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