Finite difference methods for 1st Order in time, 2nd order in space, hyperbolic systems used in numerical relativity [Elektronische Ressource] / von Mihaela Chirvasa

Chirvasa, Mihaela

Sujets

• Physik

Informations

• Publié par: universitat_potsdam
• Publié le: 01 janvier 2009
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Finite Difference Methods
for 1st Order in Time, 2nd Order in Space
Hyperbolic Systems
used in Numerical Relativity
Dissertation
von
Mihaela Chirvasa
eingereicht bei der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der Universit¨at Potsdam
durchgefu¨hrt in Golm am
Max Planck Institut fu¨r Gravitationsphysik
Albert Einstein Institut
unter der Betreuung von
Prof. Dr. Bernard Schutz
Potsdam, July 2009This work is licensed under a Creative Commons License:
Attribution - Noncommercial - Share Alike 3.0 Germany
To view a copy of this license visit
http://creativecommons.org/licenses/by-nc-sa/3.0/de/deed.en









































Published online at the
Institutional Repository of the University of Potsdam:
URL http://opus.kobv.de/ubp/volltexte/2010/4213/
URN urn:nbn:de:kobv:517-opus-42135
http://nbn-resolving.org/urn:nbn:de:kobv:517-opus-42135 Acknowledgments
I would like to thank my supervisor Prof. Bernard Schutz for his help and
enthusiasm during this work. Special thanks go also to Prof. Edward Seidel
for his guidance in the early stages of the thesis.
I am very much indebted to Sascha Husa, Manuel Tiglio, Bela Szil´agyi,
Denis Pollney and Steve White for their invaluable support and advice.
It was a privilege to work at the Max Planck Institute for Gravitational
Physics and I am grateful for having this chance.
Finally, I would like to thank Thomas and Yanis for their moral support
over the last years. Contents
1 Introduction 1
1.1 Initial (Boundary) Value Problem in Numerical Relativity . . 3
1.2 1st order time 2nd order space hyperbolic formulations . . . . 7
1.3 Numerical Discretization . . . . . . . . . . . . . . . . . . . . . 10
1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 13
2 Initial Value Problem for 2nd Order Systems in space and 1st
order in time 15
2.1 Space Discretization . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Representation of periodic functions . . . . . . . . . . . 16
2.1.2 High Order Finite Difference Operators . . . . . . . . . 19
2.1.3 Artificial Dissipation Operator . . . . . . . . . . . . . . 23
2.1.4 Fourier Symbols: Properties I . . . . . . . . . . . . . . 24
2.1.5 Fourier Symbols: Properties II . . . . . . . . . . . . . . 26
2.2 Time Integration using Runge-Kutta Methods . . . . . . . . . 31
2.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Absolute Stability of Runge-Kutta methods . . . . . . 35
2.3 Well-Posedness and Numerical Stability . . . . . . . . . . . . . 37
2.3.1 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Numerical Stability . . . . . . . . . . . . . . . . . . . . 42
2.4 Dispersion and Dissipation . . . . . . . . . . . . . . . . . . . . 46
2.4.1 Mode Splitting . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Amplification Factor and Speed Errors . . . . . . . . . 47
2.4.3 Advection Equation . . . . . . . . . . . . . . . . . . . . 48
3 Initial Value Problem for the Wave Equation 50
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Continuum Problem . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Semidiscrete Problem . . . . . . . . . . . . . . . . . . . 54
3.3.2 Courant Limits and the Role of Dissipation . . . . . . 56
3.4 Dispersion and Dissipation . . . . . . . . . . . . . . . . . . . . 60
3.5 Phase and Group Speeds . . . . . . . . . . . . . . . . . . . . . 64
3.5.1 Small Frequencies . . . . . . . . . . . . . . . . . . . . . 65
3.5.2 Scaling of the Speeds Errors with the Order of Approx-
imation when β =0. . . . . . . . . . . . . . . . . . . . 66
3.5.3 Scaling of the Speeds Errors with the Order of Approx-
imation when β =0. . . . . . . . . . . . . . . . . . . . 67
3.5.4 Speeds Errors for Different Off-Centerings at the Same
Order of Approximation . . . . . . . . . . . . . . . . . 76
3.6 Numerical Experiments in 1-D . . . . . . . . . . . . . . . . . . 85
3.6.1 Centered Scheme vs One-Point Advected Scheme . . . 86
3.6.2 Accuracy and Convergence of Higher Orders . . . . . . 87
4 Initial Boundary Value Problem for the Wave Equation 94
4.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 95
4.1.1 Well-Posedness and Strong Well-Posedness . . . . . . . 95
4.1.2 Discrete Schemes and Stability Concepts for IBVP . . 96
4.2 Continuum Problem . . . . . . . . . . . . . . . . . . . . . . . 102
4.3 Ghost-Point Method . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.1 Boundary Prescriptions . . . . . . . . . . . . . . . . . . 105
4.3.2 Equivalent Systems and Known Results . . . . . . . . . 107
64.3.3 StabilityAnalysisviaEnergyMethodforOutflowBound-
ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.4 Stability Analysis via Laplace Transform Method . . . 117
4.4 SBP-SAT Method for Inflow Boundary . . . . . . . . . . . . . 129
4.4.1 Another Continuum Energy Estimate . . . . . . . . . . 129
4.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . 131
4.5 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.5.1 General Setup . . . . . . . . . . . . . . . . . . . . . . . 136
4.5.2 Inflow-Inflow GP algorithm(|β|< 1). . . . . . . . . . . 138
4.5.3 Outflow-Completely Inflow GP algorithm(|β|> 1) . . . 139
4.5.4 Outflow-Inflow GP algorithm (β =1) . . . . . . . . . . 141
4.5.5 Inflow-Inflow SBP-SAT algorithm(|β|<1) . . . . . . . 143
4.5.6 Disscussion . . . . . . . . . . . . . . . . . . . . . . . . 147
5 BSSN System in Spherical Symmetry 149
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2 Deducing the Equations . . . . . . . . . . . . . . . . . . . . . 152
5.2.1 ADM in Spherical Symmetry . . . . . . . . . . . . . . 152
5.2.2 BSSN in Spherical Symmetry . . . . . . . . . . . . . . 153
5.2.3 Minimal System with Densitized Lapse . . . . . . . . . 161
5.2.4 Analysis of the Principal Part . . . . . . . . . . . . . . 162
5.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . 164
5.3.1 Boundary Algorithms . . . . . . . . . . . . . . . . . . . 165
5.4 Numerical Results: Linear Case . . . . . . . . . . . . . . . . . 168
5.4.1 test 1: two timelike boundaries, zero shift . . . . . . . 169
5.4.2 test 2(a): two timelike boundaries, with shift (static) . 171
5.4.3 test 2(b): two timelike boundaries, with shift (dynamic) 173
5.4.4 test 3 : spacelike inner boundary and timelike outer
boundary . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.5 Numerical Results: Nonlinear Case . . . . . . . . . . . . . . . 1756 Summary and Outlook 179
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
A Harmonic Formulation 189
B 3+1 split and ADM equations 191
C BSSN equations 193
D SBP operators 195Notations
continuum
~x = (x :::x ) space vector1 d
u;v;K;Φ vector or scalar functions
ˆ ˆuˆ;ˆv;K;Φ Fourier coefficients
~ω =(ω :::ω ) wave vector1 d
md∂ partial differential operatori :::i i1 m i1 mdx :::dx
ˆ∂ Fourier symbol associated to ∂i :::i i :::i1 m 1 m
v (v ) phase (group) speedp g
ˆ ˆ ˆ ˆP, H, T, Δ, ... expressions depending on Fourier symbols
discrete
N number of grid points
h space resolution
k time resolution
λ Courant factor
s number of offcentered points
ǫ sense of offcentering (left or right)
x grid coordinates
u;v;K;Φ grid vectors
ˆ ˆuˆ;vˆ;K;Φ discrete Fourier coefficients
ω =(ω ;:::ω ) grid wave vector1 d
ξ = (ξ ;:::ξ ) grid frequency1 d
D ;D forward, backward FDO+ −(m;n;s;ǫ) mD 2n-accurate FDO corresponding to ∂
(m;n) mD 2n-accurate CFDO corresponding to ∂
(m;n)
D 2n-accurate CFDO corresponding to ∂i :::ii :::i 1 m1 m
(m;n)(m;n;s;ǫ) (m;n)ˆ ˆ ˆ ˆD , D , D , D discrete Fourier symbols± i :::i1 m
(n;s) (n) (n;s) (n)
v ;v ;v (v ;v ;v ) numerical phase (group) speedsp p g gp g
(n;s) (n) (n;s) (n)
ǫ ;ǫ ;ǫ (ǫ ;ǫ ;ǫ ) numerical phase (group) speed errorsp p g gp g
a amplification factorp
ˆ ˆ ˆ ˆP, H, T, Δ, ... expressions depending on discrete Fourier symbols