+Thus we find that
+\begin{equation}
+\left. \frac{df}{dx} \right |_{x_i} = \frac{f_{i+1}-f_{i-1}}{2\Delta x}
+ + O((\Delta x)^2).
+\end{equation}\pause
+Similarly we find for the second derivative that
+\begin{equation}
+\left. \frac{d^2f}{dx^2} \right |_{x_i} = \frac{f_{i+1}-2 f_i+f_{i-1}}{(\Delta x)^2}
+ + O((\Delta x)^2).
+\end{equation}\pause
+These finite difference operators are second order accurate.\pause
+
+Higher order accuracy or higher order derivatives require larger stencils.\pause
+
+Another way of looking at finite differencing operators is through
+interpolating polynomials.\pause
+
+Either approach gives the same coefficients for the same stencil.\pause
+
+It is also clear from either approach that the error estimates are only
+correct if $f$ is smooth enough.
+}
\end{document}
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