testing binary black hole codes in strong field regimes

Transcription

The Pennsylvania State University
The Graduate School
Department of Physics
TESTING BINARY BLACK HOLE CODES
IN STRONG FIELD REGIMES
A Thesis in
Physics
by
David Garrison
c 2002 David Garrison
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2002
We approve the thesis of David Garrison.
Date of Signature
Jorge Pullin
Professor of Physics
Thesis Adviser
Co-Chair of Committee
Pablo Laguna
Professor of Astronomy and Astrophysics & Physics
Thesis Co-Adviser
Co-Chair of Committee
Abhay Astekar
Eberly Professor of Physics
Steinn Sigurdsson
Assistant Professor of Astronomy and Astrophysics
Jayanth R. Banavar
Professor of Physics
Head of the Department of Physics
iii
Abstract
In order to further our understanding of the instabilities which develop in numerical relativity codes, I study vacuum solutions of the cosmological type (T 3 topology).
Specifically, I focus on the 3+1 ADM formulation of Einsteins equations. This involves
testing the numerical code using the following non-trivial periodic solutions, Kasner,
Gowdy, Bondi and non-linear gauge waves. I look for constraint violating and gauge
mode instabilities as well as numerical effects such as convergence, dissipation and dispersion. I will discuss techniques developed to investigate the stability properties of the
numerical code.
iv
Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Acknowledgments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Chapter 2. Computing Environment . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.1
Cactus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.2
Maya
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Chapter 3. ADM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
3.1
Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.1.1
Gauge: Lapse Function and Shift Vector . . . . . . . . . . . .
10
3.1.2
The 3-metric evolution equation
. . . . . . . . . . . . . . . .
11
3.1.3
The extrinsic curvature evolution equation . . . . . . . . . . .
13
3.1.4
The Hamiltonian Constraint . . . . . . . . . . . . . . . . . . .
15
3.1.5
The Momentum Constraint . . . . . . . . . . . . . . . . . . .
15
v
3.2
Gauge Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.3
Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . .
17
3.4
Finite Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Chapter 4. Issues of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.1.1
L2 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.1.2
Form of Equations . . . . . . . . . . . . . . . . . . . . . . . .
21
4.1.3
Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.1.4
Defining an unstable evolution . . . . . . . . . . . . . . . . .
23
4.1.5
Consistency, Convergence, Dispersion and Dissipation . . . .
25
4.1.6
Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . .
27
4.2
Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.3
Unanswered Questions . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.4
The Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Chapter 5. Nearly Trivial Solutions: Perturbed Flat-space & Kasner . . . . . . .
33
5.1
5.2
Perturbed Flat-space . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.1.1
Error growth for different resolutions . . . . . . . . . . . . . .
33
5.1.2
Error growth for different amplitudes . . . . . . . . . . . . . .
35
5.1.3
Testing the stability of Iterated Crank-Nicholson . . . . . . .
36
Kasner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.2.2
40
vi
5.2.3
Errors in the Kasner spacetime . . . . . . . . . . . . . . . . .
42
5.2.4
Type I Stability Tests . . . . . . . . . . . . . . . . . . . . . .
43
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Chapter 6. Cosmological Spacetimes: Gowdy & Bondi . . . . . . . . . . . . . . .
49
5.3
6.1
Bondi Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
6.1.1
Non-linear Periodic Plane Waves . . . . . . . . . . . . . . . .
49
6.1.2
Introduction to Bondi Waves . . . . . . . . . . . . . . . . . .
50
6.1.3
Convergence Tests for Bondi
. . . . . . . . . . . . . . . . . .
50
6.1.4
Type I Stability Analysis . . . . . . . . . . . . . . . . . . . .
54
Gowdy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
6.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
6.2.2
Convergence Tests for Gowdy . . . . . . . . . . . . . . . . . .
59
6.2.3
60
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Chapter 7. The “Gauge” Wave Solution & Stability . . . . . . . . . . . . . . . .
65
6.2
6.3
7.1
”Gauge” Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
7.1.1
Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
7.1.2
67
7.1.3
69
7.1.4
Why Gauge Wave is unstable for large A
. . . . . . . . . . .
71
7.2
Type II Stability in all the spacetimes . . . . . . . . . . . . . . . . .
73
7.3
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
vii
Chapter 8. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
8.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
8.2
The effect of constraint violations on Stability . . . . . . . . . . . . .
82
8.3
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Appendix A. Additional plots for the Bondi System . . . . . . . . . . . . . . . .
85
Appendix B. Additional plots for the Gowdy System . . . . . . . . . . . . . . .
89
Appendix C. Additional plots for the ”Gauge” Wave System . . . . . . . . . . .
91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
viii
List of Tables
5.1
Error growth rate of perturbed flat-space for different amplitudes . . . .
37
5.2
Convergence test specifications for Kasner . . . . . . . . . . . . . . . . .
40
6.1
Convergence test specifications for Bondi1 . . . . . . . . . . . . . . . . .
51
6.2
Convergence test specifications for Bondi2 . . . . . . . . . . . . . . . . .
52
6.3
Convergence test specifications for Gowdy . . . . . . . . . . . . . . . . .
59
7.1
Convergence test specifications for Gauge Wave . . . . . . . . . . . . . .
68
ix
List of Figures
3.1
The above drawing shows how four dimensional spacetime is sliced into 3D
space-like hypersurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
10
Noise 1: Above shows the value of the L2 norm of the gxx metric component vs
time. The growth rate of the metric seems larger as the resolution is increased.
For this set of runs a grid size of L = 31 was used with resolutions of ∆x =
1.0, 0.5 and 0.25 respectively, therefore it runs less than one crossing time. Also
notice the deviation between the L2 norm of the metric and that of flat space
becomes smaller as the resolution is increased. . . . . . . . . . . . . . . . . .
5.2
34
Noise 2: The L2 norm of the constraints vs time. These runs all have a grid
size of L = 31 and the same specifications as the runs in figure 5.1. As a result
the range of this plot is less than one crossing time. Note that the constraint
values seem to noticeably fluctuate with time for the fine resolution. . . . . . .
5.3
35
Noise 3: The L2 norm of the constraint values vs time. This run only involves
the coarse resolution with a grid size of L = 10 and ∆x = 0.5, It runs for 20
crossing times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
36
Kasner 1: These are the results of the convergence tests for the Kasner spacetime. Notice that the results of the coarse/medium tests are the same for both
Test 1 and 2 as are the results for the medium/fine runs. . . . . . . . . . . .
5.5
41
Kasner 2: This is a close-up how the Kasner gyy metric component grows with
time. Notice that the numerical solution lags the analytic solution. . . . . . .
44
x
5.6
Kasner 3: Plot of the Kasner metric (with a non-trivial lapse) changing with
time. Note the metric becomes singular where the lapse varies farthest from unity.
5.7
45
Kasner 4: Plot of the normalized Kasner numerical and constraint errors versus
time using a non-trivial lapse. The normalized numerical errors are defined as
the numeric solution minus analytic solution divided by the sum of the absog
−ag
lute values of the numerical and analytic solutions (ex. negyy = |g yy|+|agyy | ).
yy
yy
The normalized constraint values are defined by dividing the constraint values by the sum the the absolute values of their components (ex. nham =
2
ij
R+K −Kij K
). Notice that the constraints begin to explode before the
|R|+|K 2 |+|Kij K ij |
numerical errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8
48
Kasner 5: The above shows how the errors in both gxx and gyy are effected by
the spike in the constraint values when random noise was added to the lapse,
metric and/or extrinsic curvature. Again, normalized variables are used here so
that all errors appear on the same relative scale. Here the errors all start out
at roughly the same order of magnitude but soon the numerical errors become
several orders of magnitude larger than the constraints. . . . . . . . . . . . .
6.1
48
Bondi Wave 1: These are the results of the convergence tests for the g yy component of the Bondi1 spacetime. The large number of hyperconvergence peaks
may be the results of truncation errors which are of the same order of magnitude
as round-off errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
xi
6.2
Bondi Wave 2: These are the results of the convergence tests for the g yy component of the Bondi2 spacetime. Notice the number of hyperconvergence peaks
has been significantly reduced. Although test 4 suggests significant convergence
this may not be a meaningful result. . . . . . . . . . . . . . . . . . . . . . .
6.3
54
Bondi Wave 3: Plot of the normalized errors in the Bondi metric v.s. time using
geodesic slicing. Notice that the constraint values decrease while the numerical
errors oscillate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
55
Bondi Wave 4: Plot of the normalized error in the Bondi metric v.s. time using
a non-trivial lapse. Notice that the constraint values begin to explode before the
numerical errors implying the instability was caused by a constraint violating
mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
57
Gowdy 1: These are the results of the convergence tests for the g yy component
of the Gowdy spacetime. Results of the convergence tests for gxx are shown in
Fig B.1.
6.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Gowdy 2: Normalized and unnormalized numerical and constraint errors in
Gowdy with amplitude A = 0.1. Here the constraint values decrease by an
order of magnitude as the numerical error increase by two orders of magnitude.
6.7
63
Gowdy 3: Normalized and unnormalized numerical and constraint errors in
Gowdy with amplitude A = 1.0. Notice that the constraint values decrease
exponentially as the numerical error increases exponentially. . . . . . . . . . .
7.1
64
Gauge Wave 1: These are the results of the conversion tests for the g xx component of the gauge wave spacetime. . . . . . . . . . . . . . . . . . . . . . .
69
xii
7.2
Gauge Wave 2: Error in the gauge wave metric versus time shown on a log-log
scale. Notice here that a straight line indicates a power law growth rate. . . .
7.3
Gauge Wave 3: Error in the gauge wave metric versus time with an amplitude
A = 1.0 using a Courant factor of 0.25. . . . . . . . . . . . . . . . . . . . .
7.4
70
71
Above shows the metric error versus time of the gauge wave system for two
extremely high resolutions (62 and 122 grid points) and small amplitudes A =
0.01. Notice the loss of convergence around t = 400. . . . . . . . . . . . . . .
7.5
74
Consistency 1: The consistency of the Kasner system. Notice the error in the
fine solution appears to explode first, this would not happen in a type II stable
system according to the Lax theorem. . . . . . . . . . . . . . . . . . . . . .
7.6
75
Consistency 2: The consistency of the Bondi system. Again the fine error
explodes first. Also the error only appears to oscillate for the coarse solution
proving that the dispersion is a function of ∆x. . . . . . . . . . . . . . . . .
7.7
Consistency 3: The consistency of the Gowdy system. Notice the fine solution
explodes first proving that the system is type II unstable. . . . . . . . . . . .
7.8
77
Consistency 4: The consistency of the Gauge wave system. Here the medium
solution explodes first followed by the fine and coarse solutions. . . . . . . . .
7.9
76
77
PSD 1: Above is the power spectral density of the analytic solution of the gauge
wave spacetime with A = 1. This graph is simply used to show the frequency
spectrum of the system without the addition of unstable modes. . . . . . . . .
78
7.10 PSD 2: The power spectral density of the coarse solution of the gauge wave
spacetime with A = 1. Notice the amplitudes of the various modes are very
similar to those in the analytic solution. . . . . . . . . . . . . . . . . . . . .
78
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7.11 PSD 3: The power spectral density of the medium solution of the gauge wave
spacetime with A = 1. Here the high frequency modes are slightly larger than
for the coarse solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
7.12 PSD 4: The power spectral density of the fine solution for the gauge wave
spacetime with A = 1. Notice the high frequency modes have a much higher
relative amplitude than in the lower resolution solutions. This may be the cause
of the instabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
A.1 Bondi Wave 1: Bondi metric at t = 1. These next two plots demonstrate the
dispersion seen in the Bondi system.
. . . . . . . . . . . . . . . . . . . . .
85
A.2 Bondi Wave 2: Bondi metric at t = 9. Notice the growing phase shift in the
solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
A.3 Bondi Wave 2: Bondi metric at t = 17. As time increases the phase error
continues to grow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
A.4 Bondi Wave 3: L2 norm of the Bondi1 and Bondi2 systems at all times. Note:
the coarse solution grows faster than the higher resolution solutions. Also the
Bondi2 system appears to grow about 100 times faster than the Bondi 1 system.
87
A.5 Bondi Wave 4: L2 norm of the constraints of the Bondi1 and Bondi2 systems
respectively at all times. Notice that for the Bondi2 system the constraint values
appear to grow late in the evolution. . . . . . . . . . . . . . . . . . . . . . .
88
B.1 Gowdy 1: These are the results of the convergence tests for the gxx component
of the Gowdy space-time. They are similar to the convergence results for the
gyy component seen in chapter 6.
. . . . . . . . . . . . . . . . . . . . . . .
89
xiv
B.2 Gowdy 2: Shows the value of egyy at different times immediately before during
and after the hyperconvergence peak at t = 4.14. Notice how the error oscillates
through a value near zero. . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
B.3 Gowdy 3: Above is the L2 norm of the Hamiltonian constraint vs time. Note
that the oscillations in the constraint value increases with time especially for
the low resolution solution. Note this run lasts for 2 crossing times. . . . . . .
90
C.1 Gauge Wave 1: Gauge wave metric at t = 13, right before the large hyperconvergence peak in the convergence results. Notice how the numerical solutions
lag the analytic solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
C.2 Gauge Wave 2: Gauge wave metric at t = 14, during the large hyperconvergence
peak in the convergence results. Notice how the numerical solutions now slightly
lead the analytic solution. . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
C.3 Gauge Wave 3: Gauge wave metric at t = 15, right after the hyperconvergence
peak. The numerical solution now clearly leads the analytic solution. This
switch between leading to lagging the analytic solution explains the origin of
the hyperconvergence peak seen in chapter 7. . . . . . . . . . . . . . . . . .
92
C.4 Gauge Wave 4: L2 norm of the gauge wave metric at all times. Notice how the
growth rate oscillates as the metric grows unlike the Bondi solution. . . . . . .
93
C.5 Gauge Wave 5: Error in the gauge wave metric versus time with an amplitude
A = 1.0 using a Courant factor of 0.125. This is the result mentioned in section
7.1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
xv
Acknowledgments
I would like to thank all the people who believed in and supported me over the past
several, very difficult years. To my adviser Jorge Pullin, without his constant support
and advise this work would not have been possible, thanks for helping me maintain
my confidence. To my family; Mom, Dad, Sandra and Michael, I know that you are
always with me. I would also like to thank Rispba who’s love and support has given
me a reason to keep going these past five years. Thanks to Pablo for helping me get
through this final year, Gabriela for working with me on the Gravity Gradient project,
Mijan and Steve for taking the time to tutor me in numerical relativity, and thanks to
the numerical relativity group at LSU: Manuel, Gioel and Olivier for working with me
on this project and helping to bring it to a successful conclusion. To my friends in the
center and elsewhere in the physics department, thanks for all the stimulating (and not
so stimulating) conversations. Thanks to my thesis committee for all the help along the
way. Finally, I would like to thank my financial supporters: Joan Centrella for NASA’s
GSRP fellowship, all the people who participated in Penn State’s Academic Computing
Fellowship Program, The Sloan Foundation and the Compact for Faculty Diversity.
xvi
Preface
Within the next few years, several highly sensitive gravitational wave interferometers will come online. These new observatories, primarily LIGO (Laser Interferometer
Gravitational-Wave Observatory) and LISA (Laser Interferometer Space Antenna) will
allow us to see the universe in an entirely new way. We will no longer be limited to
observing the universe through electromagnetic radiation, for the first time in history
we will be able to create a new form of astronomy based on the gravity fields of the
universe. This will allow us to witness events such as the binary inspiral of black holes
and neutron stars and answer questions about the origin and fate of the universe which
could never be answered by traditonal astronomy.
However, this new vision of our universe does not come easily. It will require
close cooperation between experimentalists and theorists to understand the observations.
Unlike traditional astronomy, in order to draw useful information it will be necessary to
first predict what the gravitational wave signals will look like. This can only be done
using computer simulations of the events that cause gravitational waves. One of these
events, the inspiralling collision of two black holes, is of primary importance to both
LIGO and LISA.
While LIGO is designed to look for waveforms in the frequency regime of solar
mass black holes, LISA is designed to study supermassive black holes. Simulating solar
mass or supermassive black holes is essentially the same problem. Because of this,
xvii
solving the binary black hole problem is useful for both the ground-based and spacebased observatories. Unfortunately there are still many problems to overcome. Thus
far, all attempts to develop a successful numerical code to simulate a binary black hole
inspiral have failed because of growing instabilities.
Much of my work during my time at Penn State has involved gravitational wave
detectors in some way. I’ve had the opportunity to work with different numerical relativity codes as well as some of the data analysis and experimental aspects of LIGO. My
first project involved a study of causal differencing, a method of handling the advective
terms in a numerical code while respecting causality. This project lead to a publication
with Mijan Huq and Luis Lehner. Another project with Olaf Dreyer, Lee Finn, Ramon
Lopez-Aleman, Badri Krishnan and Bernard J. Kelly dealt with using gravitational wave
detectors to test the no-hair theory in general relativity. I also did a study with Gabriela
Gonzalez on gravitational gradients and their effect on ground-based detectors such as
LIGO. Gravitational gradients are fluctuations in the earth’s gravitational field caused
by seismic activity, these fluctuations are therefore a noise source for any device designed
to observe gravitational waves.
This thesis deals with my final project here at Penn State. The purpose of this
work is to further our understanding of the instabilities which develop in the numerical
codes which are designed to study binary black hole evolutions. Specifically, I focus
on the systems which evolve the coupled non-linear partial differential equations that
describe the curvature of spacetime. These numerical equations are so complex that
they can not be fully understood analytically, so I remove the black holes and test the
stability of the equations themselves. This involves testing the numerical code (like a
xviii
black box) using different non-trivial spacetimes with periodic boundary conditions. I
study constraint violating and gauge mode instabilities as well as numerical effects such
as convergence, dissipation and dispersion.
1
Chapter 1
Introduction
Gravitational waves are ripples in the fabric of spacetime caused by significant
astronomical events such as the collision of black holes or neutron stars. Until now most
of these events have gone undetected because they emit strong gravitational radiation
but no electromagnetic radiation. However, in the next few years this will all change.
Soon several gravitational wave observatories will be brought online. They will then
allow us to see into the dark corners of the universe where no conventional telescope can
reach.
In addition to the expanding our knowledge of the heavens, these new observatories will provide us with a way of testing our current theories about gravity. However,
this can only happen if we can somehow compare theory with experiment. In order to
draw useful information from the data received by gravitational wave detectors we need
theoretical models which, because of the complexity Einstein’s equations, can only be
found using numerical simulations of astrophysical events. In the past all attempts at
simulating these events, such as the inspiralling collision of two black holes, have failed
because of growing instabilities in the code. Therefore, the purpose of this thesis is to
develop a method of testing the stability of a 3D numerical code at its most basic level
without involving the use of complex boundary conditions or excision techniques.
2
In chapter 2, I will describe the computational environment in which I completed
the work found in this thesis. I describe the computational code, which is a combination
of two different codes, Cactus and Maya. I also describe the computers used in this work.
Although this code can run on several different hardware platforms, here we use an SGI
Origin 2000.
Chapter 3 will describe the form of Einstein’s equations developed by Arnowitt,
Deser and Misner (ADM) which is used in this project. After deriving the ADM evolution
and constraint equations, I discuss how gauge conditions are chosen. I also introduce
periodic boundary conditions, the Iterated Crank-Nicholson (ICN) time integrator and
the concept of finite differencing.
Chapter 4 will summarize other work done to understand the stability of the ADM
equations. I begin by defining the various concepts associated with the form of partial
differential equations (PDE) such as well-posedness, stability and consistency. Next, I
explain how convergence tests are performed in later chapters. Constraint violating and
gauge modes are then introduced. I conclude by reviewing questions that are still left
unanswered by previous research and describe how the tests presented in this thesis can
help.
Chapters 5-7 describe the results of tests performed on the numerical code. Chapter 5 describes results for the nearly trivial spacetimes. I look at the error growth rate
associated with the perturbed flat-space spacetime and how this rate is effected by changing various parameters. The Kasner spacetime is then introduced. Convergence tests
are performed and the error growth rate is studied. Next, the stability of the system is
tested by applying different gauge choices and adding perturbations to the initial data.
3
Chapter 6 deals with cosmological spacetimes. I begin by introducing the Bondi
spacetime. After performing convergence tests, I look for numerical effects such as dispersion and dissipation. Again, I analyze the stability of the spacetime by applying a
non-trivial gauge. After an introduction to the Gowdy spacetime, I perform convergence
tests and analyze the stability of the system. Here I use different Gowdy wave amplitudes
in order to study its stability.
In chapter 7, I first introduce the ”gauge” wave spacetime. I then perform convergence tests and study the stability of the spacetime using different wave amplitudes.
Next, I further test the stability of all the spacetimes presented in this thesis. Chapter
8 will summarize all the previous chapters and provide a unifying conclusion.
4
Chapter 2
Computing Environment
All of the simulations run during the course of this work use a modified version
of the original Maya code, a code developed by the numerical relativity group at Penn
State University. Maya itself is a thorn of Cactus, a set of parallelized tools developed for
numerical relativity simulations. The resulting code was then run on an Silicon Graphics
(SGI) Origin 2000.
2.1
Software
The code used to perform the numerical simulations presented in this thesis is
based on two pieces: Cactus version 4.09 beta and an ADM version of Maya. Cactus
takes care of the parameter handling, parallelism, memory management and input and
output routines while Maya controls initial data, gauge conditions, Dirichlet boundary
conditions, error calculations, and numerical evolution.
2.1.1
Cactus
The Cactus Code [19] is a computational tool developed at the Albert Einstein
Institute (also known as the Max Planck institute for Gravitational Physics) located
in Golm Germany. It has been developed for solving Einstein’s Equations numerically
although it may be used to solve any finite differenced partial differential equations
(PDE) problem. Cactus is a modular parallel collaborative tool which means that it is
5
designed to be modified to perform a variety of different numerical simulations. Cactus
handles the overall structure of the numerical code as well as the output of data. It is
supported by several architectures including SGI, Cray T3E, Linux and many more. The
source code is a combination of several languages such as Perl, Fortran and C/C++.
Cactus is designed as a master code which provides the basic infrastructure and
several thorns which allow a user to add to or reconfigure the cactus system. A configuration list of thorns can be changed before compiling the code by editing a single file
(therefore alterning the program). Each thorn is simply a piece of source code which has
a compatible interface with the cactus system. Because most of Cactus’ development
involves changes made to the thorns, it is possible for software to be developed by many
different groups without introducing conflicts. Using thorns also allows for the use of
parameter files from which the user can select a list of active thorns and paramater values
at run time without recompiling.
The Message Passing Interface (MPI) programming model allows the code to run
on several processors in a computer or network at once. The Cactus code handles the
distribution of data for several processors and the communication between processors,
leaving the developer free to concentrate on numerics. By developing a common structure
for all thorns and using universal variables, parallelism can be accomplished with a
significantly reduced effort.
In order to parallelize the code, Cactus breaks the computational numerical grid
into several peices, each peice is controlled by a single processor. In order to allow the
processors to communicate, “ghost zones” are established between the processor regions.
6
Each ghost zone is a region which a processor can read which belongs to its neighbor.
These act as boundaries for the processor’s local grid.
2.1.2
Maya
Maya is a Cactus thorn developed by the Numerical Relativity Group at The
Pennsylvania State University [66]. Some major contributors to the development of Maya
are Pablo Laguna, Jorge Pullin, Erik Schnetter, Hisaaki Shinkai, Deirdre Shoemaker,
Kenneth Smith and David Fisk. The main goal of the Maya project is to simulate the
inspiralling collision of black holes. The version of Maya used here is based on a standard
form of the ADM. This is not the version of Maya currently used by the group at Penn
State, as the latest version of the Maya is based on the evolution system developed by
Baumgarte and Shapiro as well as Shibata and Nakamura (BSSN).
Maya works by first creating initial data from an analytically known solution or
a numeric solution to a differential equation. The initial data is then evolved using an
evolution scheme such as ADM. After each evolution step the resulting data is analyzed
by comparing the evolved data to the analytic solution and calculating the Hamiltonian and momentum constraints. Notice that boundary conditions are not mentioned
here because the simulations presented here use periodic boundary conditions that are
provided by Cactus instead of Maya.
In the version of Maya used in this research, the metric g ij and the extrinsic
curvature Kij must be specified initially. For this work, these quantities are known
analytically for all space and time so specifying initial conditions consists of looping over
the computational grid at the initial time. The gauge conditions needed to evolve this
7
data is also known analytically. The data is then evolved forward in time by pluging
the metric, extrinsic curvature and gauge values into the evolution equations. After the
evolution step a new metric and extrisic curvature are produced. The process is then
repeated therefore generating values of the metric and extrinsic curvature at later times.
The analysis section of the code calculates the errors in the evolution variables
as well as the values of the constraints. The difference between the numerical solution
calculated during the evolution and the analytic solution is used as the numeric error.
Calculating the constraints involves computing a function of the metric and extrinsic
curvature. Because the errors and constraint values are not used in the evolution equations this type of evolution is called a ”free” or unconstrained evolution. Therefore the
quantities used in the analysis do not influence the ongoing simulation.
In theory this code should be able to start with any metric and extrinsic curvature
which satisfy the constraint equations and evolve the system forever using a reasonable
set of gauge conditions. Becaue this does not happen, we monitor the numerical errors
and constraint violations during the evolution.
2.2
Hardware
The Silicon Graphics Origin 2000 used in this work has 1280 MB of RAM and
runs an Irix 64 bit operating system. It has 6 250 Mhz processors in a shared memory
configuration. In order to compile and run Cactus, the Origin 2000 requires Perl 5.0,
make, a C/C++ compiler, a F90/F77 compiler, Concurrent Versioning System (CVS) to
update software during development and Message Passing Interface for inter-processor
communications. Text editors such as vi are used to modify the code, xgraph and
8
gnuplot are used to visualize the output and MapleV was used to calculate the initial
data. Additional analysis software was used on other computers.
9
Chapter 3
ADM
In order to evolve a solution to Einstein’s equations, the spacetime must first be
decomposed into a form that can be handled by the computer. The Arnowitt-DeserMisner [12] or ADM system serves this purpose by splitting a 4 dimensional spacetime
into a 3+1 form that can be evolved forward in time. The spacetime is then composed of
a 3-metric which defines the spatial curvature at a single time and an extrinsic curvature
which defines how the 3-metric changes in time. By doing this the spacetime can be
evolved by two sets of equations which are first-order in time.
3.1
Derivation
Using the notation found in Wald [78] and a method similar to that found in
Ulrich Sperhake’s Ph.D. thesis [71], we derive the ADM equations. We begin with the
4 dimensional spacetime manifold M which has the 4-metric (4) gµν . The spacetime
(M,(4) gµν ) is then assumed to have a signature (- + + +) which represents 1 timelike
and 3 spacelike dimensions. The spacetime can then be sliced into a sequence of Cauchy
surfaces each representing a different time. The manifold M is therefore decomposed
τ where
P
The collection of all
P
into R ×
P
τ is a set of spacelike hypersurfaces that are parameterized by τ .
τ form the foliation of the spacetime manifold.
10
Σdt
β
α
αn
n
t(x ) = dt
t
Σ0
α
t(x ) = 0
Fig. 3.1. The above drawing shows how four dimensional spacetime is sliced into 3D space-like
hypersurfaces.
3.1.1
Gauge: Lapse Function and Shift Vector
The “distance” or “lapse” between time-consecutive hypersurfaces is defined by
a lapse function α. We define a lapse function in terms of the four metric and τ .
α−2 ≡ −(4) g µν ∇µ τ ∇ν τ
(3.1)
The normal to the hypersurfaces is then defined as,
nµ = −α∇µ τ
(3.2)
Because coordinates may shift between hypersurfaces, a timelike vector field can then
be defined as,
ti = αni + β i .
(3.3)
11
β i is a spacelike vector which is referred to as the shift vector. It is always tangent to the
hypersurface. ti are then tangent vectors to the worldlines of coordinate observers on
P
i
τ and α dt is the proper time between hypersurfaces. α and β are considered gauge
variables which represent the four degrees of coordinate freedom in general relativity. α
determines how the spacetime is sliced and β i describes the shift of spatial coordinates
from one hypersurface to another.
3.1.2
The 3-metric evolution equation
The decomposition is carried out using projection tensors that are defined in terms
of timelike normals to the hypersurfaces. ⊥ µ
≡ δνµ + nµ nν is the projection tensor that
ν
P
projects v µ ∈ M onto τ . Nνµ ≡ −nµ nν projects vectors in M along nµ orthogonal
to
P
(4) g
τ . The 3-metric gij then describes the geometry of
µν onto
P
P
τ and is found by projecting
τ.
⊥ν (4) gµν = (4) gij
gij = ⊥µ
i j
(3.4)
The projection of the covariant form of Einstein’s equations leads to a 12 partial differential equations which describe the time evolution of the 3-metric g ij .
P
The covariant derivative on τ is then defined as Di so that Dk gij = 0. The
covariant derivative of a tensor can be obtained by projecting all free components of the
4D covariant derivative onto
i i ...i
P
τ . Example,
ν
ν
η
...µN
Dk Tj 1j 2...jN = ⊥iµ1 . . . ⊥iµN ⊥j 1 . . . ⊥j N ⊥k ∇η Tνµ1νµ2...ν
N
1
1 2 N
1 2
N
N
1
(3.5)
12
The extrinsic curvature, Kij , describes the curvature of
P
τ relative to the manifold M.
It is often thought of as the rate of change of g ij . It is defined as
Kij ≡ −⊥∇(i nj) .
(3.6)
By looking at the gradients of the time-like normals, we can rewrite the extrinsic curvature in terms of the 3-metric.
∇i nj = δiµ δjν ∇µ nν . . . = (⊥µ
− nµ ni )(⊥νj − nν nj )∇µ nν . . . = ⊥∇i nj − ni ⊥nk ∇k nj
i
(3.7)
Here nk ∇k nj is the acceleration of the normal aj . By using the definition of the normal,
nν = (4) gνη nη , the right hand side of the equation reduces to the lie derivative the
3-metric with respect to the time-like normal vector so that,
1
Kij = − Ln gij .
2
(3.8)
Because the time-like vector field is made up of both a lapse, normal to the hypersurface,
and a shift tangent to it, the lie derivative of the time-like vector field can be written as,
Lt gij = Lαn gij + Lβ gij .
(3.9)
Using the definition of the extrinsic curvature the evolution equation for the metric
becomes:
∂t gij = −2αKij + Lβ gij .
(3.10)
13
3.1.3
The extrinsic curvature evolution equation
Starting with the Einstein tensor, we can derive the constraint equations and
evolution equations for the extrinsic curvature.
Gµν ≡
(4)
Rµν −
1 (4)
(4)
gµν R
2
(3.11)
The evolution equation for the extrinsic curvature is given by ⊥G ij while the Hamiltonian
constraint is given by Gij ni nj and the momentum constraints are given by ⊥G ij ni .
To derive these equations we need to use the Gauss-Codazzi-Ricci equations. Gauss’
equation is given by projecting the full Riemann tensor onto the hypersurface,
⊥(4) Rijkl = Rijkl + Kik Kjl − Kil Kjk
(3.12)
Codazzi’s equation is given by projecting one component of the Riemann tensor normal
to the hypersurface and the other three onto the hypersurface,
⊥(4) Rijkl nl = Dj Kik − Di Kjk .
(3.13)
Ricci’s equation is given by projecting two components of the Riemann tensor along the
normal and the other two onto the hypersurface,
⊥(4) Rikjl nk nl = Ln Kij + Kik Kjk + ai aj + Di aj .
(3.14)
14
The evolution equations for the extrinsic curvature are similar to the evolution equations
for the metric, they are both based on the lie derivative of the time-like vector field.
Decomposing this vector field into components normal and tangential to the hypersurface
yield:
Lt Kij = αLn Kij + Lβ Kij .
(3.15)
Using Ricci’s equation and the identity a i = Di lnα,
Lt Kij = α(⊥(4) Rikjl nk nl − Kik Kjk ) − Di Dj α + Lβ Kij .
(3.16)
The relationship,
(4)
(4)
(4) µν η π ξ ζ (4)
Rηπξζ
Rηπ + nξ nζ ⊥ησ ⊥π
g ⊥σ ⊥ρ ⊥µ ⊥ν Rηξπζ = ⊥ησ ⊥π
ρ
ρ
(3.17)
along with Gauss’ equations we can simplify the Riemann tensor term so that
⊥ν (4) Rµν .
⊥(4) Rikjl nk nl = Rij + KKij − Kki Kjk − ⊥µ
i j
(3.18)
µ
We now replace the expression ⊥i ⊥νj (4) Rµν with 8π(Sij − 12 gij (S − ρ)), where Sij =
µ
⊥i ⊥νj Tµν is the stress-energy tensor projected onto the hypersurface. In a vacuum this
expression simplifies to zero making the evolution equations for the extrinsic curvature,
∂t Kij = α(Rij + KKij − 2Kki Kjk ) − Di Dj α + Lβ Kij .
(3.19)
15
3.1.4
The Hamiltonian Constraint
The Hamiltonian constraint is obtained by projecting G ij along the normals.
1
1
Gij ni nj = ((4) Rij ni nj + (4) R)
2
2
(3.20)
1
(4) µν σρ η π ξ ζ (4)
g g ⊥σ ⊥ρ ⊥µ ⊥ν Rηπξζ = (4) Rηπ nη nπ + (4) R
2
(3.21)
Using the expression,
we see that the Hamiltonian constraint can be simplified to,
Gij ni nj =
3.1.5
1
(R + K 2 − Kij K ij ).
2
(3.22)
The Momentum Constraint
The momentum constraint is derived by projecting one component of G ij onto
the normal and the other onto the hypersurface.
1
⊥ik nj Gij = ⊥ik nj ((4) Rij − (4) gij (4) R)
2
(3.23)
Because ⊥ik ni = 0 this equation becomes,
⊥ik nj Gij = ⊥ik nj (4) Rij .
(3.24)
16
By using Codazzi’s equation we get the final form of the expression for the momentum
constraint,
j
⊥ik nj Gij = Dj Kk − Dk K.
(3.25)
Again because we are dealing only with the vacuum form of Einstein’s equations the
Hamilton and momentum constraints should be zero. Our evolution system is made up
of sixteen equations. Six of the evolution equations evolve the metric while the other six
evolve the extrinsic curvature.There is one equation for the Hamiltonian constraint and
three for the momentum constraint. The constraint equations are just used as a way
of checking to see if the metric and extrinsic curvature still obey Einstein’s equations.
Because the constraint equations are not used in the evolution, this is called a free or
unconstrained evolution.
3.2
Gauge Choices
The full 4-metric can itself be written as,
ds2 = −α2 dt2 + gij (dxi + β i dt)(dxj + β j dt).
(3.26)
By substituting the Lβ terms for their explicit forms, the evolution and constraint equations become,
∂t gij ≡ −2αKij + 2D(i βj)
∂t Kij ≡ α(Rij + KKij − 2Kki Kjk ) − Di Dj α + β k Dk Kij − 2Kk(i D k βj)
H ≡ R + K 2 − Kij K ij = 0
(3.27)
(3.28)
(3.29)
17
j
Mk ≡ Dj Kk − Dk K = 0.
(3.30)
Here the lapse, α, and the shift, β i , are considered gauge quantities which can (in theory)
be anything after the initial data is chosen. It is not yet known exactly why some choices
of gauge result in stable evolutions while others become unstable at the numerical level.
Our choice of gauge determines how a 4D spacetime is sliced into hypersurfaces. For
the experiments that I will be performing in this thesis, I will set the shift equal to zero
but will use different functions of the lapse. By setting the shift equal to zero, we can
avoid errors associated with the advective terms in our computations. It was previously
found that the stability of an evolution can often depend on how the advective terms
are treated in the evolution [56]. Using different functions of lapse can cause a long-term
stable system to become unstable and therefore provide information about unstable
modes.
3.3
Periodic Boundary Conditions
One of the advantages of using the Cactus code is that it makes using periodic
boundary conditions in a parallelized code possible. Each of the systems that will be
used in this thesis occupy a T 3 or 3 dimensional torus topology. This simply means that
an imaginary observer moving through one of these spacetimes in a “straight” line will
always eventually pass through the same point in space. We can create this topology
in our simulations by “tying” the opposite edges of the computational grid together.
Effectively, we are doing our calculations on a torus which has no edges as opposed to
a plane which does. With periodic boundary conditions a computational grid with N
18
points has the same value for point N + 1 and point 1. Doing this allows us to avoid
errors at the boundaries which could propagate into the interior of the computational
domain and effect the stability of the simulation. By using periodic boundary conditions
we know that our boundaries will not be that cause of any instabilities. Also other effects
of artificial boundary conditions such as the ”anchoring” of solutions to a point on the
grid, do not happen with periodic boundary conditions.
3.4
Finite Differencing
In order to solve the differential equations we use finite differencing. This method
allows us to approximate the derivatives in the differential equations as finite numerical
operations. Assuming a grid spacing of ∆x, the spatial derivatives become,
∂x u(x, t) =
u(x + ∆x, t) − u(x − ∆x, t)
+ O(∆x2 )
2∆x
(3.31)
and
∂x2 u(x, t) =
u(x + ∆x, t) − 2u(x, t) + u(x − ∆x, t)
∆x2
+ O(∆x2 )
(3.32)
for first and second derivatives respectively. Because all of our evolution equations are
first order derivatives in time, we can evolve them forward as a Cauchy problem where,
2
∂t u(x, t) = A(u, ∂x u, ∂x u, t).
(3.33)
Where A is a function of the previous solution and its derivatives. The right hand side of
the equation can be solved for by using the spatial numerical derivatives and previously
19
known values. By knowing both the value of u(x,t) and its ∂ t u(x, t), we can use a time
integrator to solve for u(x,t+∆t) the solution on the next time slice. We use a second
order Iterated Crank-Nicholson (ICN) [72] routine for this purpose. It uses a Cauchy step
and two partial steps in order to evolve the system with a truncation error of O(∆t 2 ).
1
+ ∆tA( ((2) ũn+1
= un
))
un+1
+ un
j
j
j
j
2
(3.34)
(2) n+1
ũj
1
= un
+ ∆tA( ((1) ũn+1
+ un
))
j
j
j
2
(3.35)
(1) n+1
ũj
)
+ ∆tA(un
= un
j
j
(3.36)
Although any number of iterations can be used with ICN, there is no advantage to using
more than two. We will use this technique to solve for the metric and then use its new
value to solve for a new value of the extrinsic curvature and continue back and forth
between the two, evolving the system. ∆t depends on ∆x by a constant Courant factor
∆t
∆x . Because both the time and spatial derivatives have truncation errors which vary as
∆x2 we expect the total numerical errors to vary as ∆x 2 or second order, this will be
important when we discuss convergence.
20
Chapter 4
Issues of Stability
In this section I will summarize the theoretical work that has been done to date in
order to understand the stability of systems of equations which are designed to numerically evolve systems of partial differential equations. I will also explain how the methods
used in this thesis may determine whether or not a numerical evolution system is stable
for non-linear spacetimes.
4.1
Definitions
In this section, I will define many of the terms used in this thesis.
4.1.1
L2 norm
The L2 norm is basically the average absolute value of a given series of quantities.
It is used in this thesis to study convergence as well as how the system’s metric and
numerical errors change with time. It is defined numerically as:
v
u
N
u X
u1
L2 norm = t
(uk )2 .
N
k=1
(4.1)
21
4.1.2
Form of Equations
We begin with a linear second order PDE in terms of two variables [30],
auxx + 2buxy + cuyy + dux + euy + f u = g
(4.2)
By replacing the differentials with greek indices (u xx = α2 , uxy = αβ, uyy = β 2 , ux = α
and uy = β) we get the second degree polynomial:
P (α, β) ≡ aα2 + 2bαβ + cβ 2 + dα + eβ + f
(4.3)
The polynomial (and therefore the second order PDE which it coorisponds to) is classified
as elliptic, parabolic or hyperbolic depending on wether b 2 - ac is negative, zero or
positive. Elliptic PDEs have the form ∂ t2 u + ∂x2 u = RHS. An example of this is the
Poisson equation,
∂x2 u + ∂y2 u = ρ(x, y).
(4.4)
Parabolic PDEs have the form ∂t u - ∂x2 u = RHS like in the diffusion equation,
∂t u = ∂x (D∂x u).
(4.5)
Hyperbolic PDEs have the form ∂t2 u - ∂x2 u = RHS. A common example of this is the
one dimensional wave equation,
2
2 2
∂t u = v ∂x u.
(4.6)
22
Unlike elliptic or parabolic equations, hyperbolic equations can be split into the form
∂t u ± v∂x u = RHS and evolved as a Cauchy problem. The first order ADM evolution
equations fall in the category of hyperbolic PDEs when linearized. This is apparent when
the total derivative, ∂0 uij is broken down into ∂t uij - Lβ uij . Here Lβ uij contains the
advective term β k ∂k uij .
4.1.3
Well-posedness
Much work has gone into attempting to theoretically determine whether or not
an evolution will be stable by looking at the form of the evolution equations. We can
rewrite a set of linear first order differential equations as
∂t u = A∂x u.
(4.7)
Where A is a matrix and u is a vector quantity containing information about all the
evolved variables. For a hyperbolic set of equations, A only has real eigenvalues. If
the matrix A has only real eigenvalues, is diagonalizable and a has a complete set of
eigenvalues, it is said to be strongly hyperbolic. If the matrix A has real eigenvalues, is
not diagonalizable and an incomplete set of eigenvectors, it is weakly hyperbolic. When
rewritten in a first order form the ADM equations used here are weakly hyperbolic.
Having a strongly or weakly hyperbolic system should tell us whether or not a system
is well-posed. By definition a well-posed initial value problem has a unique solution
which depends continuously on the initial data. A system is said to be well-posed if all
23
numerical solutions satisfy
αt
ku(·, t)k ≤ Ke kf (·)k.
(4.8)
K and α are constants which have no dependence on initial data or the resolution of the
system. This equation stipulates that the evolution system is only well-posed if the norm
of the numerical solution does not grow faster than exponentially, although exponential
growth is possible. An ill-posed system has solutions which do not depend on the initial
data. If the matrix A has complex eigenvalues the problem is completely ill-posed. The
growth of an ill-posed solution depends directly on the frequency or wave-number of the
modes (which are directly related to the resolution of the computational grid) and so no
constant value of α or K exists.
4.1.4
Defining an unstable evolution
If not for years of computational experience, no one would expect unstable evo-
lutions to be an issue at all. Because of this, early works on numerical methods rarely
discussed instabilities [75]. An unstable system is classically defined as one where any
perturbation of the exact solution diverges as time increases until the numerical solution
no longer depends on the initial data. Because an evolution can survive almost infinitely
long before this happens, this definition is not always practical. For this reason, I will
define an unstable evolution as one containing what I call type I or type II instabilities.
While type I instabilities are easy to witness numerically, type II instabilities are based
on a mathematical theorem which will be discussed later. Type I instabilities satisfy at
least one of the following four conditions:
24
(1) Any point in the computational grid has any associated value which grows or
shrinks exponentially faster than the value of its neighbor. This means that the
solution is no longer smooth.
(2) The spacetime becomes un-physical or singular. This usually happens when zeros
or infinities appear in the diagonal metric components. Although this is often
caused by coordinate singularities in some systems, we do not expect to see any
singularities, coordinate or otherwise, in the spacetimes studied here during stable
evolutions.
(3) The norm of the numerical errors grows exponentially fast. Although exponential
growth is allowed in well-posed solutions, for a problem that can be practically
studied on a computer an exponential growth in the numerical errors implies that
we are no longer solving a relevant problem.
(4) The norm of the constraint values grow exponentially fast. This means that we
are no longer evolving a solution to Einstein’s equations.
By using these conditions we can tell whether or not we are witnessing a type
I unstable evolution and determine when the evolution first becomes unstable. Please
note that type I unstable evolutions are what people who work with numerical codes
typically refer to as an unstable run. This definition does not prove anything about the
overall stability of the system, because we do not know if our instabilities are caused by
large truncation errors or high frequency unstable modes.
25
4.1.5
Consistency, Convergence, Dispersion and Dissipation
The numerical error associated with the finite difference method is called the
truncation error. Both the finite spatial derivatives used on the right-hand side of the
evolution equations and the Iterated Crank-Nicholson method used for the time integrator have second order truncation errors. Consistency refers to the quality that the
truncation error always varies as ∆x 2 . Because of this we expect the error in our evolutions to go to zero as ∆x, ∆t → 0.
Consistency and/or stability can be determined by testing the convergence of
the code. Because truncation errors decrease with resolution, we expect the system to
converge if the truncation errors are the only significant errors in the numerical solution.
The mathematical definition of convergence is that the absolute value of the error at
each point of spacetime goes to zero as the resolution increases.
|un
− fkn | → 0
k
∆x, ∆t → 0
(4.9)
Here un
is the numerical solution and fkn is the analytic solution. According to the
k
Lax-Richtmyer equivalence theorem, ”If the difference approximation is stable and consistent, then we obtain convergence even if the underlying continuous problem only has
a generalized solution. If the approximation is convergent, then it is stable.” [42] In
other words, while a code with a consistent discretization scheme will not always converge if it is unstable, a stable and consistent code should always converge. Examining
convergence throughout an evolution should therefore give us an insight into whether
or not the numerical code is stable. We will define a type II unstable evolution as one
26
which loses convergence during the run. Here a loss of convergence simply means that
numerical errors increase with resolution, counter to equation 4.9. Therefore, unlike type
I instabilities which are based on observations of a single run, type II instabilities depend
on a mathematical theorem and data from at least two runs. Note that a system can
only be type II unstable if it is type I unstable in at least one of the runs. Because of
this, type II is simply a subclass of type I. However, type II instabilities can suggest more
about the overall stability of the system than type I because it allows us to distinguish
between instabilities caused by large truncation errors and high frequency modes.
Dispersion is a numerical effect where the propagation velocity of a mode depends on its wavelength and frequency. Dispersion may lead to instabilities when nonperiodic boundary conditions are used because it can result in growing phase errors at
the boundaries. The phase errors in the solution can often be seen in the convergence
tests. Dissipation occurs when none of the Fourier modes grow and at least one mode decays. Dissipation also effects stability because it can damp out unstable growing modes.
This effect also depends on frequency and could also effect the convergence rate of the
evolution. Because of this, convergence testing the numerical solution is important for
understanding the stability of the numerical code.
Dispersion and dissipation can be calculated by first assuming that the numerical
solution emits a ”plane wave” and so is free to oscillate according to the equation,
un
= ûei(ωn∆t+βk∆x) .
k
(4.10)
27
Where ω = α + ib is frequency and β is wavenumber. By inserting this expression for
un
into the finite difference evolution equations, we can solve for ω as a function of β,
k
∆x and ∆t. The phase velocity of the numerical solution is then,
ω
v=− .
β
(4.11)
In a perfect evolution v = 1 but because of numerical dispersion v is not always equal
to 1. Because ω contains both a real and imaginary part so does the velocity. The real
part of the velocity provides information about dispersion while the imaginary part of ω
reveals information about dissipation. If the imaginary part of ω is positive the solution
decays with time but if it is negative it grows.
4.1.6
Convergence Tests
The convergence rates can be calculated in four different ways.
(1) Using the L2 norms of the metric’s errors to calculate the convergence. This method
should give us a rough indication of how fast the numerical solution is converging
to the analytic solution.
2q =
k E∆ k
k E∆ k
(4.12)
2
Here E is the numerical error or difference between the numerical and analytic
solutions. Because of the way this calculation is performed I derive two values
of the convergence factor q for each output time, one for the coarse and medium
28
resolutions and another for the medium and fine resolutions. This gives us a range
of possible convergence factors.
(2) Using the L2 norm of the ratio of the metric errors to calculate the convergence.
This is much more time consuming than the first method but also much more
accurate. Again this gives us an idea of how fast the solution is converging to the
analytic solution.
E∆ 2 =
E∆ q
(4.13)
2
Notice that I take the L2 norms of the error ratio before calculating the convergence
rate, therefore each set of error ratios yields one convergence rate. Because there
are two sets of error ratios, I again derive two values of the convergence factor q
for each time used in the calculation.
(3) Using only the numerical solutions of the metric to calculate the convergence, this
is referred to as the self-convergence of the system. This will tell us if the metric is
converging to a solution even if that solution is different than the analytic solution.
Again this calculation is time consuming but accurate.
u∆ − u ∆ 2 2 =
u − u ∆
∆
q
2
(4.14)
4
(4) Using the Hamiltonian constraints to calculate the convergence. Because all of the
spacetimes are vacuum solutions to Einstein’s equations, any nonzero value of the
constraints can be interpreted as an error. This gives us an idea if the solution is
29
converging to a solution of Einstein’s equations.
H∆ 2 =
H ∆
q
(4.15)
2
Where H is the value of the Hamiltonian constraint. Again, the L2 norms of the
constraint ratios are used for calculating the convergence rate. Because of the way
this calculation is performed we can derive two values of the convergence factor q
for each time used in the calculation.
4.2
Modes
When linearized, the ADM system is weakly hyperbolic. This implies that the
system is not well-posed and may also become unstable because of the lower order nonlinear terms. In past experiences evolving these equations almost all the linear systems
appear to be stable while many nonlinear systems quickly go unstable. Also, it is difficult
to determine what nonlinear terms may cause the evolution to become unstable. The
instabilities which cause the code to become unstable manifest as modes that can be
classified as either constraint violating or gauge modes. Gauge modes cause the evolution to satisfy the first, second and/or third conditions for unstable evolutions without
generating a solution which has diverging constraint values. Constraint violating modes
satisfy the fourth (and possibly the first, second and/or third) condition(s) for unstable
evolutions.
According to work done by Alcubierre [4], a major cause of numerical instabilities
is zero speed constraint violating modes. While constraint violating modes may exist in
30
stable numerical codes, the fact that they leave the computational domain after a finite
amount of time keeps the code stable. With periodic boundary conditions unstable modes
can no longer leave the computational domain. The propagation of constraint values has
been the subject of several papers in recent years [81, 80, 21]. This hypothesis about the
velocities of code crashing constraint violating modes can be tested using the analysis
methods presented here. By studying what happens to a spacetime as it goes unstable,
we can determine whether we are witnessing gauge or constraint violating modes and
determine the propagation velocity of these modes. We can also determine how to
produce each kind of mode. This experimentation may one day help us understand
conclusively which lower order nonlinear terms lead to numerical instabilities.
4.3
Unanswered Questions
Several alternatives to the standard ADM system using strongly hyperbolic sys-
tems have been developed [48, 52, 14, 69, 38]. Although these systems are believed to
be well-posed, simulations using these evolution systems still become unstable and crash.
There are only two possible explanations for this. Either well-posedness in the linearized
form of the equations does not yield a stable code or some other factor such as a poor
choice of boundary or gauge conditions causes the code to become unstable. If the former is true, we will have to re-examine our numerical theories about hyperbolicity and
well-posedness. Specifically, we need to know how lower order nonlinear terms can cause
a well-posed system to become unstable. If the latter is true, we can proceed knowing
that our evolution scheme is correct and that our problem lies elsewhere in the numerical code. Testing these codes using well defined techniques can tell us if the evolution
31
methods are unstable and may make it possible to expose poor parameter choices which
could lead to unstable evolutions.
4.4
The Test
The purpose of this work is to develop a test to determine whether or not an
evolution system is stable. By eliminating all possible causes of instabilities outside of
an unstable evolution system and attempting to stimulate gauge and constraint violating
modes, it may be possible to determine whether or not a system is stable. An unstable
system, such as ADM, should be susceptible to unstable modes while a stable system
should be immune to them. In order to remove all the possible causes of instabilities we
should use periodic boundary conditions and relatively simple spacetimes. The systems
that we should use for these tests should therefore be periodic and lack singularities. By
using periodic boundary conditions we can avoid errors and reflections at the boundaries.
By using spacetimes without singularities, we can avoid having to use special “inner”
boundary conditions, excision methods or singularity avoiding gauge choices. The general
form of a diagonal four metric satisfying these conditions is,
ds2 = −α(x, t)2 dt2 + eM (x,t) dx2 + eN (x,t) dy 2 + eP (x,t) dz 2 .
(4.16)
With the appropriate choice of α, this basic metric can be modified to form any periodic
diagonal solution to Einstein’s equations. In the case where α = 1, M = N = P = 0, the
result is simply a flat space metric. If M, N and P are constant in x but not in time and
α = 1, the result is a Kasner spacetime. If α, M, N and P are periodic in x but not in
32
time we get the Gowdy spacetime. If N and P are periodic in x and t but M = 0 and α
= 1, the result is a plane gravitational wave. We found that this spacetime can only be
periodic in the linearized form of Einstein’s equations and is then called a Bondi Wave.
The last possibility is that flat space is coordinate transformed so N = P = 0 and M
and α are both periodic in x and time. The result is what appears to be a longitudinal
gravitational wave or “gauge” wave.
The first goal of this project is to determine what conditions stimulate type I
instabilities (such as gauge and constraint violating modes) in the ADM system. Because
of the different characteristics of the Kasner, Gowdy, Bondi and ”gauge” wave systems,
it is possible to determine several conditions which stimulate each type of mode. Next, I
will use the numerical techniques mentioned above to study the consistency, convergence,
dissipation and dispersion associated with the ADM system. Each spacetime will be
convergence tested and an analysis of its type I stability characteristics will be performed.
Our goal is to determine both the characteristics of the unstable modes and possible
causes of the instabilities. Once it is determined how to produce unstable evolutions for
each system, we can test the type II stability of each spacetime under the same conditions
which create type I instabilities. If the system ever fails to converge, then the evolution
is both type I and type II unstable and according to the Lax Theorem the system is
unstable. If the system converges at all times then it is stable.
In the future, this method can be applied to other evolution systems in order
to determine whether or not they are stable and what kinds of instabilities they are
susceptible to. A stable evolution system should not reproduce the unstable modes seen
in the ADM system or fail to converge.
33
Chapter 5
Nearly Trivial Solutions: Perturbed Flat-space & Kasner
Both perturbed flat-space and the Kasner solution presented here are variations
of the trivial flat spacetime. Because of the simplicity of these spacetimes, we can learn
much about how the evolution equations work on a basic level.
5.1
Perturbed Flat-space
The ”noisy” spacetime is simply flat space with random noise added to its metric.
This is used because the flat-space solution by itself does not tell us anything interesting.
Since we are using random noise that produces different solutions for different resolutions
the notion of convergence does not make sense. Because to this, the noisy spacetime is
used primarily to study the growth of small amplitude perturbations in the metric and
to see how constraint violations change with time. Note that since periodic boundary
conditions are used the solution is free to grow because it is not anchored to an analytic
solution at any point.
5.1.1
Error growth for different resolutions
To save computational resources and since we are using a generic, 3D code, each
run was carried out in a long thin computational grid extending along the x-axis and a
geodesic (α = 1) gauge condition is used to evolve the equations. Initially three different
runs are performed, each with an amplitude of 10 −6 in the random noise but all having
34
different resolutions. The fine resolution had ∆x = 0.125, the medium resolution had
∆x = 0.25 and the coarse resolution had ∆x = 0.5. All the runs also had a common
Courant factor of 0.25.
By running the simulation with different resolutions, we see that the growth rate
of the metric does depend on resolution. The finer the resolution the faster the growth
rate appears to be. This may be because the growth rate of the metric depends on
the number of iterations and not on the coordinate time. It could also mean that finer
resolution grids are more susceptible to growing instabilities because they allow for higher
frequency modes. Also the growth rate is not exponential in time so we can conclude
that this metric appears long-term stable. In the plot below we see the L2 norms of three
different resolutions of perturbed flat space. The most interesting effect witnessed here is
L2 norm of metric vs time
1.00000004
1.00000002
metric value
1
0.99999998
coarse grid
medium grid
fine grid
0.99999996
0.99999994
0.99999992
0.9999999
0
5
10
15
20
25
time
Fig. 5.1. Noise 1: Above shows the value of the L2 norm of the gxx metric component vs time.
The growth rate of the metric seems larger as the resolution is increased. For this set of runs a
grid size of L = 31 was used with resolutions of ∆x = 1.0, 0.5 and 0.25 respectively, therefore it
runs less than one crossing time. Also notice the deviation between the L2 norm of the metric
and that of flat space becomes smaller as the resolution is increased.
35
that the constraint violations appear to shrink exponentially with time. This is based on
runs of the noisy spacetime which lasted for 20 or more crossing times. For some of the
runs we see what appears to be a beat frequency in the L2 norm of the constraint values
as they decrease with time. As the amplitude of the noise perturbations is increased,
the beats seen if figure 5.3 become much less noticeable.
L2 norm of constraint values vs time
0.00006
0.00005
constraints
0.00004
coarse grid
0.00003
medium grid
fine grid
0.00002
0.00001
0
0
5
10
15
20
25
Time
Fig. 5.2.
Noise 2: The L2 norm of the constraints vs time. These runs all have a grid size of
L = 31 and the same specifications as the runs in figure 5.1. As a result the range of this plot
is less than one crossing time. Note that the constraint values seem to noticeably fluctuate with
time for the fine resolution.
5.1.2
Error growth for different amplitudes
In order to test how the growth rate of the errors was effected by the amplitude
of the noise perturbations, I completed six runs with amplitudes ranging from 10 −6 to
10−1 . Each run lasted for 20 crossing times and had a domain length of 10 units and a
resolution of ∆x = 0.5, ∆t = 0.125.
36
L2 norm of Constraints vs Time
0.000007
0.000006
Constraint Violation
0.000005
0.000004
Ham
0.000003
0.000002
0.000001
0
0
20
40
60
80
100
120
140
160
180
200
Time
Fig. 5.3. Noise 3: The L2 norm of the constraint values vs time. This run only involves the
coarse resolution with a grid size of L = 10 and ∆x = 0.5, It runs for 20 crossing times.
The constraint violations decreased at a faster rate as the initial amplitude of
the perturbations was increased. For runs 1 and 2 the magnitude of the perturbations
increased with time while the L2 norm of the metric’s error remained fairly constant.
For runs 3-6 the perturbations appeared to die off although the L2 norm of the metric
continued to grow at a constant rate. Despite the high power-law growth rate, run 6
survived for over 2000 crossing times.
5.1.3
Testing the stability of Iterated Crank-Nicholson
When using two ICN steps, the constraints seemed to grow and then shrink quickly
before becoming unstable and crashing the code. Instead of a constant power law growth
as seen in the case with three ICN steps, the metric using two ICN steps grew exponentially. This is consistent with the results of Saul Teukolsky’s paper on Iterated CrankNicholson [72], if our method of counting ICN steps is adjusted. Our counting convention
37
Table 5.1.
Error growth rate of perturbed flat-space for different amplitudes
Run
Perturbation Amplitude
Growth rate of gxx error (L2 norm)
1
10−6
t0 throughout the evolution
2
10−5
t0 throughout the evolution
3
10−4
t0 until around t = 60 then t0.85
4
10−3
t0 until around t = 20 then t1.23
5
10−2
t1.40 throughout the evolution
6
10−1
t1.58 throughout the evolution
is different than Teukolsky’s so 2 of Teukolsky’s steps is the equivalent of 3 of ours. In
other words. Teukolsky starts at 0 while we start at 1. Because of this I will only report
on data runs which used three ICN steps (Cauchy step plus two of Teukolsky’s ICN
steps).
5.2
Kasner
The Kasner spacetime used here can be thought of as a flat spacetime in a time
varying coordinate system. It allows us to look at the simplified evolution system (and
time integrator) without the complications of spatial derivatives. By removing the spatial
dependence the ADM equations simplify to:
∂t gij = −2αKij
(5.1)
k
(5.2)
∂t Kij = α(KKij − 2Kik Kj ) − Di Dj α.
38
When using geodesic slicing the system becomes extremely simple.
5.2.1
Introduction
The Kasner metric has the general form,
ds2 = −dt2 + t2p1 dx2 + t2p2 dy 2 + t2p3 dz 2 .
Where p1 , p2 and p3 must meet the condition that,
P
pi = 1 and
(5.3)
P 2
pi = 1 to yield a
vacuum solution to Einstein’s equations. In order to satisfy these conditions, p i is given
by the equations.
p1 =
p2 =
p3 =
k2 − 1
k2 + 3
2(1 + k)
k2 + 3
2(1 − k)
k2 + 3
Here k can be any number and will be referred to as the Kasner parameter. Setting
the Kasner parameter equal to 1 causes the yy component of the metric to grow while
the xx and zz components remain constant. In cases like this where there is only one
non-trivial component of the metric and extrinsic curvature in an exact vacuum solution
to Einstein’s equations, the constraints will always be identically satisfied. Consider an
example, such as this, with only non-trivial yy components of the metric and extrinsic
39
curvature each with numerical errors.
gyy = t2 + 1
(5.4)
Kyy = −t + 2
(5.5)
Here 1 and 2 are numerical errors. The non-trivial component of the inverse of the
metric can be written as:
g yy = t−2 + 3 .
(5.6)
This makes the Kyy and K yy components:
Kyy = −t−1 + 2 t−2 − 3 t + O( · )
(5.7)
K yy = −t−3 + 2 t−4 − 23 t−1 + O( · )
(5.8)
This means that Kij K ij and trK 2 will reduce to,
Kij K ij = trK 2 = t−2 − 22 t−3 + 23 + O( · )
(5.9)
The Ricci tensor, Rij , will reduce to zero because all spatial derivatives of the metric
are equal to zero. As a result the Hamiltonian constraint is identically zero.
H = R + trK 2 − Kij K ij ≡ 0
(5.10)
40
The momentum constraint is also identically zero because in
Mi = ∇j Kij − ∇i trK
(5.11)
j
Ki = trK when summing over all values of i and j.
By setting k not equal to unity, we recover non-zero constraint values. Also in
order to make the Kasner system more interesting, we can add random noise to the
system and use different gauge choices to study the results.
5.2.2
Convergence Tests
The convergence tests were performed by completing three runs with different
resolutions.
Table 5.2.
Convergence test specifications for Kasner
Resolution
Iterations
Output Freq
∆t
Total Outputs
Coarse
100
every 4 iters
0.08333 . . .
25
Medium
200
every 8 iters
0.04166 . . .
25
Fine
400
every 16 iters
0.02083 . . .
25
Notice that no specifications about the spatial resolution of the grid are included
because there is no spatial variation in the grid. A Kasner parameter of k = 1 was used
41
for this simulation in order to simplify the results. Einstein’s equations are satisfied
identically so there was no need for test 4 (see Chapter 3). Tests 1 and 2 suggest
a convergence factor which grew from q = 1.4 to 2.0. Test 3 showed similar results,
starting with a q = 1.25 and growing quickly to almost 2.0 after several iterations. We
believe that the convergence rate starts around 1.0 because the initial Cauchy step in
the ICN routine dominates the solution. After several iterations the Cauchy step no
longer dominates the rest of the routine, this results in an convergence rate of about 2.0.
Because the convergence rate eventually approached 2.0, it is fair to say that this system
converges like we expect it to. However the simple form of the metric begs the question,
why is the error not zero at all times?
Convergence results for Kasner
2.1
2
1.9
Convergence Factor
1.8
1.7
Test 1 coarse/medium
Test 1 medium/fine
1.6
Test 2 medium/fine
Test 3
1.5
1.4
1.3
1.2
1.1
1
2
3
4
5
6
7
8
9
10
Time
Fig. 5.4.
Kasner 1: These are the results of the convergence tests for the Kasner spacetime.
Notice that the results of the coarse/medium tests are the same for both Test 1 and 2 as are the
results for the medium/fine runs.
42
5.2.3
Errors in the Kasner spacetime
ICN is second order accurate as expected but its errors do not have a simple
predictable form. Second order accurate methods typically have errors for all higher
order terms even for functions where the higher orders of its expansion vanish. This is
the case with the Kasner metric. The time integration method fails to yield vanishing
values for higher order terms and therefore it does not give the exact result for a fixed
resolution. The error in the ICN method after one full iteration, to third order is given
by:
1
1
1
error (3) = [ (F 0 )2 u,t − u,ttt + F 00 u,2t ]∆t3 .
4
6
8
(5.12)
Notice that this is a generalized solution and does not involve the ADM equations. Here
u,t = F (u) and primes denote the derivatives of F with respect to u. In the case where k =
√
1, u = t2 and u,t = F (u) = 2 u form the solution to the gyy Kasner metric component.
The error (3) term is non-vanishing even though there are no Taylor expansion terms for
u with orders higher than two. This error then grows with each new iteration.
If for the Kasner case u = t2 and u,t = F (u) = 2t was the solution, the ICN
process would yield an exact result because the error (3) term will vanish. Because
the ADM evolution equations for Kasner k = 1 are of the form g yy ,t = −2Kyy and
2 /g
(3) term does not vanish. Notice that
Kyy ,t = −Kyy
yy we can see that the error
equation 5.13 applies already at the level of ODEs, and it is also true in PDEs solved by
the method of lines.
When using the ADM equations for the Kasner spacetime explicitly and solving
for the metric error in the ICN scheme, we find that the error in the g yy component after
43
one full iteration is given by:
egyy (t0 , t) = ∆t2 (1 −
1
(t + t0 )2 ).
4t t0
(5.13)
Here t0 is the initial time and t is the time after one full iteration. Notice that the ∆t 2
term is not written in terms of t and t 0 because it appears explicitly in the evolution. We
find the error after an arbitrary number of iterations is given by integrating eg yy (t0 , t)
with respect to t.
egyy (t) =
Z
egyy (t0 , t)dt = −
1 2
(t + 2t20 ln(t) − 4t t0 )∆t2
8t0
(5.14)
For t t0 = 1
1
egyy (t) ≈ − t2 ∆t2
8
(5.15)
Notice that at late times the Kasner metric error is a solution to the Einstein equations
and therefore also satisfies the constraints. If a higher order time integrator was used the
Kasner metric error may not be a solution to the Einstein equations but the constraints
should still be identically satisfied because of the proof shown in section 5.2.1.
5.2.4
Type I Stability Tests
The type I stability tests for Kasner involved a total of 14 tests. 7 used a Kasner
parameter of 1 while the other 7 used a Kasner parameter of 0.5. All runs used the same
specifications unless stated otherwise. Each computational grid consisted of a long thin
box with 22 points along the x-axis and only 3 along the y and z directions. For these
44
Kasner metric vs time
63.5
63.45
63.4
63.35
L2 norm gyy
63.3
Analytic
Coarse
63.25
Medium
Fine
63.2
63.15
63.1
63.05
63
7.93
7.94
7.95
7.96
7.97
7.98
Time
Fig. 5.5. Kasner 2: This is a close-up how the Kasner gyy metric component grows with time.
Notice that the numerical solution lags the analytic solution.
simulations ∆x = 0.5 and ∆t = 0.125 so that each grid has a length of 10 units along
the x-axis.
The first set of runs used geodesic slicing. For the k = 1 case, the metric error
grew as egyy ∝ t2.04 and the constraints were identically satisfied. The k = 0.5 case
resulted in egxx ∝ t and egyy ∝ t2.09. Also much like with the noisy spacetime the
constraint violations shrank to zero.
For the next set of runs, I introduced a non-trivial lapse function: α = 1.05 - 0.05
cos( 2π
L x) where L = 10.0. The k = 1 case crashed around t = 10 (or one crossing time).
The metric error grew as egyy ∝ t2.37 (measured using curve fits) until the crash but the
constraint violations oscillated until t = 8 then grew as C ∝ e 4.4t before exploded as the
code crashed. An examination of the metric revealed that the metric became un-physical
before the crash. After a few iterations, the metric developed a curvature similar to the
lapse function. The part of the metric which corrisponds to a α = 1.1, continued to
45
shrink until it became negative, resulting in large constraint violations. The k = 0.5
case crashed around t = 7 and experienced similar behavior. The metric error grew as
egxx ∝ t2.30 and egyy ∝ t2.40 . And again the constraint violations oscillated until t =
6 and then grew as C ∝ e6.5t before exploding and crashing the code. These seemed
to suggest constraint violating modes since the constraints grew exponentially while the
numerical errors do not. The next 3 sets of tests involved adding noise perturbations
Kasner metric with non-trivial lapse
1000
100
t=1.0
10
t=2.0
log of gyy
t=3.0
t=4.0
t=5.0
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
t=6.0
t=7.0
t=8.0
t=9.0
0.1
t=9.875
0.01
0.001
X
Fig. 5.6. Kasner 3: Plot of the Kasner metric (with a non-trivial lapse) changing with time.
Note the metric becomes singular where the lapse varies farthest from unity.
with an amplitude of 10−6 to the metric and/or extrinsic curvature. The results were
pretty much the same whether noise was added to the metric, extrinsic curvature or both.
For the k = 1 case the metric error grew as eg yy ∝ t2.02 and the constraint violations
shrank to zero. In the k = 0.5 case the g xx metric error started off constant and then
began to experience a small power law growth eg xx ∝ t0.02 after t = 465 or 46.5 crossing
46
times. The gyy metric error initially grew as egyy ∝ t2.1 but later this rate changed to
egyy ∝ t after t = 465. There was also a spike in the constraint values which occurred at
the time t = 465. At the same time the g xx metric component switched from shrinking
to growing and the gzz metric component stopped growing and became nearly constant.
In the next set of tests, noise with an amplitude of 10 −6 was added to the geodesic
lapse function. For the k = 1 case, the errors were eg xx ∝ t1.61 and egyy ∝ t2.04 . For
k = 0.5, there is again a spike in the constraint values, this time at t = 400. Before
this spike there is no noticeable growth in any of the metric errors but afterwards they
both grow as egxx ∝ egyy ∝ t. The interesting thing here is that the growth rate of the
errors is actually smaller (before the spike) with the addition of random perturbations
to the lapse than it was without them. This suggests that a good choice of lapse may
actually reduce numerical errors. For the last set of tests, I again added noise to the
lapse function but this time I used a 3D grid with 22 points in each direction. For the k
= 1 case, egxx ∝ t1.70 , egyy ∝ t2.12 and the constraint values shrank to zero and then
slowly grew to a constant value O(10 −5 ). The k = 0.5 case resulted in egyy ∝ t2.28
until t = 90 and then the growth rate increased to eg yy ∝ t7.95 . Meanwhile there was
no growth in the gxx metric error or constraint violations until t = 40 where they both
began growing at the rate egxx ∝ C ∝ e0.01t . Instead of a spike in the constraint values,
the code simply crashed at t = 266.
5.3
Conclusion
For long-term stable evolutions, it appears that the constraint violations decrease
exponentially with time. Also we found that the evolution system produces power-law
47
(often quadratic) error growth even when the spacetime is long-term stable. From our
stability analysis we found that we can produce (type I) unstable modes more efficiently
by introducing a strange lapse than by adding errors into our initial data. The unstable
modes resulting from gauge instabilities appear to eventually lead to constraint violating
modes. Also, in the case of unstable evolutions, the growth rate in the error of at least
one metric component appears to change throughout the evolution.
48
Normalized Errors in Kasner: nontrivial lapse
1
0.9
0.8
Normalized Errors
0.7
0.6
negyy
nham
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
Time
Fig. 5.7. Kasner 4: Plot of the normalized Kasner numerical and constraint errors versus
time using a non-trivial lapse. The normalized numerical errors are defined as the numeric
solution minus analytic solution divided by the sum of the absolute values of the numerical and
g
−ag
analytic solutions (ex. negyy = |g yy|+|agyy | ). The normalized constraint values are defined
yy
yy
by dividing the constraint values by the sum the the absolute values of their components (ex.
nham =
errors.
2
ij
R+K −Kij K
). Notice that the constraints begin to explode before the numerical
|R|+|K 2 |+|Kij K ij |
Normalized Errors in Kasner: Perturbed
1.E+01
0
100
200
300
400
500
600
700
800
900
1000
1.E+00
1.E-01
Normalized Errors
1.E-02
negxx
negyy
nham
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
Time
Fig. 5.8. Kasner 5: The above shows how the errors in both gxx and gyy are effected by
the spike in the constraint values when random noise was added to the lapse, metric and/or
extrinsic curvature. Again, normalized variables are used here so that all errors appear on the
same relative scale. Here the errors all start out at roughly the same order of magnitude but
soon the numerical errors become several orders of magnitude larger than the constraints.
49
Chapter 6
Cosmological Spacetimes: Gowdy & Bondi
The Gowdy and Bondi spacetimes represent cosmological solutions to Einstein’s
equations. Unlike the Kasner spacetime, the numerical Gowdy solution slightly violates
the constraint equations. The Bondi spacetime is only a solution to the linearized Einstein equations so constraint violations do not have the same meaning as they do for
solutions to the full non-linear Einstein equations.
6.1
6.1.1
Bondi Waves
Non-linear Periodic Plane Waves
One of the major reasons for using the Bondi wave solution is that non-linear
periodic traveling plane-wave solutions to the vacuum Einstein’s equations do not exist
given simple gauge conditions . This is demonstrated in the following proof by Gioel
Calabrese [20]. Suppose that a non-linear plane wave solution to Einstein’s equations
can be written in the following form.
ds2 = −dt2 + dx2 + eM (v) dy 2 + eN (v) dz 2 .
(6.1)
50
Where v = t ± x. When solving for the constraints we see that we must satisfy the
following equation in order to get a vacuum solution to Einstein’s equations.
1
∂v2 (M + N ) + ((∂v M )2 + (∂v N )2 ) = 0
2
(6.2)
The above equation can only be satisfied if ∂ v2 (M + N ) ≤ 0. This is only true if M + N is
constant. If M and N are periodic, then M + N is also periodic and the above equation
can not be solved. Therefore there is no periodic exact plane wave solution to Einstein’s
equations.
6.1.2
Introduction to Bondi Waves
Bondi waves may be thought of as linear gravitational plane waves. This space-
time is only an exact solution to Einstein’s linearized equations and therefore is not
restricted by the proof in the last section. The Bondi wave metric is given by,
ds2 = −dt2 + dx2 + (1 − A sin(2πn(x + t)/L))dy 2 + (1 + A sin(2πn(x + t)/L))dz 2 . (6.3)
Here A is the amplitude of the wave, n is the number of wave cycles in the computational
domain and L is wavelength. Because these are linear waves, the amplitudes used are
much less than unity.
6.1.3
Convergence Tests for Bondi
Bondi was tested using two different sets of parameters; the Bondi1 parameter
set used a finer grid spacing with an amplitude of A = 10 −5 and two wave cycles while
51
the Bondi2 parameter set used a much more coarse grid with an amplitude of A = 10 −4
and only one wave cycle. The specifications of each are shown in the following tables
Table 6.1.
Convergence test specifications for Bondi1
Resolution
Iterations
Output Freq
Crossing Times
Total Outputs
Coarse
160
every 8 iters
2
20
Medium
320
every 16 iters
2
20
Fine
640
every 32 iters
2
20
Resolution
xmin
xmax
xlow
xhigh
L
nx
∆x
∆t
Coarse
-5.25
5.25
-5.0
5.0
10.0
22
0.5
0.125
Medium
-5.25
5.0
-5.125
4.875
10.0
42
0.25
0.0625
Fine
-5.125
5.0
-5.0625
4.9375
10.0
82
0.125
0.03125
Because all of the interesting behavior in the Bondi spacetime occurs along the
x-axis, we can treat the system as a one-dimensional problem along the x-axis. For
computational efficiency, the grids used are long thin boxes which stretch along the
x-axis.
In tables 6.1 & 6.2, xmax and xmix are the maximum and minimum values input
into Cactus for the x-axis while xlow and xhigh are the output minimum and maximum
values. They are not the same because of the way Cactus uses ghostzones for periodic
boundary conditions. nx is the total number of points in the x-axis. ∆x is the grid
spacing of the x-axis. And L is the length of the x axis used in the periodic functions
52
Table 6.2.
Convergence test specifications for Bondi2
Resolution
Iterations
Output Freq
Crossing Times
Total Outputs
Coarse
40
every 2 iters
2
20
Medium
80
every 4 iters
2
20
Fine
160
every 8 iters
2
20
Resolution
xmin
xmax
xlow
xhigh
L
nx
∆x
∆t
Coarse
-6.75
5.25
-5.75
4.25
10.0
7
2.0
0.5
Medium
-5.75
5.25
-5.25
4.75
10.0
12
1.0
0.25
Fine
-5.25
5.25
-5.0
5.0
10.0
22
0.5
0.125
cos(2πnx/L) and sin(2πnx/L). For a periodic function with nx grid points ranging from
1 to nx, the value of point nx should always be equal to that of point 0, if it existed
as part of the computational grid. By substituting in discrete quantities, the periodic
functions mentioned above become cos(2πnk/nx) and sin(2πnk/nx) where k ranges from
1 to nx. Note the version of Maya used for these tests does not output the values of the
ghostzones. Because of this points nx + 1 and 0 many be used in the grid, but they do
not appear in the output.
The Bondi1 spacetime starts off converging as a second order system and then
experiences several bouts of hyperconvergence, where the solutions may converge with
factor as great as seven. Upon inspection of the Bondi metric vs time, it appears that
the coarser solutions of the metric propagate with a slower velocity than the analytic
solution. This results in a growing phase shift between the different resolutions. This
effect is demonstrated in Appendix A and can be explained by numerical dispersion. It
53
Convergence of Bondi1 system
8
7
6
Convergence Rate
5
Test 1 medium/fine
4
Test 2 medium/fine
Test 3
3
Test 4 medium/fine
2
1
0
-1
0
2.5
5
7.5
10
12.5
15
17.5
20
Time
Fig. 6.1. Bondi Wave 1: These are the results of the convergence tests for the g yy component
of the Bondi1 spacetime. The large number of hyperconvergence peaks may be the results of
truncation errors which are of the same order of magnitude as round-off errors.
is apparent that the velocity of the wave depends on a function of the grid spacing. Also
by observing the L2 norm of the spacetime metric versus time (Fig A.4), we notice that
a growth is occurring. This effect is accompanied by a decrease in the violation of the
Hamiltonian constraint.
We believe that the hyperconvergence may be caused by a mixture of truncation
and phase errors. Because the truncation errors are smaller than the phase errors, we
cannot expect an accurate calculation of the convergence rate. Because of this, I also
studied the Bondi2 system which included much larger truncation errors and smaller
relative phase errors.
I expect that by further increasing the truncation errors and decreasing the relative dispersion of the numerical solutions (by further increasing ∆x) we can get rid of
the hyperconvergence peaks altogether. However, in the process of increasing the errors
54
Convergence of Bondi2 system
6
5
Convergence Rate
4
Test 1 medium/fine
3
Test 2 medium/fine
Test 3
2
Test 4 medium/fine
1
0
-1
0
2.5
5
7.5
10
12.5
15
17.5
20
Time
Fig. 6.2. Bondi Wave 2: These are the results of the convergence tests for the g yy component
of the Bondi2 spacetime. Notice the number of hyperconvergence peaks has been significantly
reduced. Although test 4 suggests significant convergence this may not be a meaningful result.
we also make the code less stable. If we look at a plot of the L2 norm of the Hamiltonian
constraint vs time for the Bondi2 system (Fig A.5), we see that instead of the constraint
values decreasing throughout the evolution they eventually begin to increase. This may
be an early sign of a growing constraint violating mode.
6.1.4
Type I Stability Analysis
As part of our analysis of the long term stability of the Bondi spacetime, we
first examine the numerical dispersion and dissipation of the system. I used the method
described in section 4.1.5 for this analysis. I found that for the Bondi system in ADM
55
Normalized Errors in Bondi: trivial lapse
1.E+00
0
200
400
600
800
1000
1200
1.E-01
1.E-02
Normalized Errors
1.E-03
negyy
nham
1.E-04
1.E-05
1.E-06
1.E-07
1.E-08
Time
Fig. 6.3.
Bondi Wave 3: Plot of the normalized errors in the Bondi metric v.s. time using
geodesic slicing. Notice that the constraint values decrease while the numerical errors oscillate.
the frequency, ω, which was defined in equation 4.8, is given by:
ω =
i
2
2
2
2
2
2
[8η∆x − 4η∆t + η∆t cos(β∆x) + 2∆t cos(β∆x) + 8∆x − 8∆t ]
ξ∆t
ξ = 8∆x2 − 4∆t2 + ∆t2 cos(β∆x)
8
− 2 (∆t2 cos(β∆x)+4∆x2 −4∆t2 )
]
η = LambertW [− ∆x2 e ξ
ξ
(6.4)
Here the LambertW function is defined by the formula,
LambertW (x)eLambertW (x) = x.
(6.5)
For a reasonable choice of ∆x, ∆t and β, such as ∆x = 0.5, ∆t = 0.125 and β = 3, the
velocity of the numerical solution of the Bondi wave spacetime can be calculated as v =
-ω
β = 0.942 - 0.027 i where ω = -2.826 + 0.082 i. What this means is that the numerical
56
solution will travel slower than the analytic solution and dissipate energy as we evolve
the system forward in time. This is pretty much what we see in the simulations, proving
that the code contains both dissipation and dispersion. An additional effect is that the
dissipation is completely cancelled out by the numerical growth mentioned in the last
chapter. This results in an error growth rate slightly lower than that of the Kasner
system.
I test the effect of two different gauge conditions on the Bondi spacetime. The
first run featured a geodesic gauge with the same specifications as the coarse Bondi1
run except that this run lasted for 125 crossing times. During the run the constraint
violations oscillated as they shrank to zero and the metric error (eg yy ) oscillated like a
damped sine wave with a period of about 27.5 crossing times. The growth rate of the
metric error was found by subtracting the L2 norm of the analytic solution from the L2
norm of the numerical solution. This resulted in (L2 norm g yy - L2 norm agyy ) ∝ t1.88 .
A
2π
In the next Bondi run a non-trivial lapse was introduced (α = 1 + A
2 - 2 cos( L x)).
Here A = 10−5 therefore the lapse and wave-form have the same amplitude. This is similar to the lapse used to test the Kasner spacetime. This run seemed almost identical to
the last until around t = 800. Around t = 834, a large spike developed in the constraint
values. At t = 839 the L2 norm of gyy began to oscillate wildly and spikes began to
develop in the metric where α = 1 + A. Throughout the evolution (L2 norm g yy - L2
norm agyy ) ∝ t1.88 just like in the last run. Also like in the Kasner case, some of the
metric values became un-physical before the spikes developed. Again a poor choice of
lapse resulted in what appears to be constraint violating modes.
57
Normalized Errors in Bondi: nontrivial lapse
1.E+00
0
100
200
300
400
500
600
700
800
900
1.E-01
Normalized Errors
1.E-02
1.E-03
negyy
nham
1.E-04
1.E-05
1.E-06
1.E-07
Time
Fig. 6.4. Bondi Wave 4: Plot of the normalized error in the Bondi metric v.s. time using a
non-trivial lapse. Notice that the constraint values begin to explode before the numerical errors
implying the instability was caused by a constraint violating mode.
6.2
6.2.1
Gowdy
Introduction
The Gowdy spacetime used here is the same as the polarized Gowdy solution used
by Berger and Hern for the T 3 topology spacetime [15, 44]. Gowdy is an expanding
spacetime which consists of standing gravitational waves which depend on the value of
x. Its metric is given by,
ds2 = eλ/2 t−1/2(−dt2 + dx2 ) + t(eP dy 2 + e−P dz 2 ).
(6.6)
λ is a function of P and its derivatives. The time derivative of λ is the equivalent of the
Hamiltonian constraint and the spatial derivatives form the momentum constraints. P
58
and λ must satisfy the following differential equations,
P,tt = P,xx −(1/t)P,t
λ,t = t(P,2t +P,2x )
λ,x = 2t(P,t P,x )
(6.7)
In order to satisfy the first equation P must be a periodic function of x that varies with
time by a Bessel function. To satisfy the other equations a function P, which is a solution
of the first equation, is choosen and λ is then solved for using the other equations. P is
chosen to be,
P = AJ0 (t) cos(x)
(6.8)
1
1
λ = A2 (−tJ0 (t)J1 (t) cos(x)2 + (tJ0 (t))2 + (tJ0 (t))2 ).
2
2
(6.9)
therefore λ is given by
Here A is the amplitude of the standing waves and J 0 (t) and J1 (t) are bessel functions.
An amplitude of A = 0.1 was chosen as a default for the tests shown below. As a result
the yy and zz components of metric expand while the xx component contracts until it
reaches a minimum value and then begins to grow..
59
6.2.2
Convergence Tests for Gowdy
Again three different resolutions were used for the convergence tests. Many of the
specifications for each run are the same as that of Bondi1. Additional specifications for
each resolution are shown in table 6.3.
Table 6.3.
Convergence test specifications for Gowdy
Resolution
xmin
xmax
xlow
xhigh
L
nx
∆x
∆t
Coarse
−1.05π
1.05π
−π
π
2π
22
π/10
π/40
Medium
−1.05π
π
−1.025π
0.975π
2π
42
π/20
π/80
Fine
−1.025π
π
−1.0125π
0.9875π
2π
82
π/40
π/160
The Gowdy metric shows second order convergence with several hyperconvergence
peaks. Upon further inspection of the errors of the Gowdy metric, it was found that the
metric error is a standing wave which oscillates through zero, Fig B.2. This is most likely
related to the numerical dispersion discussed in the last section. Because the frequency
of this oscillation appears to be the same for all resolutions, round-off errors dominate
truncation errors as the total numerical error oscillates near zero. Hyperconvergence
results when the convergence test effectively divides a small number by a tiny number.
When studying the value of the constraints vs time, we see that shortly after the
initial step the constraint values drop dramatically and then oscillate around a constant
value throughout the evolution (Fig. B.3). As time moves on these oscillations become
60
Convergence of Gowdy gyy
5.5
5
4.5
Convergence Rate
4
Test 1 medium/fine
3.5
Test 2 medium/fine
Test 3
3
Test 4 medium/fine
2.5
2
1.5
1
1
3
5
7
9
11
13
Time
Fig. 6.5. Gowdy 1: These are the results of the convergence tests for the g yy component of the
Gowdy spacetime. Results of the convergence tests for gxx are shown in Fig B.1.
larger suggesting that they may explode into constraint violating modes. This effect is
most easily seen in the coarse grid resolution.
6.2.3
Here I studied the stability of the Gowdy spacetime by examining long evolutions
using different amplitudes. The first run used an amplitude of A = 0.1 and the same
specifications as the coarse grid run shown above. This run lasted for over 125 crossing
times. The growth rate of the errors were consistent for the yy and zz components but
varied for the xx component. The yy and zz errors where eg yy ∝ egzz ∝ t1.86. The xx
errors initially decreased until t = 8 as eg xx ∝ t−1 . From t = 8 until t = 250 the growth
rate of egxx was almost impossible to determine and then from t = 250 until the end of the
run egxx ∝ t2.77 . As with the Bondi case the constraint values decreased and oscillated
61
throughout the evolution. For the next run I increased the amplitude of the standing
waves to A = 1. This run crashed around t = 106 after the metric became un-physical.
The yy and zz metric errors grew faster in this case eg yy ∝ egzz ∝ t2.13. The xx metric
error grew as egxx ∝ t2.39 until t = 6 and then it became exponential eg xx ∝ e0.19t
until the crash. Much like in the unstable Bondi and Kasner cases, spikes developed in
the metric. However throughout the evolution the constraint values continued to shrink
exponentially to zero. This implies that we just witnessed an unstable gauge mode. In
the final Gowdy run, an amplitude of A = 1 was again used but the Courant factor was
lowered to 0.125. This run still crashed around t = 106 but the growth rate of the metric
errors was much easier to track. This run proves that the Courant factor is not always
an major element in the stability of the code. From t = 1 until t = 1.9 eg xx ∝ t3.3 . This
grew to egxx ∝ t4.8 from t = 1.9 to 2.9. Then shrank to egxx ∝ t1.5 from t = 2.9 to
3.5. From t = 3.9 to 4.3 the error continued to shrink like eg xx ∝ t−0.04 and the grew
like egxx ∝ t2.81 from t = 4.5 to 6. After t = 6 the error began to grow exponentially
as egxx ∝ e0.19t .
6.3
Conclusion
Our evolution system has both dissipation and dispersion. Dispersion can lead
to errors at the boundaries when using traditional boundary conditions. This may be a
cause of instabilities in other numerical codes. The Gowdy spacetime only appears to
be long-term stable when the amplitude of the standing waves is small. In this regime,
the Gowdy solution approaches that of the Kasner spacetime. As the waves become
strongly non-linear, the longevity of the evolution is decreased. As the code becomes
62
(type I) unstable the only interesting behavior occurs in the xx component of the metric.
The error oscillates before growing exponentially, giving us an early indication that the
code is going unstable. Meanwhile the constraint violations remain low suggesting that
the unstable modes caused by large non-linearities are gauge modes.
63
Normalized Errors for Gowdy A=0.1
0.5
0.45
0.4
Normalized Errors
0.35
0.3
negxx
nham
0.25
0.2
0.15
0.1
0.05
0
0
100
200
300
400
500
600
700
800
Time
Gowdy Error with A = 0.1
1.E-02
0
100
200
300
400
500
600
700
800
Error
1.E-03
egxx
1.E-04
constraints
1.E-05
1.E-06
Time
Fig. 6.6. Gowdy 2: Normalized and unnormalized numerical and constraint errors in Gowdy
with amplitude A = 0.1. Here the constraint values decrease by an order of magnitude as the
numerical error increase by two orders of magnitude.
64
Normalized Errors in Gowdy A=1.0
1
0.9
0.8
Normalized Errors
0.7
0.6
negxx
nham
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
Time
Gowdy error with A = 1.0
1.E+08
1.E+06
1.E+04
1.E+02
Error
1.E+00
0
20
40
60
80
100
egxx
constraints
1.E-02
1.E-04
1.E-06
1.E-08
1.E-10
Time
Fig. 6.7. Gowdy 3: Normalized and unnormalized numerical and constraint errors in Gowdy
with amplitude A = 1.0. Notice that the constraint values decrease exponentially as the numerical
error increases exponentially.
65
Chapter 7
The “Gauge” Wave Solution & Stability
7.1
”Gauge” Wave
The ”gauge” wave spacetime is simply a coordinate transformation of flat space
which appears as traveling longitudinal waves of arbitrary amplitude [22]. This spacetime
allows us to simulate traveling non-linear waves as a solution to the vacuum Einstein
equations.
7.1.1
Derivation
We begin with the flat spacetime metric,
ds2 = −dt2 + dx2 + dy 2 + dz 2
(7.1)
and define the new coordinates v and u so that
dudv = dx2 − dt2 .
(7.2)
If we define a new coordinate V so that
dudv = eA sin(2πnu/L) dudV = eA sin(2πnu/L) (−dT 2 + dX 2 ).
(7.3)
66
Substituting dT and dX for dt and dx leads to the “gauge” wave solution which we will
be using.
ds2 = eA sin(2πn(x+t)/L) (−dt2 + dx2 ) + dy 2 + dz 2
(7.4)
Here A is the amplitude of the wave and n is the number of wave cycles in the computational domain. Unlike the Bondi wave spacetime, there are no limit to the range of
possible wave amplitudes. Also, like the Kasner spacetime this system only has one nontrivial metric component and therefore the constraints are always identically satisfied.
However, unlike Kasner, this system can be highly non-linear. Consider the proof below
with only one non-trivial metric and extrinsic curvature component.
α(x, t) = ef (x,t)/2
(7.5)
gxx = ef (x,t) + 1
(7.6)
1
Kxx = −∂t gxx /2α = − ∂t f (x, t)ef (x,t)/2 + 2
2
(7.7)
Here α(x, t) is the lapse function, f(x,t) is any continuous function and 1 and 2 are
numerical errors. The non-trivial component of the inverse of the metric can be written
as:
g xx = e−f (x,t) + 3 .
(7.8)
This makes the Kxx and K xx components:
1
1
Kxx = − ∂t f (x, t)e−f (x,t)/2 + 2 e−f (x,t) − 3 ∂t f (x, t)ef (x,t)/2 + O( · )
2
2
(7.9)
67
1
K xx = − ∂t f (x, t)e−3f (x,t)/2 + 2 e−2f (x,t) − 3 ∂t f (x, t)e−f (x,t)/2 + O( · ) (7.10)
2
This means that Kij K ij and trK 2 will reduce to,
1
1
Kij K ij = trK 2 = (∂t f (x, t))2 e−f (x,t) − 2 ∂t f (x, t)e−3f (x,t)/2 + 3 ∂t f (x, t) + O( · )
4
2
(7.11)
The Ricci tensor, Rii , will reduce to zero because,
Rii = ∂i Γkik − ∂i Γkik = 0.
(7.12)
Therefore, because there is only a non-trivial dependence on the xx component of the
metric, the Ricci scalar is zero. As a result the Hamiltonian constraint is identically zero.
2
H = R + trK − Kij K
ij
≡0
(7.13)
The momentum constraint is also identically zero because in
Mi = ∇j Kij − ∇i trK
(7.14)
j
∇j Ki reduces to ∇i trK.
7.1.2
Convergence Tests
In order to study convergence I again used three different resolutions, each with
A = 0.1, n = 2 and a wavelength of 5 units. The specifications for each run are shown
in the following tables.
68
Table 7.1.
Convergence test specifications for Gauge Wave
Resolution
Iterations
Output Freq
Crossing Times
Total Outputs
Coarse
640
every 8 iters
8
80
Medium
1280
every 16 iters
8
80
Fine
2560
every 32 iters
8
80
Resolution
xmin
xmax
xlow
xhigh
L
nx
∆x
∆t
Coarse
-5.25
5.25
-5.0
5.0
10.0
22
0.5
0.125
Medium
-5.25
5.0
-5.125
4.875
10.0
42
0.25
0.0625
Fine
-5.125
5.0
-5.0625
4.9375
10.0
82
0.125
0.03125
See chapter 6 for an explaination of the quantities presented here. The gauge
wave convergence results were similar to that of the Bondi system. The convergence rate
started off around 2 for all tests. After about 1.4 crossing times a large hyperconvergence
peak developed. Because of this, I computed the convergence rate of the Gauge Wave
system for 8 crossing times instead of the 2 crossing times used for the other systems.
Upon further inspection of the metric, we see that the coarse solution appears to switch
between leading and lagging the analytic solution. The hyperconvergence (and minimalconvergence) peaks occur when the coarse solution switches between lagging and leading
as shown in Appendix C. It was later confirmed that this switching was caused by
the mixing of standing wave modes with the traveling wave mode that the system was
supposed to model. This effect was observed in another numerical code as well [22].
By observing the L2 norm of the spacetime metric versus time, we notice a nonlinear and non-constant growth rate, Fig C.4. The slight dips in the L2 norm of the
69
Convergence of Gauge Waves
10
9
8
Convergence Rate
7
6
Test 1 medium/fine
Test 2 medium/fine
5
Test 3
4
3
2
1
0
10
20
30
40
50
60
70
80
Time
Fig. 7.1. Gauge Wave 1: These are the results of the conversion tests for the g xx component
of the gauge wave spacetime.
metric are caused by oscillations in the standing wave mode. Again much like the Gowdy
system, the errors appear in the form of standing wave modes.
7.1.3
Much like with Gowdy, I studied the stability of the gauge wave spacetime by
examining long evolutions using different amplitudes. The first run used the same specifications as the coarse grid but lasted for 125 crossing times. This run yielded an error
growth rate of egxx ∝ t2.2 with no signs of numerical instabilities. The constraints remained identically satisfied throughout the evolution and the growth rate of the metric
error never changed. The next run featured an amplitude of A = 1. It crashed around t
= 9 and developed several spikes in the metric solution after parts of the metric became
un-physical. However the constraint values remained identically satisfied. It seems that
70
Error in Gauge Wave with A = 0.1
1.E+03
1.E+02
egxx
1.E+01
1.E+00
1.E+00
egxx
1.E+01
1.E+02
1.E+03
1.E-01
1.E-02
1.E-03
Time
Fig. 7.2. Gauge Wave 2: Error in the gauge wave metric versus time shown on a log-log scale.
Notice here that a straight line indicates a power law growth rate.
the standing wave modes became more dominate when the amplitude was increased.
The error growth rate started as egxx ∝ t1.86 until t = 1.75. Then from t = 1.75 to
4.75 the rate decreased to egxx ∝ t0.93 . The growth then became exponential. It was
egxx ∝ e0.65t from t = 4.75 to 6.0 then egxx ∝ e0.45t till t = 7.125 where it grew to
egxx ∝ e1.72t until the crash at t = 9.375. A final run using a Courant factor of 0.125
and an amplitude of A = 1 was also attempted, Fig. C.5. The results were the same
as in the last run except that the growth rate at 7.4 was slightly lower eg xx ∝ e1.18t .
Because of this, instead of the code crashing at 9.375, the error continued to grow till
13.6 and then slowed to egxx ∝ e0.15t until the crash at t = 36.
71
Error in Gauge Wave with A = 1.0
1.E+03
1.E+02
egxx & gxx
1.E+01
egxx
1.E+00
0
1
2
3
4
5
6
7
8
9
10
gxx
1.E-01
1.E-02
1.E-03
Time
Fig. 7.3. Gauge Wave 3: Error in the gauge wave metric versus time with an amplitude A =
1.0 using a Courant factor of 0.25.
7.1.4
Why Gauge Wave is unstable for large A
In order to understand why systems become unstable faster as the wave amplitude
is increased we must first examine the difference between the long-term stable Kasner
system and the unstable non-linear gauge wave system. Both systems have the exact
same evolution equations because both use geodesic slicing and only have one non-trivial
metric component. The evolution equations for both are:
∂t gxx = −2Kxx
(7.15)
∂t Kxx = −Kxx trK.
(7.16)
This is perhaps the simplest version of the ADM evolution equations which allow for nonlinearities. For the Kasner system the metric errors are proportional to t 2 ∆t2 . Using
72
the same analysis as for the Kasner system in Chapter 5, the gauge wave system has an
error equivalent to,
egxx (t) ≈ A2 ψ 2 t2 ∆t2 .
(7.17)
Where ψ is a constant with dimensions of inverse time squared. Note, the spatial dependence of the metric error is not included for simplicity. A major difference between the
errors in gauge wave and Kasner is that the magnitude of the metric error in the gauge
wave system depends on amplitude as well as the amount of time which has passed since
the initial time-step. For the Kasner system the evolution variables look like,
gyy = t2 + 1 (t2 )
(7.18)
Kyy = −t + 2 (t)
(7.19)
trK = −t−1 + 3 (t−1 )
(7.20)
Where the terms are proportional to the terms in the parentheses. The evolution
equations then become,
∂t gyy = −2t + 2 (t)
(7.21)
∂t Kyy = −1 + 2 + 3 − 2 3 .
(7.22)
Notice here that the error in the extrinsic curvature evolution equation does not grow
with time because the errors in trK and K yy balance out. For the gauge wave system
73
the evolution variables are:
gxx = eA sin(k(x+t)) + 1 (A2 , t2 )
(7.23)
1
Kxx = − Ak cos(k(x + t))eA sin(k(x+t)) + 2 (A, t)
2
(7.24)
1
trK = − Ak cos(k(x + t)) + 3 (A, t)
2
(7.25)
Where k = 2πn
L . The evolution equations are then,
1
∂t gxx = Ak cos(k(x + t))eA sin(k(x+t)) − 2 (A, t)
2
(7.26)
∂t Kxx = − 41 A2 k 2 cos(k(x + t))2 eA sin(k(x+t)) + 21 Ak cos(k(x + t))[2 (A, t) +
3 (A, t)eA sin(k(x+t)) ] + 2 (A, t)3 (A, t)
(7.27)
We see here that the error in the extrinsic curvature evolution equation continues to
grow as a function of amplitude and time. Eventually this error becomes as large as
the non-linear term causing the run to go unstable. For different runs, the longevity is
decreased as the wave amplitude is increased. This statement is true for any numerical
evolution of the ADM equations involving non-linearities.
7.2
Type II Stability in all the spacetimes
By studying the long term convergence of each of the spacetimes we can test the
type II stability of the ADM system. We first complete a run of the gauge wave spacetime
74
Gauge Wave error vs time for A = 0.01
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
Error gxx
1.E+01
1.E+00
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Fine
Coarse
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
Time
Fig. 7.4. Above shows the metric error versus time of the gauge wave system for two extremely
high resolutions (62 and 122 grid points) and small amplitudes A = 0.01. Notice the loss of
convergence around t = 400.
using two resolutions of 62 and 122 grid points. Each of these runs used an amplitude
of A = 0.01 and had a computational domain ranging from −π to π. The coarse run
went (type I) unstable after about 133 crossing times while the fine run went unstable
around 60 crossing times. According to the definition in chapter 4 this system is type
II unstable.
Now that we know how to demonstrate type I unstable modes in all of
the spacetimes we use this knowledge to test their type II stability. According to a loose
interpretation of the Lax theorem, increasing the resolution of a consistent numerical
code should decrease its numerical errors, therefore increasing its longevity at the cost
of computational efficiency. In previous chapters we have shown that this code has a
consistent finite difference approximation so whether or not it is convergent depends
only on its (type II) stability. To do this test I complete three runs of each spacetime
model using three different resolutions, much like we did for the earlier convergence
75
Kasner metric error vs time
1.E+10
1.E+09
1.E+08
1.E+07
1.E+06
L2 norm of egyy
1.E+05
Coarse
Medium
Fine
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
1
2
3
4
5
6
7
1.E-01
1.E-02
1.E-03
Time
Fig. 7.5.
Consistency 1: The consistency of the Kasner system. Notice the error in the fine
solution appears to explode first, this would not happen in a type II stable system according to
the Lax theorem.
tests. However, unlike previous convergence tests I will only use cases where I know
that type I numerical instabilities exist. If the code is type II stable then we expect the
system to converge throughout the run.
As you can see from the consistency plots,
each of the systems which is type I unstable is also type II unstable. An unexpected
result was that the medium resolution run of the gauge wave solution exploded before
either the fine or coarse ones. For the gauge wave solution, I also performed a spectral
analysis of the results so that we can get a better idea of what is causing the code to go
unstable.
The most striking difference between the power spectral densities (PSD) or
frequency distributions of the various resolutions is the number of dominant modes. As
we increase the resolution of the run, the number of modes increases until they merge
into a single peak. The most important of these are the high frequency modes with
frequencies greater than 3. According to Tiglio [23], these modes are what destroy the
76
Bondi metric error vs time
1.E+00
0
100
200
300
400
500
600
700
800
900
1.E-01
1.E-02
L2 norm of egyy
1.E-03
Coarse
Medium
Fine
1.E-04
1.E-05
1.E-06
1.E-07
1.E-08
Time
Fig. 7.6.
Consistency 2: The consistency of the Bondi system. Again the fine error explodes
first. Also the error only appears to oscillate for the coarse solution proving that the dispersion
is a function of ∆x.
stability of the system. It is these extra modes which cause the code to go unstable
faster given finer resolutions.
7.3
Conclusion
The usual way in which a code can go unstable is through frequency dependent
growing modes, whose rate of growth increases with resolution. The presence of these
modes would suggest that the system is unstable. This is confirmed by figures 7.4 through
7.12. Therefore this test suggests that the ADM evolution system is not stable in the
strictest sense. If the system were really stable, then solution would be type II stable
even in the presence of growing modes.
77
Gowdy metric error vs time
1.E+10
1.E+09
1.E+08
1.E+07
1.E+06
1.E+05
L2 norm of egxx
1.E+04
1.E+03
Coarse
Medium
Fine
1.E+02
1.E+01
1.E+00
1.E-01
0
20
40
60
80
100
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
Time
Fig. 7.7.
Consistency 3: The consistency of the Gowdy system. Notice the fine solution
explodes first proving that the system is type II unstable.
Gauge Wave metric error vs time
1.E+13
1.E+12
1.E+11
1.E+10
1.E+09
1.E+08
1.E+07
L2 norm of egxx
1.E+06
1.E+05
Coarse
Medium
Fine
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
1.E-01 0
2
4
6
8
10
12
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
Time
Fig. 7.8. Consistency 4: The consistency of the Gauge wave system. Here the medium solution
explodes first followed by the fine and coarse solutions.
78
PSD of Gauge Wave analytic solution
0.003
0.0025
0.002
0.0025-0.003
0.002-0.0025
0.0015-0.002
0.001-0.0015
0.0005-0.001
0-0.0005
PSD 0.0015
0.001
0.0005
4.25
-0.25 X
6
6.5
5
5.5
4
4.5
3
3.5
2
2.5
1
1.5
0
-4.75
0.5
0
freq
Fig. 7.9. PSD 1: Above is the power spectral density of the analytic solution of the gauge wave
spacetime with A = 1. This graph is simply used to show the frequency spectrum of the system
without the addition of unstable modes.
PSD of Gauge Wave coarse resolution
0.0025
0.002
0.0015
0.002-0.0025
0.0015-0.002
0.001-0.0015
0.0005-0.001
0-0.0005
PSD
0.001
0.0005
1.75
X
-4.75
6
6.5
5
5.5
4
4.5
3
3.5
2
2.5
1
1.5
0
0.5
0
freq
Fig. 7.10. PSD 2: The power spectral density of the coarse solution of the gauge wave spacetime
with A = 1. Notice the amplitudes of the various modes are very similar to those in the analytic
solution.
79
PSD of Gauge Wave medium resolution
0.0007
0.0006
0.0005
0.0004
0.0006-0.0007
0.0005-0.0006
0.0004-0.0005
0.0003-0.0004
0.0002-0.0003
0.0001-0.0002
0-0.0001
PSD
0.0003
0.0002
0.0001
1.75
X
-4.75
6.5
6
5.5
5
4
4.5
3
3.5
2
2.5
1
1.5
0
0.5
0
freq
Fig. 7.11. PSD 3: The power spectral density of the medium solution of the gauge wave
spacetime with A = 1. Here the high frequency modes are slightly larger than for the coarse
solution.
PSD of Gauge Wave fine resolution
0.000035
0.00003
0.000025
0.00002
0.00003-0.000035
0.000025-0.00003
0.00002-0.000025
0.000015-0.00002
0.00001-0.000015
0.000005-0.00001
0-0.000005
PSD
0.000015
0.00001
0.000005
1.75
X
-4.75
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
freq
Fig. 7.12. PSD 4: The power spectral density of the fine solution for the gauge wave spacetime
with A = 1. Notice the high frequency modes have a much higher relative amplitude than in the
lower resolution solutions. This may be the cause of the instabilities.
80
Chapter 8
Results
This work was an effort to better understand the stability of numerical codes
developed for the purpose of solving the problem of the inspiralling collision of a binary
black hole system. In particular, I focused on the system of equations which were designed to solve the nonlinear partial differential equations. Below is a chapter-by-chapter
summary of this work along with comments which may be helpful in the future.
8.1
Summary
In Chapter 2, I gave a brief description of the computing environment in which
these tests were performed. The software consists of two parts, Cactus and Maya. Cactus
is a computational tool developed at the Albert Einstein Institute also known as the Max
Planck institute for Gravitational Physics located in Golm, Germany. Maya is a code
that was developed by the Numerical Relativity group at the Center for Gravitational
Physics and Geometry at the Pennsylvania State University. The code used in this thesis
is an early version of the Maya code which is based on the ADM evolution system. The
code was run on a 6 node SGI Origin 2000.
Chapter 3 provided an introduction to the ADM evolution system, periodic boundary conditions and finite differencing. The ADM system was developed by Arnowitt,
81
Deser and Misner and was once the most popular decomposition of 4 dimensional spacetime used in numerical relativity. In this chapter I derive the ADM equations from
Einstein’s equations and show how they are evolved numerically on computers.
Chapter 4 focuses on work which has been done by other researchers in this field
in order to determine why the codes used in numerical relativity become unstable. In
the first section, I define several terms and concepts which are used in this area of
research. These include descriptions of the form of the equations, well-posedness, type
I and type II instabilities and consistency. I then went on to explain how I conducted
convergence tests and what the various results of these tests mean. In the next section
I summarized the still unanswered questions concerning the stability of these computer
codes and concluded by describing the tests that would be performed in the following
chapters.
In chapter 5, I covered nearly trivial solutions to Einstein’s equations. In the
presence of periodic boundary conditions the perturbed flat-space solution experienced
a power-law growth rate but never crashed. Also this growth rate appeared to increase
with both resolution and amplitude. Another interesting effect was that the norm of the
constraint violations decreased exponentially with time. Tests of the Kasner spacetime
revealed a quadratic error growth inherent in the evolution system. As the code went
type I unstable, this growth rate changed unpredictably. We also found that introducing
a poor choice of gauge resulted in an unstable evolution. Before crashing the code the
constraint violations increased before the numerical errors exploded.
82
Chapter 6 covered cosmological spacetimes such as Gowdy and Bondi. The Bondi
spacetime proved that the numerical code was subject to the effects of numerical dispersion and dissipation. Again, a poor choice of lapse resulted in a type I unstable Bondi
spacetime. The Gowdy spacetime was found to go type I unstable as the amplitude of
the standing waves was increased. Also the errors in the Gowdy spacetime appeared
in the form of standing waves which may be related to numerical dispersion. For both
systems the constraint violations decreased exponentially as a function of time.
The last chapter focused on the ”gauge” wave spacetime and the overall stability
of all the spacetimes studied here. At coarser resolutions standing waves seemed to
mix with the gauge wave solutions which we are evolving. Also much like the Gowdy
case, the spacetime becomes type I unstable more quickly as the amplitude of the waves
is increased. This effect was explained by examining the errors in non-linear term of
the evolution. The type II stability of all the spacetimes was then tested and it was
found that in every case, the finest resolution goes unstable before the coarsest. Further
confirming that type II instabilities are a subclass of type I instabilities and that the
ADM system is not inherently stable. By inspecting the power spectral density of the
unstable gauge wave solutions we see that the higher resolution runs yield larger relative
amplitude, high frequency modes.
8.2
The effect of constraint violations on Stability
An interesting and unexpected result of this work is that we where able to study
systems which identically satisfy the constraints, Kasner and gauge wave. I found that
the Kasner system could essentially last forever when there is no constraint violation.
83
Likewise the gauge wave system could also essentially last forever (even though there is
a temporary loss of stability see Fig. 7.4). In other systems where there is a significant
constraint violation the numerical errors eventually grow to infinity (Fig. 7.7) causing
the evolution to crash.
The perturbed flat-spacetime evolution also appears to last forever even though it
contains small constraint violations. Its constraint violations start off at the same order
of magnitude as the numerical errors and decrease exponentially as the numerical error
grows. This decrease appears to occur faster as the amplitude of the perturbations is
increased. This suggests a connection between the small constraint violation and the
longevity of the evolution.
When a non-trivial lapse is used for the Kasner system the constraint violations
are significant and the numerical errors grow to infinity causing the code to crash. When
the lapse, metric and/or extrinsic curvature of the Kasner system is perturbed and the
Kasner parameter is set to unity so there is very little constraint violation, we observe
no change in the error growth rate of the system. In this case the run could essentially
last forever. Even when the Kasner parameter is not set to unity the system appears
to recover after the constraint spike (Fig. 5.8), again suggesting that the run could
essentially last forever.
The Bondi system with a trivial lapse yields a shrinking constraint violation and
lasts essentially forever. When a non-trivial lapse is used, large constraint violations
result and the numerical error grows to infinity causing the code to crash. This also
happens in the Gowdy spacetime, significant constraint violations lead to numerical
errors which eventually grow to infinity and cause the code to crash. Because the gauge
84
wave errors rarely seem to grow to infinity and crash the code, all of this suggests that by
controlling the constraint errors we can increase the longevity of the evolution. However,
large errors may still develop in the numerical solution.
8.3
Conclusion
We have shown that the ADM system often yields type I unstable solutions. These
solutions also appear to lose convergence as errors become large and are therefore type II
unstable. We have also shown that the evolution system generates constraint violating
modes which result from gauge instabilities and gauge modes given large nonlinearities.
In addition we have shown that the code contains numerical dispersion which can cause
errors given a poor choice of boundary conditions. Because of all this we see that it
would be extremely difficult, if not impossible, to stably evolve a nonlinear system using
the ADM equations for a significant amount of time. Therefore the ADM system may
not be useful in binary black hole evolutions.
85
Appendix A
Additional plots for the Bondi System
This appendix shows additional plots related to my analysis of the Bondi spacetime performed in chapter 6. Although many of these plots are useful for understanding
the performance of the system, there was not enough room to include them in the body
of the chapter.
Bondi at t=1
1.000015
1.00001
1.000005
gyy
Analytic
Coarse
1
Medium
Fine
0.999995
0.99999
0.999985
-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Fig. A.1. Bondi Wave 1: Bondi metric at t = 1. These next two plots demonstrate the
dispersion seen in the Bondi system.
86
Bondi at t=9
1.000015
1.00001
1.000005
gyy
Analytic
Coarse
1
Medium
Fine
0.999995
0.99999
0.999985
-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Fig. A.2. Bondi Wave 2: Bondi metric at t = 9. Notice the growing phase shift in the solutions.
Bondi at t=17
1.000015
1.00001
1.000005
gyy
Analytic
Coarse
1
Medium
Fine
0.999995
0.99999
0.999985
-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Fig. A.3. Bondi Wave 2: Bondi metric at t = 17. As time increases the phase error continues
to grow.
87
L2 norm of Bondi1 system
1.0000000016
1.0000000014
1.0000000012
L2 norm gyy
1.0000000010
Analytic
Coarse
1.0000000008
Medium
Fine
1.0000000006
1.0000000004
1.0000000002
1.0000000000
0
2
4
6
8
10
12
14
16
18
20
Time
L2 norm of the Bondi2 system
1.00000014
1.00000012
L2 norm of gyy
1.0000001
1.00000008
Analytic
Coarse
Medium
Fine
1.00000006
1.00000004
1.00000002
1
0
2
4
6
8
10
12
14
16
18
20
Time
Fig. A.4. Bondi Wave 3: L2 norm of the Bondi1 and Bondi2 systems at all times. Note: the
coarse solution grows faster than the higher resolution solutions. Also the Bondi2 system appears
to grow about 100 times faster than the Bondi 1 system.
88
Constraints vs time for Bondi1
2E-10
1.8E-10
L2 norm of Constraints
1.6E-10
Coarse
Medium
Fine
1.4E-10
1.2E-10
1E-10
8E-11
0
2
4
6
8
10
12
14
16
18
20
Time
Constraints vs Time for Bondi2
5.0E-09
L2 norm of Constraints
4.5E-09
4.0E-09
Coarse
Medium
Fine
3.5E-09
3.0E-09
2.5E-09
0
2
4
6
8
10
12
14
16
18
20
Time
Fig. A.5.
Bondi Wave 4: L2 norm of the constraints of the Bondi1 and Bondi2 systems
respectively at all times. Notice that for the Bondi2 system the constraint values appear to grow
late in the evolution.
89
Appendix B
Additional plots for the Gowdy System
This appendix shows additional plots related to my analysis of the Gowdy spacetime performed in chapter 6. Although many of these plots are useful for understanding
the performance of the system, there was not enough room to include them in the body
of the chapter.
Convergence of Gowdy gxx
5.5
5
4.5
Convergence Rate
4
Test 1 medium/fine
3.5
Test 2 medium/fine
Test 3
3
Test 4 medium/fine
2.5
2
1.5
1
1
3
5
7
9
11
13
Time
Fig. B.1.
Gowdy 1: These are the results of the convergence tests for the g xx component of
the Gowdy space-time. They are similar to the convergence results for the g yy component seen
in chapter 6.
90
Gowdy Error
0.005
0.004
0.003
0.002
t = 2.88
0.001
egyy
t = 3.51
t = 4.14
t = 4.77
0
-3.15
-2.15
-1.15
-0.15
0.85
1.85
2.85
t = 5.40
-0.001
-0.002
-0.003
-0.004
X
Fig. B.2. Gowdy 2: Shows the value of egyy at different times immediately before during and
after the hyperconvergence peak at t = 4.14. Notice how the error oscillates through a value near
zero.
L2 norm of Ham vs time
0.00018
0.00016
0.00014
ham_nm2
0.00012
0.0001
Coarse
Medium
Fine
0.00008
0.00006
0.00004
0.00002
0
1
3
5
7
9
11
13
Time
Fig. B.3. Gowdy 3: Above is the L2 norm of the Hamiltonian constraint vs time. Note that the
oscillations in the constraint value increases with time especially for the low resolution solution.
Note this run lasts for 2 crossing times.
91
Appendix C
Additional plots for the ”Gauge” Wave System
This appendix shows additional plots related to my analysis of the ”gauge” wave
spacetime performed in chapter 7. Although many of these plots are useful for understanding the performance of the system, there was not enough room to include them in
the body of the chapter.
Gauge Wave at t=13
1.1
1.05
Analytic
gxx
Coarse
Medium
1
Fine
0.95
0.9
-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Fig. C.1. Gauge Wave 1: Gauge wave metric at t = 13, right before the large hyperconvergence
peak in the convergence results. Notice how the numerical solutions lag the analytic solution.
92
Gauge Wave metric at t=14
1.1
1.05
Analytic
gxx
Coarse
Medium
1
Fine
0.95
0.9
-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Fig. C.2.
Gauge Wave 2: Gauge wave metric at t = 14, during the large hyperconvergence
peak in the convergence results. Notice how the numerical solutions now slightly lead the analytic
solution.
Gauge Wave at t=15
1.15
1.1
Analytic
1.05
gxx
Coarse
Medium
Fine
1
0.95
0.9
-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Fig. C.3. Gauge Wave 3: Gauge wave metric at t = 15, right after the hyperconvergence peak.
The numerical solution now clearly leads the analytic solution. This switch between leading to
lagging the analytic solution explains the origin of the hyperconvergence peak seen in chapter 7.
93
Gauge Wave Metric vs Time
1.09
1.08
1.07
L2 norm of gxx
1.06
Analytic
1.05
Coarse
Medium
1.04
Fine
1.03
1.02
1.01
1
0
2
4
6
8
10
12
14
16
18
20
Time
Fig. C.4.
Gauge Wave 4: L2 norm of the gauge wave metric at all times. Notice how the
growth rate oscillates as the metric grows unlike the Bondi solution.
Gauge Wave error with A=1.0 & dt/dx=0.125
1.E+05
1.E+04
1.E+03
1.E+02
egxx
1.E+01
egxx
gxx
1.E+00
0
5
10
15
20
25
30
35
1.E-01
1.E-02
1.E-03
1.E-04
Time
Fig. C.5. Gauge Wave 5: Error in the gauge wave metric versus time with an amplitude A =
1.0 using a Courant factor of 0.125. This is the result mentioned in section 7.1.3.
94
References
[1] Miguel Alcubierre. The appearance of coordinate shocks in hyperbolic formalisms
of General Relativity. Phys.Rev.D55:5981-5991, 1997.
[2] Miguel Alcubierre, Gabriella Allen, Bernd Brügmann, Thomas Dramlitsch,
José A. Font, Phillippos Papadopoulos, Edward Seidel, Nikolaos Stergioulas,
Wai-Mo Suen, and Ryoji Takahashi. Towards a Stable Numerical Evolution of
Strongly Gravitating Systems in General Relativity: The Conformal Treatments.
Phys.Rev.D62:044034, 2000.
[3] Miguel Alcubierre, Gabriella Allen, Bernd Brügmann, Gerd Lanfermann, Edward
Seidel, Wai-Mo Suen, and Malcom Tabias. Gravitational collapse of gravitational
waves in 3D numerical relativity. Phys.Rev.D61:041501, 2000.
[4] Miguel Alcubierre, Gabriella Allen, Bernd Brügmann, Edward Seidel, and Wai-Mo
Suen. Towards an understanding of the stability properties of the 3+1 evolution
equations in general relativity. Phys.Rev.D62:124011, 2000.
[5] Serge Alinhac. Blowup for Nonlinear Hyperbolic Equations. Birkhäuser, 1995.
[6] Yu. E. Anikonov. Formulas in Inverse and Ill-Posed Problems. VSP, 1997.
[7] Peter Anninos, Joan Centrella, and Richard A. Matzner. Nonlinear wave solutions
to the planar vacuum Einstein equations. Phys.Rev.D43:1825-1838, 1991.
95
[8] Peter Anninos, Joan Massó, Edward Seidel, Wai mo Suen, and Malcolm Tobias.
THE NEAR LINEAR REGIME OF GRAVITATIONAL WAVES IN NUMERICAL
RELATIVITY. Phys.Rev.D54:6544-6547, 1996.
[9] Peter Anninos, Joan Massó, Edward Seidel, Wai mo Suen, and Malcolm Tobias.
DYNAMICS OF GRAVITATIONAL WAVES IN 3-D: FORMULATIONS, METHODS, AND TESTS. Phys.Rev.D56:842-858, 1997.
[10] A. Arbona, C. Bona, J. Massó, and J. Stela. Robust evolution system for Numerical
Relativity. Phys.Rev.D60:104014, 1999.
[11] George B. Arfken and Hans J. Weber. Mathematical Methods for Physicists. Academic Press, 1995.
[12] R. Arnowitt, S. Deser, and C. W. Misner. The dynamics of general relativity. John
Wiley, 1962. In L. Witten (Ed.), Gravitation an introduction to current research,
pp. 227–265.
[13] J.M. Bardeen and L.T. Buchman. NUMERICAL TESTS OF EVOLUTION SYSTEMS, GAUGE CONDITIONS, AND BOUNDARY CONDITIONS FOR 1-D
COLLIDING GRAVITATIONAL PLANE WAVES. Phys.Rev.D65:064037, 2002.
[14] Thomas W. Baumgarte and Stuart L. Shapiro. On the Numerical Integration of
Einstein’s Field Equations. Phys.Rev.D59:024007, 1999.
[15] Beverly K. Berger and David Garfinkle. Phenomenology of the Gowdy Universe on
T3 × R. Phys.Rev.D57:4767-4777, 1998.
96
[16] Beverly K. Berger and Vincent Moncrief. Numerical investigation of cosmological
singularities. Phys.Rev.D48:4676-4687, 1993.
[17] David H. Bernstein, David W. Hobill, and Larry L. Smarr. Black Hole Spacetimes:
Testing Numerical Relativity. Cambridge University Press, 1989. In C. Evans, L.
Finn, and D. Hobill (Ed.), Frontiers in Numerical Relativity, pp. 57-73.
[18] Carles Bona, Joan Massó, Edward Seidel, and Paul Walker.
THREE-
DIMENSIONAL NUMERICAL RELATIVITY WITH A HYPERBOLIC FORMULATION. gr-qc/9804052.
[19] The
Cactus
Development
Team.
The
Cactus
Computational
Toolkit.
http://www.cactuscode.org.
[20] Gioel Calabrese. The Metric cannot be periodic. internal document for PSU NR
group.
[21] Gioel Calabrese, Luis Lehner, and Manuel Tiglio. Constraint-preserving boundary
conditions in numerical relativity. gr-qc/0111003.
[22] Gioel Calabrese, Jorge Pullin, Olivier Sarbach, and Manuel Tiglio. Private communication with the numerical relativity group at LSU.
[23] Gioel Calabrese, Jorge Pullin, Olivier Sarbach, and Manuel Tiglio. Stability and
convergence in numerical relativity, 2002. Numerical Relativity lunch talk at PSU.
[24] Karen Camarda. 1D Black Hole Code.
[25] Sean M. Carroll. Lecture Notes on General Relativity. gr-qc/9712019.
97
[26] Joan Centrella and James R. Wilson. PLANAR NUMERICAL COSMOLOGY. 1.
THE DIFFERENTIAL EQUATIONS. Ap.J.273:428-435, 1983.
[27] Joan Centrella and James R. Wilson. PLANAR NUMERICAL COSMOLOGY. 2.
THE DIFFERENCE EQUATIONS AND NUMERICAL TESTS. Ap.J.Sup.54:229249, 1984.
[28] Joan M. Centrella. Nonlinear Gravitational Waves and Inhomogeneous Cosmologies.
In IN *PHILADELPHIA 1985, PROCEEDINGS, DYNAMICAL SPACETIMES
AND NUMERICAL RELATIVITY* 123-150, 1985.
[29] M.W. Choptuik. Consistency of finite-difference solutions of Einstein’s equations.
Phys.Rev.D44:3124, 1991.
[30] Paul DuChateau and David W. Zachmann. Schaum’s Outline on Theory and Problems of Partial Differential Equations. McGraw-Hill, Inc, 1986.
[31] T. M. R. Ellis, Ivor R. Philips, and Thomas M. Lahey. Fortran 90 Programming.
Addison-Wesley Publishing Company, 1994.
[32] Giampiero Esposito and Cosimo Stornaiolo. ON THE ADM EQUATIONS FOR
GENERAL RELATIVITY. Found.Phys.Lett.13:279-288, 2000.
[33] Lee Samuel Finn. A Numerical Approach to Binary Black Hole Coalescence. grqc/9603004.
[34] George E. Forsythe and Wolfgang R. Wasow. Finite-Difference Methods for Partial
Differential Equations. John Wiley & Sons, Inc., 1960.
98
[35] Helmut Friedrich.
HYPERBOLIC REDUCTIONS FOR EINSTEIN’S EQUA-
TIONS. Class.Quant.Grav.13:1451-1469, 1996.
[36] Helmut Friedrich and Gabriel Nagy. The Initial Boundary Value Problem for Einstein’s Vacuum Equation. Commun.Math.Phys.201:619-655, 1999.
[37] Simonetta Frittelli and Roberto Gómez. Ill-posedness in the Einstein equations.
J.Math.Phys.41:5535-5549, 2000.
[38] Simonetta Frittelli and Oscar Reula. Well-posed forms of the 3+1 conformallydecomposed Einstein equations. J.Math.Phys.40:5143-5156, 1999.
[39] Robert H. Gowdy. Gravitational Waves in Closed Universes. Phys.Rev.Lett.27:826829, 1971.
[40] The Grand Challenge Alliance. The Binary Black Hole Collision Grand Challenge.
http://jean-luc.ncsa.uiuc.edu/GC/GC.html.
[41] Carsten Gundlach and Paul Walker. Causal differencing of flux-conservative equations applied to black hole spacetimes. Class.Quant.Grav.16:991-1010, 1999.
[42] Bertil Gustafsson, Heinzo-Otto Kress, and Joseph Oliger. Time Dependent Problems
and Difference Methods. John Wiley & Sons, Inc., 1995.
[43] S. I. Harihan and T. H. Moulden, editors. Numerical methods for partial differential
equations. Longman Scientific & Technical, 1986.
[44] Simon David Hern. Numerical Relativity and Inhomogeneous Cosmologies. PhD
thesis, Wolfson College, 1999.
99
[45] Mijan Huq and Richard Matzner. The Binary Black Hole Grand Challenge Alliance
Cauchy code, 1999.
[46] Hans G. Kaper and Marc Garbey, editors. asymptotic analysis and the numerical
solution of partial differential equations. Marcel Dekker, Inc., 1991.
[47] Bernard Kelly, Pablo Laguna, Keith Lockitch, Jorge Pullin, Deirdre Shoemaker,
and Manuel Tiglio. A cure for unstable numerical evolutions of single black holes:
adjusting the standard ADM equations. Phys.Rev.D64:084013, 2001.
[48] Lawrence E. Kidder, Mark A. Scheel, and Saul A. Teukolsky. Extending the lifetime
of 3D black hole computations with a new hyperbolic system of evolution equations.
Phys.Rev.D64:064017, 2001.
[49] A. M. Knapp, E. J. Walker, and T. W. Baumgarte. Illustrating Stability Properties
of Numerical Relativity in Electrodynamics. Phys.Rev.D65:064031, 2002.
[50] Heinz O. Kreiss and Omar E. Ortiz. SOME MATHEMATICAL AND NUMERICAL QUESTIONS CONNECTED WITH FIRST AND SECOND ORDER TIME
DEPENDENT SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS.
gr-
qc/0106085.
[51] Andrei G. Kulikovskii, Nikolai V. Pogorelov, and Andrei Yu. Semenov. Mathematical
Aspects of Numerical Solution of Hyperbolic Systems. Chapman & Hall/CRC, 2001.
[52] Pablo Laguna and Deirdre Shoemaker. Numerical stability of a new conformaltraceless 3+1 formulation of the Einstein equation. gr-qc/0202105.
100
[53] V. Lakshmikantham, S. Leela, and A. A. Martynyuk. Stability Analysis of Nonlinear
Systems. Marcel Dekker, Inc., 1989.
[54] Luis Lehner. Numerical Relativity: Status and Prospects. gr-qc/0202055.
[55] Luis Lehner, Mijan Huq, Matt Anderson, Erin Bonning, Doug Schaefer, and Richard
Matzner. Towards stable evolutions of excised black hole spacetimes via the ADM
equations: A spherically symmetric test. Phys.Rev.D.62:044037, 2000.
[56] Luis Lehner, Mijan Huq, and David Garrison. Causal Differencing in ADM and
Conformal ADM Formulations: A COMPARISON IN SPHERICAL SYMMETRY.
Phys.Rev.D62:084016, 2000.
[57] Mark Miller. On the Numerical Stability of the Einstein Equations. gr-qc/0008017.
[58] Mark Miller. Coalescing Black Holes and Neutron Stars in Numerical Relativity,
2002. Numerical Relativity lunch talk at PSU.
[59] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W.H.
Freeman and Company, 1997.
[60] David Neilsen. Black hole physics on your laptop: Simple tests of a hyperbolic
formulation, 2002. Talk at PSU.
[61] Kimberly C.B. New, Keith Watt, Charles W. Misner, and Joan M. Centrella. STABLE THREE LEVEL LEAPFROG INTEGRATION IN NUMERICAL RELATIVITY. Phys.Rev.D58:064022, 1998.
101
[62] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery.
Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge
University Press, 1992.
[63] Alan D. Rendall. APPLICATIONS OF THE THEORY OF EVOLUTION EQUATIONS TO GENERAL RELATIVITY. gr-qc/0109028.
[64] Mark A. Scheel, Thomas W. Baumgarte, Gregory B. Cook, Stuart L. Shapiro,
and Saul A. Teukolsky. Numerical Evolution of Black Holes with a Hyperbolic
Formulation of General Relativity. Phys.Rev.D56:6320-6335, 1997.
[65] Erik Schnetter. Separating Constraint-Violating Modes and Gauge Modes in Spherical Symmetry. unpublished notes.
[66] Erik
Schnetter.
The
Maya
Project:
General
Concepts,
2000.
http://www.astro.psu.edu/nr.
[67] Edward Seidel. Numerical Relativity: Towards Simulations of 3D Black Hole Coalescence. gr-qc/9806088.
[68] Edward Seidel and Wai-Mo Suen. Towards a Singularity-Proof Scheme in Numerical
Relativity. Phys.Rev.Lett.69:1845-1848, 1992.
[69] Masaru Shibata and Takashi Nakamura. Evolution of three-dimensional gravitational wave: Harmonic slicing case. Phys.Rev.D52:5428-5444, 1995.
[70] Hisaaki Shinkai. Comments on “hyperbolic formulation and NR” , 1999. Numerical
Relativity lunch talk at PSU.
102
[71] Ulrich Sperhake. Non-linear numerical Schemes in General Relativity. PhD thesis,
University of Southampton, 2001.
[72] Saul A. Teukolsky.
ON THE STABILITY OF THE ITERATED CRANK-
NICHOLSON METHOD IN NUMERICAL RELATIVITY. Phys.Rev.D61:087501,
2000.
[73] J. W. Thomas. Numerical Partial Differential Equations: Conservation Laws and
Elliptic Equations. Springer-Verlag, 1999.
[74] J. W. Thomas. Numerical Partial Differential Equations: Finite Difference Methods.
Springer-Verlag, 1999.
[75] Lloyd N. Trefethen. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/
pdetext.html.
[76] Bernd Brügmann. Numerical Relativity in 3+1 Dimensions. Annalen Phys.9:227246, 2000.
[77] Dumitru N. Vulcanov and Miguel Alcubierre. Testing The Cactus code on exact
solutions of the Einstein field equations. gr-qc/0110031.
[78] Robert M. Wald. General Relativity. The University of Chicago Press, 1984.
[79] Jeffrey Winicour. A New Way to Make Waves. gr-qc/0003029.
103
[80] Gen Yoneda and Hisaaki Shinkai. Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system. grqc/0204002.
[81] Gen Yoneda and Hisaaki Shinkai. Constraint propagation in the family of ADM
systems. Phys.Rev.D63:124019, 2001.
Vita
David Garrison was born in Chicago, Il in 1975. In 1984, he and his family
moved to O’Fallon, MO where he finished grammar school and later graduated from
Fort Zumwalt North High School. Following graduation, he attended the Massachusetts
Institute of Technology in Cambridge, MA with the help of the University Club of St.
Louis Scholarship and the Class of 1961 Clarke E. Swannack Scholarship. In 1997 he
was awarded the Bachelor of Science Degree in Physics from MIT along with a minor in
Earth Atmosphereric and Planetary Science. In the fall on 1997, David Garrison began
his graduate work in the physics department at The Pennsylvania State University. For
the past four years he has been a Ph.D candidate working in the Center for Gravitational
Physics and Geometry.
During his time at Penn State, he worked on several projects in the center covering
many aspects of gravitational physics. Some of these include: a study on Causal Differencing in ADM and Conformal ADM Formulations with Luis Lehner and Mijan Huq
(published in 2000. Physical Review D. Volume 62, 084016), a project involving Gravity
Gradients in LIGO with Gabriela Gonzalez and work on Black Hole Spectroscopy with
Olaf Dreyer, Lee Finn, Ramon Lopez-Aleman, Badri Krishnan and Bernard J. Kelly.
In addition to teaching and research funding from Penn State, Mr. Garrison also
received funding from several fellowships. These included the NASA GSRP Fellowship
in 2001, Penn State’s Academic Computing Fellowship in 2001, the Bayer Fellowship in
1997 and a Minority Scholars Award in 1997. He is a Sloan Scholar as well as a member
of the American Physical Society and the National Society of Black Physicists.

testing binary black hole codes in strong field regimes

Transcription

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