Axisymmetric Numerical Relativity

2.1 MB·

by  Oliver Rinne

Axisymmetric Numerical Relativity
6
0
0
2

n
a Oliver Rinne
J

7
1 Trinity College, Cambridge

1
v
4
6
0
1
0
6
0
/
c A dissertation submitted to the
q
-
r University of Cambridge
g
:
v
for the degree of
i
X
r
a Doctor of Philosophy
13 September 2005

Preface
This thesis is based on research carried out under the supervision of Dr. John
M. Stewart at the Department of Applied Mathematics and Theoretical
Physics from November 2002.
Chapters2,3and6containworkdoneincollaborationwithmysupervisor
and published in a joint paper [119]. The dynamical shift conditions in
chapter 6 are a later addition by myself. The remaining chapters are my own
work.
Allcomputer programmes were written by myself unless otherwise stated.
c Oliver Rinne, 2005
(cid:13)
i
Abstract
This thesis is concerned with formulations of the Einstein equations in axi-
symmetric spacetimes which are suitable for numerical evolutions. The com-
mon basis for our formulations is provided by the (2+1)+1 formalism. Gen-
eral matter sources and rotational degrees of freedom are included.
Afirstevolutionsystemadoptsellipticgaugeconditionsarisingfrommax-
imal slicing and conformal flatness. The numerical implementation is based
on the finite-difference approach, using a Multigrid algorithm for the elliptic
equations and the method of lines for the hyperbolic evolution equations.
Problems with both constrained and free evolution are explained from an
analytical as well as a numerical viewpoint.
The second half of the thesis is concerned with a strongly hyperbolic first-
order formulation of the axisymmetric Einstein equations. Hyperbolicity
is achieved by combining the (2+1)+1 formalism with the Z4 formalism.
The system is supplemented with generalized harmonic gauge conditions. A
careful study of the behaviour of regular axisymmetric tensor fields enables
us to cast the equations in a form that is well-behaved on the axis.
Aclass ofexact solutions oflinearized theoryareusedasatest problemin
order to demonstrate the accuracy of our implementation. We derive various
outer boundary conditions of dissipative and of differential type based on the
Newman-Penrose scalars and the constraint and gauge propagation systems.
The stability of these boundary conditions is examined both analytically and
numerically.
The code is applied to the evolution of strong Brill waves close to the
threshold of black hole formation. As a novel ingredient, a nonzero twist is
included. Adaptive mesh refinement is found to be crucial in order to resolve
the highly distorted waveforms that occur if harmonic slicing is used.
ii
Acknowledgements
I would like to express my gratitude to Dr. John Stewart, my research su-
pervisor, for his advice and encouragement throughout this project.
I would also like to thank Dr. Nikolaos Nikiforakis of the Laboratory of
Computational Dynamics and fellow students Dr. Anita Barnes and Joshua
Horwoodforhelpfuldiscussions, andDr.StuartRankin,theRelativityGroup’s
computer officer, for helping me with my computing problems.
Iappreciated thehospitality ofthenumericalrelativity groupsattheMax
Planck Institute for Gravitational Physics (Albert Einstein Institute), Golm,
Germany, and at the University of Southampton.
Financial support from the Gates Cambridge Trust, the Engineering and
Physical Sciences Research Council and Trinity College Cambridge is grate-
fully acknowledged.
My final thanks go to my friends at Cambridge and beyond and, above
all, to my parents, for all their sympathy and support.
iii
Contents
1 Introduction 1
1.1 Numerical relativity . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Axisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Evolution formalisms . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Numerical methods and implementation . . . . . . . . . . . . 7
1.5 Gravitational waves and critical collapse . . . . . . . . . . . . 9
1.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Notation and conventions . . . . . . . . . . . . . . . . . . . . 11
2 Implications of axisymmetry 13
2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Vectors and covectors . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Symmetric 2-tensors . . . . . . . . . . . . . . . . . . . . . . . 16
3 The (2+1)+1 formalism 20
3.1 The Geroch decomposition . . . . . . . . . . . . . . . . . . . . 21
3.2 The ADM decomposition . . . . . . . . . . . . . . . . . . . . . 26
3.3 Matter evolution equations . . . . . . . . . . . . . . . . . . . . 32
4 Numerical methods 35
4.1 The finite difference technique . . . . . . . . . . . . . . . . . . 36
iv
4.1.1 The numerical grid . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Centred finite difference operators . . . . . . . . . . . . 37
4.1.3 The ghost cell technique . . . . . . . . . . . . . . . . . 38
4.2 The method of lines . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1 Properties of Runge-Kutta and ICN schemes . . . . . . 41
4.2.2 Numerical dissipation . . . . . . . . . . . . . . . . . . . 49
4.3 The Multigrid method . . . . . . . . . . . . . . . . . . . . . . 51
4.3.1 Relaxation Methods . . . . . . . . . . . . . . . . . . . 51
4.3.2 The Multigrid idea . . . . . . . . . . . . . . . . . . . . 54
4.3.3 Nonlinear Multigrid . . . . . . . . . . . . . . . . . . . . 58
4.3.4 Extension to systems and multidimensions . . . . . . . 60
4.4 Alternative methods . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1 Finite volume methods . . . . . . . . . . . . . . . . . . 61
4.4.2 Conjugate gradient methods . . . . . . . . . . . . . . . 63
4.5 Adaptive mesh refinement . . . . . . . . . . . . . . . . . . . . 65
4.5.1 The grid hierarchy . . . . . . . . . . . . . . . . . . . . 66
4.5.2 Time-stepping the grid hierarchy . . . . . . . . . . . . 67
4.5.3 Adapting the grid hierarchy . . . . . . . . . . . . . . . 69
5 A mixed hyperbolic-elliptic system 73
5.1 Elliptic gauge conditions . . . . . . . . . . . . . . . . . . . . . 74
5.2 Regularity on axis . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Final equations . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4 Alternate evolution schemes . . . . . . . . . . . . . . . . . . . 82
5.4.1 A free evolution scheme . . . . . . . . . . . . . . . . . 82
5.4.2 A constrained evolution scheme . . . . . . . . . . . . . 83
5.4.3 A partially constrained evolution scheme . . . . . . . . 84
5.5 Solvability of the elliptic equations . . . . . . . . . . . . . . . 84
v
5.5.1 Analytical considerations . . . . . . . . . . . . . . . . . 85
5.5.2 Numerical considerations . . . . . . . . . . . . . . . . . 90
5.6 Evolution of the constraints . . . . . . . . . . . . . . . . . . . 91
5.7 Evolutions of Brill waves . . . . . . . . . . . . . . . . . . . . . 94
5.7.1 Initial data . . . . . . . . . . . . . . . . . . . . . . . . 94
5.7.2 Boundary conditions . . . . . . . . . . . . . . . . . . . 95
5.7.3 Numerical method . . . . . . . . . . . . . . . . . . . . 97
5.7.4 Weak Brill waves with twist . . . . . . . . . . . . . . . 98
5.7.5 Strong Brill waves . . . . . . . . . . . . . . . . . . . . 100
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 The Z(2+1)+1 system 107
6.1 The Z4 extension of the (2+1)+1 formalism . . . . . . . . . . 109
6.2 Dynamical gauge conditions . . . . . . . . . . . . . . . . . . . 112
6.3 First-order reduction . . . . . . . . . . . . . . . . . . . . . . . 115
6.4 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4.1 Generalities, well-posedness of the IVP . . . . . . . . . 120
6.4.2 The dynamical shift case . . . . . . . . . . . . . . . . . 122
6.4.3 The vanishing shift case . . . . . . . . . . . . . . . . . 127
6.5 Regularity on axis . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.5.1 The main regularization procedure . . . . . . . . . . . 129
6.5.2 Choice of gauge source functions . . . . . . . . . . . . 131
6.5.3 Regularized conservation forms . . . . . . . . . . . . . 132
6.5.4 Hyperbolicity and the characteristic transformation . . 134
6.6 Equation checks and code generation . . . . . . . . . . . . . . 137
6.6.1 Checking the equations with exact solutions . . . . . . 137
6.6.2 Code generation . . . . . . . . . . . . . . . . . . . . . . 140
vi
7 A test problem in linearized theory 141
7.1 The linearized Z(2+1)+1 equations . . . . . . . . . . . . . . . 142
7.2 Transverse-traceless gauge . . . . . . . . . . . . . . . . . . . . 144
7.3 Teukolsky’s quadrupole solution . . . . . . . . . . . . . . . . . 147
7.3.1 The even-parity solution . . . . . . . . . . . . . . . . . 147
7.3.2 The odd-parity solution . . . . . . . . . . . . . . . . . 150
7.4 An even-parity twisting octupole solution . . . . . . . . . . . . 151
7.5 Numerical evolutions . . . . . . . . . . . . . . . . . . . . . . . 153
7.5.1 Numerical method . . . . . . . . . . . . . . . . . . . . 154
7.5.2 Snapshots of the evolution . . . . . . . . . . . . . . . . 155
7.5.3 Convergence tests . . . . . . . . . . . . . . . . . . . . . 157
7.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 159
8 Outer boundary conditions 162
8.1 Linearized characteristic variables . . . . . . . . . . . . . . . . 163
8.2 Dissipative boundary conditions . . . . . . . . . . . . . . . . . 166
8.2.1 Well-posedness of the IBVP . . . . . . . . . . . . . . . 167
8.2.2 Absorbing boundary conditions . . . . . . . . . . . . . 168
8.2.3 Zero-Z boundary conditions . . . . . . . . . . . . . . . 169
8.3 Outgoing-radiation boundary conditions . . . . . . . . . . . . 170
8.3.1 Newman-Penrose scalars and the peeling theorem . . . 170
8.3.2 Construction of the NP tetrad . . . . . . . . . . . . . . 173
8.3.3 Computation of Ψ . . . . . . . . . . . . . . . . . . . . 175
0
8.4 Constraint-preserving boundary conditions . . . . . . . . . . . 177
8.5 Gauge boundary conditions and summary . . . . . . . . . . . 181
8.6 Fourier-Laplace analysis . . . . . . . . . . . . . . . . . . . . . 185
8.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 193
8.7.1 Numerical method . . . . . . . . . . . . . . . . . . . . 194
vii
8.7.2 Numerical results . . . . . . . . . . . . . . . . . . . . . 195
8.7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 199
9 Evolutions of nonlinear Brill waves 201
9.1 Initial data and gauge choices . . . . . . . . . . . . . . . . . . 202
9.2 Convergence test . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.3 Apparent horizon finder . . . . . . . . . . . . . . . . . . . . . 213
9.4 Adaptive collapse simulations . . . . . . . . . . . . . . . . . . 220
10 Conclusions and outlook 227
10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.2 Outlook on future work . . . . . . . . . . . . . . . . . . . . . . 231
A Perfect fluid 233
A.1 Conservation form . . . . . . . . . . . . . . . . . . . . . . . . 233
A.2 Matter model . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
A.3 Characteristic decomposition . . . . . . . . . . . . . . . . . . . 236
A.4 From conserved to primitive variables . . . . . . . . . . . . . . 238
B Regularized conservation form 241
B.1 Fluxes in the r direction . . . . . . . . . . . . . . . . . . . . . 241
B.2 Fluxes in the z direction . . . . . . . . . . . . . . . . . . . . . 245
B.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
viii