[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 148)
knarf at cct.lsu.edu
knarf at cct.lsu.edu
Sun Oct 16 22:46:26 CDT 2011
User: knarf
Date: 2011/10/16 10:46 PM
Added:
/
plot_defaults.py
/examples/tov/
plot_defaults.py
Removed:
/examples/tov/
plot_defaults.py
Modified:
/
ET.tex
/examples/tov/
rho_max.pdf, rho_max.py
Log:
smothen tov section and minor issues here and there
Directory Changes:
Directory: /svn:executable/
===========================
+ *
Directory: /svn:special/
========================
+ *
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +28 -24
===================================================================
--- ET.tex 2011-10-13 05:00:20 UTC (rev 147)
+++ ET.tex 2011-10-17 03:46:25 UTC (rev 148)
@@ -2593,7 +2593,7 @@
and an initial central density of $\varrho_0=1.28\times10^{-3}$. This model can
be taken to represent a non-rotating NS with a mass of
$M=1.4\mathrm{M}_\odot$. The computational domain is a cube of length
-$640\mathrm{M}$ and a base resolution of $2\mathrm{M}$ ($4\mathrm{M}$,
+$640\mathrm{M}$ with a base resolution of $2\mathrm{M}$ ($4\mathrm{M}$,
$8\mathrm{M}$) in each dimension. Four additional grids refine the region
around the star located at the origin, each doubling the resolution, with sizes
of $120\mathrm{M}$, $60\mathrm{M}$, $30\mathrm{M}$ and $15\mathrm{M}$,
@@ -2601,12 +2601,12 @@
$0.5\mathrm{M}$) across the entire star.
In figure~\ref{fig:tov_rho_max} we show the evolution of the central density of
-the star, over an evolution time of $1300\mathrm{M}$. The initial spike is due
-to the perturbation of the solution resulting from the interpolation onto the
-evolution grid. The remaining oscillations are due to the interaction of the
-star and the artificial atmosphere and are present during the whole evolution.
-Given enough evolution time, the frequencies of these oscillations can be
-measured with satisfactory accuracy.
+the star, over an evolution time of $1300\mathrm{M}$ ($6.5\mathrm{ms}$). The
+initial spike is due to the perturbation of the solution resulting from the
+interpolation onto the evolution grid. The remaining oscillations are mainly
+due to the interaction of the star and the artificial atmosphere and are
+present during the whole evolution. Given enough evolution time, the
+frequencies of these oscillations can be measured with satisfactory accuracy.
\begin{figure}
\label{fig:tov_rho_max}
@@ -2628,41 +2628,45 @@
trend, averaging over Hanning windows overlapping half the signal length after
padding the signal to five time it's length. The agreement of the frequencies
of the fundamental mode and the first three overtones is clearly visible, but
-we are not able to show the same for the fourth overtone, at this resolution.
+we are not able to show the same for higher overtones, at this resolution.
We expect to be able to resolve higher overtones when using even higher
-resolution, but did not try because of the involved computational cost.
+resolution, but did not pursue further because of the involved computational
+cost.
\begin{figure}
\label{fig:tov_mode_spectrum}
\includegraphics[width=0.9\textwidth]{examples/tov/mode_spectrum}
- \caption{Eigenfrequency mode spectrum of a TOV star. Shown is the power spectral density
- of the central density, computed from a full 3D relativistic hydrodynamics simulation compared
- to the values obtained by perturbation theory.\todo{improve caption}}
+ \caption{Eigenfrequency mode spectrum of a TOV star. Shown is the power
+ spectral density of the central matter density, computed from a full 3D
+ relativistic hydrodynamics simulation and compared to the values obtained by
+ perturbation theory. The agreement of the frequencies of the fundamental mode
+ and the first three overtones is clearly visible.}
\end{figure}
Within this test it is also interesting to study the convergence behavior of
the coupled curvature and matter evolution code. One of the variables often
used for this test is the Hamiltonian constraint violation. This violation
vanishes for the continuum problem, but is non-zero and resolution-dependent in
-discrete simulations. The expected rate of convergence if the hydrodynamics
-code is between $1$ and $2$. It cannot be higher than $2$ because of the used
-directional flux-split algorithm which is of second order, and in some cases
-the hydrodynamics code is only of first order, in particular at extrema (like
-the center of the star), or at the star surface.
+discrete simulations. The expected rate of convergence of the hydrodynamics
+code lies between $1$ and $2$. It cannot be higher than $2$ due to the
+directional flux-split algorithm which is of second order. Depending on
+solution itself, the hydrodynamics code is only of first order in particular
+regions, e.g., at extrema (like the center of the star), or at the star
+surface.
Figure~\ref{fig:tov_ham_conv} shows the order of convergence of the Hamiltonian
constraint violation, using the two highest-resolution runs, at the stellar
-center and a coordinate radius of $r=5$ which is about half-way between the
-center and the surface. The observed convergence rate is most of the time
-between $1.4$ and $1.5$ at the center, and between $1.6$ and $2$ inside the
-star. This difference is expected because of the mentioned data-dependent
-convergence orders of the underlying hydrodynamics evolution schemes.
+center and a coordinate radius of $r=5\mathrm{M}$ which is about half-way between the
+center and the surface. The observed convergence rate for most of the
+simulation time lies between $1.4$ and $1.5$ at the center, and between $1.6$ and
+$2$ at $r=5\mathrm{M}$, consistent with the expected data-dependend convergence
+order of the underlying hydrodynamics evolution scheme.
\begin{figure}
\label{fig:tov_ham_conv}
\includegraphics[width=0.9\textwidth]{examples/tov/ham_conv}
- \caption{Convergence factor of Hamiltonian constraint violation at $r=0$ and
- $r=5$. The observed convergence order of about $1.5$ at the center of
+ \caption{Convergence factor of Hamiltonian constraint violation at $r=0\mathrm{M}$ and
+ $r=5\mathrm{M}$. The observed convergence order of about $1.5$ at the center of
the star is lower then the general second order of the hydrodynamics
evolution scheme. This is expected because the scheme's convergence rate drops to first
order at extrema or shocks, like the stellar center or the star surface.
File [added]: plot_defaults.py
Delta lines: +43 -0
===================================================================
--- plot_defaults.py (rev 0)
+++ plot_defaults.py 2011-10-17 03:46:25 UTC (rev 148)
@@ -0,0 +1,43 @@
+#!/usr/bin/python
+
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.ticker as mticker
+
+from matplotlib import rc
+
+# stuff
+fontsize = 10
+linewidth = 2
+rc('text', usetex=True)
+rc('font', family='serif')
+rc('font', serif='palatino')
+rc('font', weight='bolder')
+rc('mathtext', default='sf')
+rc("lines", markeredgewidth=1)
+rc("lines", linewidth=linewidth)
+rc('axes', labelsize=fontsize)
+rc("axes", linewidth=(linewidth+1)//2)
+rc('xtick', labelsize=fontsize)
+rc('ytick', labelsize=fontsize)
+rc('legend', fontsize=fontsize)
+rc('xtick.major', pad=8)
+rc('ytick.major', pad=8)
+
+def set_tick_sizes(ax, major, minor):
+ for l in ax.get_xticklines() + ax.get_yticklines():
+ l.set_markersize(major)
+ for tick in ax.xaxis.get_minor_ticks() + ax.yaxis.get_minor_ticks():
+ tick.tick1line.set_markersize(minor)
+ tick.tick2line.set_markersize(minor)
+ ax.xaxis.LABELPAD=10.
+ ax.xaxis.OFFSETTEXTPAD=10.
+
+# constants
+G = 6.673e-11
+c = 299792458
+M_sol = 1.98892e30
+# convertion factors
+M_to_ms = 1./(1000*M_sol*G/(c*c*c))
+# fix this - what is 7e10?
+M_to_rho_g_cm3 = M_sol*1000/7e10/7e10/7e10
Property changes on: plot_defaults.py
___________________________________________________________________
Directory: /examples/tov/
=========================
File [removed]: plot_defaults.py
Delta lines: +1 -0
===================================================================
--- examples/tov/plot_defaults.py (rev 0)
+++ examples/tov/plot_defaults.py 2011-10-17 03:46:25 UTC (rev 148)
@@ -0,0 +1 @@
+link ../../plot_defaults.py
\ No newline at end of file
Property changes on: examples/tov/plot_defaults.py
___________________________________________________________________
File [added]: plot_defaults.py
Delta lines: None
None
File [modified]: rho_max.pdf
Delta lines: +0 -0
===================================================================
(Binary files differ)
File [modified]: rho_max.py
Delta lines: +1 -1
===================================================================
--- examples/tov/rho_max.py 2011-10-13 05:00:20 UTC (rev 147)
+++ examples/tov/rho_max.py 2011-10-17 03:46:25 UTC (rev 148)
@@ -33,7 +33,7 @@
ax2.set_xlim((ax.get_xlim()[0]/M_to_ms, ax.get_xlim()[1]/M_to_ms))
ax2.xaxis.set_major_locator(mticker.MaxNLocator(7))
ax2.xaxis.set_minor_locator(mticker.MaxNLocator(14))
-ax.set_ylabel(r'$\varrho_c/\rho_c(0)$')
+ax.set_ylabel(r'$\varrho_c/\varrho_c(0)$')
ax.yaxis.set_major_locator(mticker.MaxNLocator(5))
ax.yaxis.set_minor_locator(mticker.MaxNLocator(10))
ax.yaxis.grid(False)
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