[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 70)

Wed Apr 27 13:54:54 CDT 2011

User: cott
Date: 2011/04/27 01:54 PM

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

Log:
 * more text on McLachlan

File Changes:

Directory: /
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File [modified]: ET.tex
Delta lines: +27 -25
===================================================================

--- ET.tex	2011-04-27 18:44:41 UTC (rev 69)
+++ ET.tex	2011-04-27 18:54:54 UTC (rev 70)
@@ -1157,18 +1157,18 @@
 
 \subsubsection{Boundary Conditions}
 
-During time evolution, we apply a Sommerfeld-type radiative boundary
-condition to all components of the evolved BSSN variables as described
-in \cite{Alcubierre2000}. The main feature of this boundary condition
-is that it assumes approximate spherical symmetry of the solution,
-while applying the actual boundary condition on the boundary of a
-cubic grid where the face normals are not aligned with the radial
-direction. This boundary condition defines the right hand side
-of the BSSN state vector on the outer boundary, which is then
-integrated in time as well, so that the boundary and interior are
-calculated with the same order of accuracy.
+During time evolution, a Sommerfeld-type radiative boundary condition
+is applied to all components of the evolved BSSN variables as
+described in \cite{Alcubierre2000}. The main feature of this boundary
+condition is that it assumes approximate spherical symmetry of the
+solution, while applying the actual boundary condition on the boundary
+of a cubic grid where the face normals are not aligned with the radial
+direction. This boundary condition defines the right hand side of the
+BSSN state vector on the outer boundary, which is then integrated in
+time as well, so that the boundary and interior are calculated with
+the same order of accuracy.
 
-The main part of the boundary condition assumes that we have an
+The main part of the boundary condition assumes that one has an
 outgoing radial wave with some speed $v_0$:
 \begin{eqnarray}
   X & = & X_0 + \frac{u(r - v_0 t)}{r}
@@ -1182,17 +1182,18 @@
 \end{eqnarray}
 where $v^i = v_0\, x^i/r$. The spatial derivatives $\partial_i$ are
 evaluated using centered finite differencing where possible, and
-one-sided finite differencing elsewhere. We use second order stencils
-in our implementation.
+one-sided finite differencing elsewhere.  Second order stencils
+are used in the current implementation.
 
-In addition to this main part, we also account for those parts of the
-solution that do not behave as a pure wave, e.g., Coulomb type terms
-caused by infall of the coordinate lines. We assume that these parts
-decay with a certain power $p$ of the radius. We implement this by
-considering the radial derivative of the source term above, and
-extrapolating according to this power-law decay.
+In addition to this main part, it is also necessary to account for
+those parts of the solution that do not behave as a pure wave, e.g.,
+Coulomb type terms caused by infall of the coordinate lines. The
+assumption is made that these parts decay with a certain power $p$ of
+the radius. This is implemented by considering the radial derivative of
+the source term above, and extrapolating according to this power-law
+decay.
 
-Given a source term $(\partial_t X)$, we define the corrected source
+Given a source term $(\partial_t X)$, one defines the corrected source
 term $(\partial_t X)^*$ via
 \begin{eqnarray}
   (\partial_t X)^* & = & (\partial_t X) + \left( \frac{r}{r - n^i
@@ -1201,8 +1202,8 @@
 where $n^i$ is the normal vector of the corresponding boundary face.
 The spatial derivatives $\partial_i$ are evaluated by comparing
 neighbouring grid points, corresponding to a second-order stencil
-evaluated in the middle between the two neighbouring grid points. We
-assume a second-order decay, i.e., we choose $p=2$.
+evaluated in the middle between the two neighbouring grid points. 
+Second-order decays is assumed, hence $p=2$.
 
 As with the initial conditions above, this boundary condition is
 evaluated on several layers of grid points, starting from the
@@ -1212,19 +1213,20 @@
 
 This boundary condition is only a coarse approximation of the actual
 decay behavior of the BSSN state vector, and it does not capture the
-correct behavior of the evolved variables. However, we observe that
+correct behavior of the evolved variables. However, one finds that
 this boundary condition leads to stable evolutions if applied
 sufficiently far from the source. Errors introduced at the boundary
 (both errors in the geometry and constraint violations) propagate
 inwards with the speed of light \cite{ES-Brown2007b}. Gauge changes
 introduced by the boundary condition, which are physically not
-observable, propagate faster, with a speed up to $\sqrt{2}$ for our
-gauge conditions.
+observable, propagate faster, with a speed up to $\sqrt{2}$ for the
+ gauge conditions used in \codename{McLachlan}.
 
 
 
 
 
+
 \subsubsection{Hydrodynamics: \codename{GRHydro}}
 \label{sec:GRHydro}
 

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

File [modified]: ott_references.bib
Delta lines: +25 -0
===================================================================
--- local_bibtex/ott_references.bib	2011-04-27 18:44:41 UTC (rev 69)
+++ local_bibtex/ott_references.bib	2011-04-27 18:54:54 UTC (rev 70)
@@ -140,3 +140,28 @@
   doi =          {10.1088/0264-9381/27/16/167001},
 }
 
+
+ at article{Alcubierre2000,
+  author = "Alcubierre, M. and Br{\"u}gmann, B. and Dramlitsch, T. and Font, J. A. and Papadopoulos, P. and Seidel, E. and Stergioulas, N. and Takahashi, R.",
+  title = "Towards a Stable Numerical Evolution of Strongly Gravitating Systems in General Relativity: The Conformal Treatments",
+  journal = "Phys. Rev. D",
+  year = "2000",
+  volume = "62",
+  pages = "044034"}
+
+
+
+ at Article{ES-Brown2007b,
+  status =       {refereed},
+  author =       {David Brown and Peter Diener and Olivier Sarbach and
+                  Erik Schnetter and Manuel Tiglio},
+  title =        {Turduckening black holes: an analytical and
+                  computational study},
+  journal =      {Phys. Rev. D},
+  year =         2009,
+  volume =       79,
+  pages =        044023,
+  receiveddate = {2008-09-20},
+  fulltexturl =  {http://link.aps.org/abstract/PRD/v79/e044023},
+  doi =          {10.1103/PhysRevD.79.044023},
+}

