[Commits] [svn:einsteintoolkit] Workshop_Spring_2012/numerical_relativity/ (Rev. 41)
diener at cct.lsu.edu
diener at cct.lsu.edu
Wed Apr 4 09:46:17 CDT 2012
User: diener
Date: 2012/04/04 09:46 AM
Modified:
/numerical_relativity/
numerical_relativity.tex
Log:
Add command /mypause to easily turn on or off all pauses.
Change the date.
Remove the Dr's.
File Changes:
Directory: /numerical_relativity/
=================================
File [modified]: numerical_relativity.tex
Delta lines: +51 -49
===================================================================
--- numerical_relativity/numerical_relativity.tex 2012-04-04 14:42:26 UTC (rev 40)
+++ numerical_relativity/numerical_relativity.tex 2012-04-04 14:46:16 UTC (rev 41)
@@ -3,12 +3,14 @@
\usecolortheme[RGB={200,200,200}]{structure}
\subtitle[Introduction to numerical methods \& numerical Relativity]{{\large Einstein Toolkit Workshop}\\*[0.3em] Numerical methods for spacetime and matter evolution}
-\author[\mbox{Peter Diener \& Bruno Mundim}]{Dr Peter Diener \& Dr Bruno Mundim}
-\date{Apr 05 2012}
+\author[\mbox{Peter Diener \& Bruno Mundim}]{Peter Diener \& Bruno Mundim}
+\date{Apr 4 2012}
\begin{document}
\newcommand{\code}[1]{\texttt{#1}}
+%\newcommand{\mypause}{\pause}
+\newcommand{\mypause}{}
\frame{\titlepage}
@@ -34,11 +36,11 @@
\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}\pause
-These equations were used a lot in the early years of numerical relativity.\pause
+\end{align}\mypause
+These equations were used a lot in the early years of numerical relativity.\mypause
However, it was later discovered that the ADM evolution equations are only
-weakly hyperbolic.\pause
+weakly hyperbolic.\mypause
In the mid to late 90's a new formulation was introduced that proved to be
much more robust and stable: The BSSN formulation.
@@ -49,18 +51,18 @@
Introduce a conformal rescaling of the three metric\vspace{\vs}
\begin{equation}
\gamma_{ij} = \psi^4 \tg_{ij}.\vspace{\vs}
-\end{equation}\pause
+\end{equation}\mypause
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}\pause
+\end{equation}\mypause
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}\pause
+\end{align}\mypause
We then promote the following variables to evolution variables\vspace{\vs}
\begin{align}
\phi &= \ln \psi = \frac{1}{12} \ln \gamma,
@@ -78,17 +80,17 @@
\begin{equation}
\tG^i = \tg^{jk} \tG^i{}_{jk} = - \partial_j \tg^{ij},
\end{equation}
-to evolved variables as well.\pause
+to evolved variables as well.\mypause
The final set of evolution variables are $\phi$, $K$, $\tg_{ij}$, $\tA_{ij}$
-and $\tG^i$.\pause
+and $\tG^i$.\mypause
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}\pause
+\end{equation}\mypause
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
@@ -144,18 +146,18 @@
\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}\pause
+\end{align}\mypause
The constraints $\tilde{{\cal G}}$ and $\tilde{{\cal A}}$ are enforced actively
-at each timestep.\pause
+at each timestep.\mypause
The other constraints ($\tilde{{\cal H}}$, $\tilde{{\cal M}}^i$ and
-$\tilde{{\cal L}}^{i}$) are not enforced.\pause
+$\tilde{{\cal L}}^{i}$) are not enforced.\mypause
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.\pause
+ directly.\mypause
\item Where $\tG^{i}$ are needed without taking derivatives
$\tg^{jk}\tG^{i}{}_{jk}$ are used instead.
\end{itemize}
@@ -163,7 +165,7 @@
\frame{\frametitle{The BSSN formulation (continued)}
In order to evolve a spacetime with the BSSN equations, you have to specify
-the gauges $\alpha$ and $\beta^{i}$.\pause
+the gauges $\alpha$ and $\beta^{i}$.\mypause
Most codes use the ``moving puncture'' gauges
\begin{align}
@@ -181,12 +183,12 @@
\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.\pause \\
+to be tracefree.\mypause \\
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}\pause
+\end{align}\mypause
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 -
@@ -203,29 +205,29 @@
black holes.
\begin{equation}
\psi = 1+\sum_{i=1}^N\frac{m_i}{2|\vec{r}-\vec{r}_i|}.
-\end{equation}\pause
+\end{equation}\mypause
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}\pause
+\end{equation}\mypause
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}\pause
+\end{equation}\mypause
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
+collection of grid points.\mypause
-The grid points are usually (but not necessarily) equally spaced. \pause
+The grid points are usually (but not necessarily) equally spaced. \mypause
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
+stencil) of the grid point. \mypause
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
@@ -245,7 +247,7 @@
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
+\end{eqnarray} \mypause
Requiring that the weighted sum approximates the derivative yields
the following equations for $a_{-1}$, $a_0$ and $a_1$
\begin{eqnarray}
@@ -261,20 +263,20 @@
\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
+\end{equation}\mypause
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
+\end{equation}\mypause
+These finite difference operators are second order accurate.\mypause
-Higher order accuracy or higher order derivatives require larger stencils.\pause
+Higher order accuracy or higher order derivatives require larger stencils.\mypause
Another way of looking at finite differencing operators is through
-interpolating polynomials.\pause
+interpolating polynomials.\mypause
-Either approach gives the same coefficients for the same stencil.\pause
+Either approach gives the same coefficients for the same stencil.\mypause
It is also clear from either approach that the error estimates are only
correct if $f$ is smooth enough.
@@ -285,20 +287,20 @@
\begin{equation}
\partial_t \mathbf{q}+\mathbf{A}^i(\mathbf{q})\partial_i
\mathbf{B}(\mathbf{q}) = \mathbf{S}(\mathbf{q}).
-\end{equation}\pause
+\end{equation}\mypause
The idea then is to discretize in space first, i.e.\ write the equations as
\begin{equation}
\partial_t \mathbf{q} = \mathbf{L}(\mathbf{q}),
\end{equation}
where $\mathbf{L}(\mathbf{q})$ is a discrete approximation to the equations
-(e.g.\ using finite differencing).\pause
+(e.g.\ using finite differencing).\mypause
This then turns the equations into a set of coupled ODE's with respect to
-time.\pause
+time.\mypause
If the spatial discretization (including boundary conditions) is stable
we we can then evolve the system of equations using any stable ODE time
-integrator.\pause
+integrator.\mypause
Often Runge-Kutta integration schemes are the scheme of choice.
}
@@ -307,37 +309,37 @@
Advantages of using the method of lines:
\begin{itemize}
\item It is easy to change the time integration scheme (e.g.\ going to higher
- order).\pause
+ order).\mypause
\item It is easy to couple different evolution equations maintaining
- high order coupling.\pause
+ high order coupling.\mypause
\end{itemize}
The method of lines is implemented in Cactus in the thorn
-\code{CactusNumerical/MoL}.\pause
+\code{CactusNumerical/MoL}.\mypause
To use:
\begin{itemize}
\item Schedule a routine to register the evolution variables and RHS
- variables with \code{MoL}.\pause
- \item Schedule routines to set the RHS variables.\pause
+ variables with \code{MoL}.\mypause
+ \item Schedule routines to set the RHS variables.\mypause
\item Set parameters when launching job to choose the time integration
scheme.
\end{itemize}
}
\frame{\frametitle{McLachlan}
-\code{McLachlan} is the Einstein Toolkit implementation of BSSN.\pause
+\code{McLachlan} is the Einstein Toolkit implementation of BSSN.\mypause
\code{McLachlan} is named in honor of the Canadian Singer/Song writer Sarah
-McLachlan.\pause
+McLachlan.\mypause
-It is based on finite differencing and the method of lines.\pause
+It is based on finite differencing and the method of lines.\mypause
-It supports high order finite differencing (8th order).\pause
+It supports high order finite differencing (8th order).\mypause
Since it is generated from the tensor equations by Kranc it is easy to
-maintain and modify.\pause
+maintain and modify.\mypause
It is highly optimized (supports vectorization and OpenMP) and an effort is
-ongoing to make it be able to run on GPU's as well.\pause
+ongoing to make it be able to run on GPU's as well.\mypause
In the future it may be necessary to generate different versions optimized
for specific computer architectures.
@@ -348,7 +350,7 @@
Once we have established the spacetime evolution and initial data
equations, we need to obtain the evolution equations for the
matter fields and the magnetic field evolution equation in
-ideal MHD case. \pause
+ideal MHD case. \mypause
These equations can be expressed as the local conservation laws of
baryon number and energy momentum. For baryon number we have:
@@ -357,7 +359,7 @@
\nabla_\nu J^\nu=0,
\end{equation}
where $J^\mu=\rho u^\mu$ is the rest-mass current, $\rho$ the rest-mass
-density and $u^\mu$ is the four-velocity of a fluid comoving observer.\pause
+density and $u^\mu$ is the four-velocity of a fluid comoving observer.\mypause
The conservation of energy-momentum is given by:
%
@@ -385,7 +387,7 @@
\end{equation}
where we define $p^*=p+b^2/2$, $h^*=h+b^2/\rho$,
$\varepsilon^*=\varepsilon+b^2/(2\rho)$, that results into
-$h^*=1+\varepsilon^*+p^*/\rho$. \pause
+$h^*=1+\varepsilon^*+p^*/\rho$. \mypause
Note that in the expression above we have used the magnetic field
$b^{\mu}$ as measured by the comoving observer. It can expressed
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