[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 102)

[cott at tapir.caltech.edu]

User: cott
Date: 2011/08/10 01:53 PM

Modified:
 /
  ET.tex
 /local_bibtex/
  ott_references.bib

Log:
 * some EOS text, more to come.

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--- ET.tex	2011-08-08 01:40:52 UTC (rev 101)
+++ ET.tex	2011-08-10 18:53:50 UTC (rev 102)
@@ -1265,15 +1265,44 @@
 \subsection{Hydrodynamics Evolution}\pages{Christian and Josh}
 \label{sec:GRHydro}
 
-Hydrodynamic evolution in the Einstein Toolkit is designed so that it interacts with the metric curvature evolution through a small set of variables, allowing for maximum modularity in implementing, editing, or replacing either evolution scheme. 
+Hydrodynamic evolution in the Einstein Toolkit is designed so that it
+interacts with the metric curvature evolution through a small set of
+variables, allowing for maximum modularity in implementing, editing,
+or replacing either evolution scheme.
 
-The primary hydrodynamics evolution routine in the Einstein Toolkit is \codename{GRHydro}, a public version derived from the \codename{Whisky} code designed primarily by researchers at AEI and their collaborators \cite{Baiotti:2004wn,Hawke:2005zw,Baiotti:2010zf,Whisky:web}.  It includes a high resolution shock capturing (HRSC) scheme to evolve hydrodynamic quantities, with several different reconstruction methods and Riemann solvers, as we discuss below.  In such a scheme, we define a set of ``conserved'' hydrodynamic variables, defined in terms of the ``primitive'' physical variables such as mass and internal energy density, pressure, and velocity.  Wherever derivatives of hydrodynamic terms appear in the evolution equations for the conserved variables, they are restricted to appear only inside divergence terms, which are referred to as fluxes.  By calculating fluxes at cell faces, we may obtain a consistent description of the intercell values using reconstruction techniques that account for the fact that hydrodynamic variables are not smooth and may not be finite differences accurately.  All other source terms in the evolution equations may contain only the hydrodynamic variables themselves as well as the metric variables and derivatives of the latter, since they must formally be smooth and thus differentiable using finite differencing techniques.  Summarizing these methods briefly, the following stages occur every timestep:
+The primary hydrodynamics evolution routine in the Einstein Toolkit is
+\codename{GRHydro}, an updated version of the public \codename{Whisky}
+code \cite{Baiotti:2004wn,Hawke:2005zw,Baiotti:2010zf,Whisky:web}.  It
+includes a high resolution shock capturing (HRSC) scheme to evolve
+hydrodynamic quantities, with several different reconstruction methods
+and Riemann solvers, as we discuss below.  In such a scheme, we define
+a set of ``conserved'' hydrodynamic variables, defined in terms of the
+``primitive'' physical variables such as mass and internal energy
+density, pressure, and velocity.  Wherever derivatives of hydrodynamic
+terms appear in the evolution equations for the conserved variables,
+they are restricted to appear only inside divergence terms, which are
+referred to as fluxes.  By calculating fluxes at cell faces, we may
+obtain a consistent description of the intercell values using
+reconstruction techniques that account for the fact that hydrodynamic
+variables are not smooth and may not be finite differences accurately.
+All other source terms in the evolution equations may contain only the
+hydrodynamic variables themselves as well as the metric variables and
+derivatives of the latter, since they must formally be smooth and thus
+differentiable using finite differencing techniques.  Summarizing
+these methods briefly, the following stages occur every timestep:
 \begin{itemize}
-\item The primitive variables are ``reconstructed'' at cell faces using shock-capturing techniques, with total variation diminishing (TVD),  piecewise parabolic (PPM),  and essentially non-oscillatory (ENO) methods currently implemented.
-\item A Riemann problem is solved at each cell face using an approximate solver.  Currently implemented versions include HLL (Harten-Lax-van Leer), Roe, and Marquina solvers.
-\item The conserved variables are advanced a timestep, and used to recalculate the new values of the primitive variables.
+\item The primitive variables are ``reconstructed'' at cell faces
+  using shock-capturing techniques, with total variation diminishing
+  (TVD), piecewise parabolic (PPM), and essentially non-oscillatory
+  (ENO) methods currently implemented.
+\item A Riemann problem is solved at each cell face using an
+  approximate solver.  Currently implemented versions include HLL
+  (Harten-Lax-van Leer), Roe, and Marquina solvers.
+\item The conserved variables are advanced a timestep, and used to
+  recalculate the new values of the primitive variables.
 \end{itemize}
-We discuss the GRHD formalism,  the stages within a timestep, and the other aspects of the code below.
+We discuss the GRHD formalism, the stages within a timestep, and the
+other aspects of the code below.
 
 \subsubsection{Ideal general relativistic hydrodynamics (GRHD)}
 The equations of ideal GR hydrodynamics evolved by \codename{GRHydro} are
@@ -1356,6 +1385,70 @@
 \subsection{Equations of State}\pages{1 Christian}
 \label{sec:eoss}
 
+An equation of state connecting the primitive state variables is
+needed to close the system of GR hydrodynamics equations.  The module
+\codename{EOS\_Omni} provides a unified general equation of state
+(EOS) interface and backend for simple analytic and complex
+microphysical EOS. 
+
+The polytropic EOS,
+\begin{equation}
+P = K\rho^\Gamma\,\,,
+\end{equation}
+where $K$ is the polytropic constant and $\Gamma$ is the adiabatic
+index, is appropriate for adiabatic (= isentropic) evolution without
+shocks. When using the polytropic EOS, one does not need to evolve the
+total fluid energy equation, since the specific internal energy
+$\epsilon$ is fixed to
+\begin{equation}
+\epsilon = \frac{K\rho^\Gamma}{(\Gamma - 1)\rho}\,.
+\end{equation} 
+Note that the adiabatic index $\Gamma = d\ln{P}/
+d\ln{\rho}$ is related to the frequently used polytropic index $n$ via
+$n = 1 / (\Gamma - 1)$.
+
+The gamma-law EOS\footnote{For historic reasons, this EOS is referred to
+as the ``ideal fluid'' EOS in \codename{GRHydro}.},
+\begin{equation}
+P = (\Gamma - 1) \rho \epsilon\,\,,
+\end{equation}
+allows for non-adiabatic flow, but still assumes fixed microphysics, which
+is encapsulated in the constant adiabatic index $\Gamma$. This EOS has been
+used extensively in simulations of NSNS and NSBH mergers.
+
+The hybrid EOS, first introduced by \cite{janka:93}, is a 2-piece
+piecewise polytropic with a thermal component designed for the
+application in simple models of stellar collapse. At densities below
+nuclear, a polytropic EOS with $\Gamma = \Gamma_1 \approx 4/3$ is
+used.  To mimick the stiffening of the nuclear EOS at nuclear density,
+the low-density polytrope is fitted to a second polytrope with 
+$\Gamma = \Gamma_2 \gtrsim 2$. To allow for thermal contributions to the
+pressure due to shock heating, a gamma-law with $\Gamma = \Gamma_\mathrm{th}$
+is used. The full EOS then reads
+\begin{eqnarray}
+  P & = & \frac{\Gamma - \Gamma_{\rm th}}{\Gamma - 1}
+  K \rho_{\rm nuc}^{\Gamma_1 - \Gamma}
+  \rho^{\Gamma} - \frac{(\Gamma_{\rm th} - 1) (\Gamma - \Gamma_1)}
+  {(\Gamma_1 - 1) (\Gamma_2 - 1)}
+  K \rho_{\rm nuc}^{\Gamma_1 - 1} \rho \nonumber \\
+  & & + (\Gamma_{\rm th} - 1) \rho \epsilon\,.
+  \label{eq:hybrid_eos}
+\end{eqnarray}%
+mIn this, $\epsilon$ denotes the total specific internal energy which
+consists of a polytropic and a thermal contribution. In iron core
+collapse, the pressure below nuclear density is dominated by the
+pressure of relativistically degenerate electrons. For this, one sets
+$K = 4.897 \times 10^{14}$ [cgs] in the above. The termal index
+$\Gamma_{\rm th}$ is usually set to $1.5$, corresponding to a mixture
+of relativistic ($\Gamma=4/3$) and non-relativistic ($\Gamma=5/3$)
+gas. Provided appropriate choices of EOS parameters (e.g.,
+\cite{dimmelmeier:07}), the hybrid EOS leads to qualitatively correct
+collapse and bounce dynamics in stellar collapse.
+
+
+{\color{red} continue here}
+
+
 \subsection{Analysis\pagesdone{Tanja}}
 \label{sec:analysis}
 It is often beneficial and sometimes necessary to evaluate analysis quantities

Directory: /local_bibtex/
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File [modified]: ott_references.bib
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===================================================================
--- local_bibtex/ott_references.bib	2011-08-08 01:40:52 UTC (rev 101)
+++ local_bibtex/ott_references.bib	2011-08-10 18:53:50 UTC (rev 102)
@@ -177,3 +177,25 @@
    number = 6,
     pages = {064008},
 }
+
+ at Article{	  janka:93,
+  author	= {{Janka}, H.-T. and {Zwerger}, T. and {M\"onchmeyer}, R.},
+  title		= "{Does artificial viscosity destroy prompt type-II
+		  supernova explosions?}",
+  journal	= {Astron. Astrophys.},
+  year		= 1993,
+  month		= feb,
+  volume	= 268,
+  pages		= {360}
+}
+
+ at ARTICLE{dimmelmeier:07,
+   author = {{Dimmelmeier}, H. and {Ott}, C.~D. and {Janka}, H.-T. and {Marek}, A. and 
+	{M{\"u}ller}, E.},
+    title = "{Generic Gravitational-Wave Signals from the Collapse of Rotating Stellar Cores}",
+  journal = {Phys. Rev. Lett},
+     year = 2007,
+    month = jun,
+   volume = 98,
+    pages = {251101},
+}