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

Modified: / ET.tex

Log: * add hydro equations

File Changes:

Directory: / ============

File [modified]: ET.tex Delta lines: +83 -3 =================================================================== --- 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}