[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 96)

[cott at tapir.caltech.edu]

User: cott
Date: 2011/08/01 10:23 AM

Modified:
 /
  ET.tex

Log:
 * add hydro equations

File Changes:

Directory: /
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File [modified]: ET.tex
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--- ET.tex	2011-08-01 14:55:33 UTC (rev 95)
+++ ET.tex	2011-08-01 15:23:53 UTC (rev 96)
@@ -966,9 +966,9 @@
 one may also apply a uniform velocity to the neutron star, though this does not affect 
 the ODE solution nor the resulting density profile.
 
-\subsection{Spacetime Curvature and Hydrodynamics Evolution}
+\subsection{Spacetime Curvature Evolution}
 \label{sec:evol}
-\todo{Christian in charge}
+\todo{Josh and Christian in charge}
 
 In the following, we assume that the reader is familiar with the
 basics of numerical relativity and GR hydrodynamics. Detailed
@@ -1266,11 +1266,91 @@
  gauge conditions used in \codename{McLachlan}.
 
 
-\subsection{Hydrodynamics: \codename{GRHydro}}\pages{Christian}
+\subsection{Hydrodynamics: \codename{GRHydro}}\pages{Christian and Josh}
 \label{sec:GRHydro}
 
+{\color{red} This text is copied from Reisswig et al. It needs to be expanded a bit and we probably want to talk about the numerical methods a bit more.}
 
+The equations of ideal GR hydrodynamics evolved by \codename{GRHydro} are
+derived from the local GR conservation laws of mass and
+energy-momentum,
+\begin{equation}
+  \nabla_{\!\mu} J^\mu = 0, \qquad \nabla_{\!\mu} T^{\mu \nu} = 0\,\,,
+  \label{eq:equations_of_motion_gr}
+\end{equation}
+where $ \nabla_{\!\mu} $ denotes the covariant derivative with respect
+to the 4-metric. $ J^{\,\mu} = \rho u^{\,\mu} $ is the mass current
+with the 4-velocity $ u^{\,\mu} $ and the rest-mass density $\rho$.  $
+T^{\mu \nu} = \rho h u^{\,\mu} u^{\,\nu} + P g^{\,\mu \nu} $ is the
+stress-energy tensor.  The quantity $ h = 1 + \epsilon + P / \rho $ is
+the specific enthalpy, $P$ is the fluid pressure and $\epsilon$ is the
+specific internal energy.
 
+We choose a definition of the 3-velocity that corresponds to the
+velocity seen by an Eulerian observer at rest in the current spatial
+3-hypersurface \cite{york:83},
+\begin{equation}
+v^i = \frac{u^i}{W} + \frac{\beta^i}{\alpha}\,\,,
+\label{eq:vel}
+\end{equation}
+where $W = (1-v^i v_i)^{-1/2}$ is the Lorentz factor. In terms of
+the 3-velocity, the contravariant 4-velocity is then given by
+\begin{equation}
+u^0  = \frac{W}{\alpha}\,,\qquad
+u^i = W \left( v^i - \frac{\beta^i}{\alpha}\right)\,\,,
+\end{equation}
+and the covariant 4-velocity is
+\begin{equation}
+u_0  = W(v^i \beta_i - \alpha)\,,\qquad
+u_i = W v_i\,\,.
+\end{equation}
+
+The {\tt GRHydro} scheme is written in a first-order hyperbolic
+flux-conservative evolution system for the conserved variables
+$\hat{D}$, $\hat{S}^i$, and $\hat{\tau}$ in terms of the primitive
+variables $\rho, \epsilon, v^i$,
+\begin{eqnarray}
+  \hat{D} &=& \sqrt{\gamma} \rho W,\nonumber\\
+  \hat{S}^i &=& \sqrt{\gamma} \rho h W^{\,2} v^i,\nonumber\\
+  \hat{\tau} &=& \sqrt{\gamma} \left(\rho h W^{\,2} - P\right) - D\,,
+\end{eqnarray}
+where $ \gamma $ is the determinant of $\gamma_{ij} $.
+The evolution system then becomes
+\begin{equation}
+  \frac{\partial \mathbf{U}}{\partial t} +
+  \frac{\partial \mathbf{F}^{\,i}}{\partial x^{\,i}} =
+  \mathbf{S}\,\,,
+  \label{eq:conservation_equations_gr}
+\end{equation}
+with
+\begin{eqnarray}
+  \mathbf{U} & = & [\hat{D}, \hat{S}_j, \hat{\tau}], \nonumber\\
+  \mathbf{F}^{\,i} & = & \alpha
+  \left[ \hat{D} \tilde{v}^{\,i}, \hat{S}_j \tilde{v}^{\,i} + \delta^{\,i}_j P,
+  \hat{\tau} \tilde{v}^{\,i} + P v^{\,i} \right]\!, \nonumber \\
+  \mathbf{S} & = & \alpha
+  \bigg[ 0, T^{\mu \nu} \left( \frac{\partial g_{\nu j}}{\partial x^{\,\mu}} - 
+  \Gamma^{\,\lambda}_{\mu \nu} g_{\lambda j} \right), \nonumber\\
+  & &\qquad\alpha \left( T^{\mu 0}
+  \frac{\partial \ln \alpha}{\partial x^{\,\mu}} -
+  T^{\mu \nu} \Gamma^{\,0}_{\mu \nu} \right) \bigg]\,.
+\end{eqnarray}%
+%
+Here, $ \tilde{v}^{\,i} = v^{\,i} - \beta^i / \alpha $ and $
+\Gamma^{\,\lambda}_{\mu \nu} $ are the 4-Christoffel symbols.  The
+above equations are solved in semi-discrete fashion. The spatial
+discretization is performed by means of a high-resolution
+shock-capturing (HRSC) scheme employing a second-order accurate
+finite-volume discretization. We make use of the Marquina flux formula
+for the local Riemann problems and piecewise-parabolic cell interface
+reconstruction (PPM\@). For a review of such methods in the GR context,
+see~\cite{font:08}. The time integration and coupling with curvature
+are carried out with the Method of
+Lines~\cite{Hyman-1976-Courant-MOL-report}.
+
+
+
+
 \subsection{Equations of State}\pages{1 Christian}
 \label{sec:eoss}