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Computing and Analyzing Gravitational Radiation in Black Hole Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2007 Computing and analyzing gravitational radiation in black hole simulations using a new multi-block approach to numerical relativity Ernst Nils Dorband Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Physical Sciences and Mathematics Commons Recommended Citation Dorband, Ernst Nils, "Computing and analyzing gravitational radiation in black hole simulations using a new multi-block approach to numerical relativity" (2007). LSU Doctoral Dissertations. 148. https://digitalcommons.lsu.edu/gradschool_dissertations/148 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. COMPUTING AND ANALYZING GRAVITATIONAL RADIATION IN BLACK HOLE SIMULATIONS USING A NEW MULTI-BLOCK APPROACH TO NUMERICAL RELATIVITY A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Ernst Nils Dorband Vordiplom, Julius-Maximilians-Universit¨at, W¨urzburg,1999 M.S., Rutgers - The State University of New Jersey, 2001 May 2007 Acknowledgements I want to thank my adviser Manuel Tiglio. With his deep knowledge of physics and numerical analysis, he gave me a whole new perspective about numerical relativity. He always took his time to discuss projects and problems and motivated me to go on during occasionally frustrating times. Over the last years, he became a precious friend. Regards go to Ed Seidel who supported me since I first started to work in the field and pulled a lot of strings in the background to make my move to Louisiana possible. I am grateful to my parents Monika and Immo for everything they do for me, which is more than would fit on this page. Thanks to my lovely girlfriend Jee Yeoun. You were always there for me and made my final year in Baton Rouge a very special one. A work like this can only be done within a strong collaboration and I was lucky to work closely together with many talented scientists. I explicitly want to mention Emanuele Berti, Peter Diener, Alessandro Nagar, Carlos Palenzuela, Enrique Pazos and Erik Schnetter. Thanks to all of them for contributing to the research presented here. I thank Steve Brandt, Vitor Cardoso, Mark Carpenter, Barrett Deris, Heinz-Otto Kreiss, Luis Lehner, Jos´eM. Mart´ın-Garc´ıa, Ken Mattsson, Frank Muldoon, Harald Pfeiffer, Jorge Pullin, Olivier Sarbach, Magnus Sv¨ard, Saul Teukolsky, Jonathan Thornburg and Burkhard Zink for many helpful discussions and suggestions. Special thanks go to Jorge Pullin for help with organizational issues at the department of physics and to Bernd Br¨ugmann, who introduced me to the field of numerical relativity back in 2001. I want to acknowledge Saul Teukolsky and the relativity group at Cornell University for their hospitality. Some crucial parts of this work were accomplished during a number of research retreats that were held there. This research was supported in part by NSF grants PHY-0505761, PHY-0244699, PHY- 0326311, PHY-0554793, NASA grant NAG5-1430, and the National Center for Supercomputer Applications grant MCA02N014 to Louisiana State University. It also employed the resources ii of the Center for Computation & Technology at Louisiana State University, which is supported by funding from the Louisiana legislature’s Information Technology Initiative. The numerical calculations used the Cactus framework [1, 2] with a number of locally de- veloped thorns, the LAPACK [3] and BLAS [4] libraries from the Netlib Repository [5], and the LAM [6, 7, 8] and MPICH [9, 10, 11] MPI [12] implementations. iii Table of Contents Acknowledgements ................................. ii Abstract ........................................ vi 1 Introduction .................................... 1 1.1 Background and Outline . 1 1.2 Notation and Conventions . 5 2 A Multi-Block Infrastructure for Numerical Relativity ......... 7 2.1 Introduction . 7 2.2 Topologies and Coordinate Systems for Multi-Block Simulations . 8 2.2.1 How Can Numerical Relativity Profit from Multi-Block Methods? . 8 2.2.2 Design Choices for Multi-Block Grid Structures . 13 2.2.3 Examples for Multi-Block Grid Structures . 15 2.2.4 The Cubed Sphere Coordinates . 17 2.2.5 Coordinates and Tensor Bases . 20 2.3 Stable Discretization of Hyperbolic Partial Differential Equations . 23 2.3.1 The Summation by Parts Condition . 23 2.3.2 Penalty Boundary Conditions . 26 2.3.3 Construction of Summation-by-Parts Finite-Differencing Operators 29 2.3.4 Artificial Dissipation . 33 2.4 Time Integrator . 35 2.5 Computational Infrastructure and Parallelization . 36 3 Numerical Accuracy and Convergence ................... 41 3.1 Linear Scalar Wave Equation on a Minkowski Background . 41 3.2 Formulation of the Evolution Equations, Initial Data and Boundary Con- ditions . 42 3.3 Grid Structure . 44 3.4 Results . 46 3.5 Local vs. Global Coordinate Basis . 49 4 Quasinormal Mode Excitation of Kerr Black Holes ........... 53 4.1 Introduction . 53 4.2 The Continued Fractions Method for Computing Quasinormal Modes . 56 4.3 Evolution Equations and Background Geometry . 59 4.4 Initial and Boundary Conditions . 60 4.5 Multi-Block Setup . 61 4.6 Specifications for the Simulations . 62 4.7 Overview of Quasinormal Mode Extraction . 63 4.8 The Time-Shift Problem . 69 4.9 Optimal Choice of Fitting Interval . 72 4.10 Rotational Mode Mixing . 76 iv 4.11 Excitation Factors for Co- and Counterrotating Modes . 79 4.12 Rapidly Spinning Black Holes and Overtones . 87 4.13 Summary . 89 5 Comparison of Wave Extraction Methods in Non-Linear Black Hole Evolutions ....................................... 93 5.1 Introduction . 93 5.2 Generalized Harmonic System . 96 5.2.1 Harmonic Coordinates . 97 5.2.2 Gauge Sources . 97 5.2.3 Relation Between Standard 3+1 Quantities and Evolution Variables of the Generalized Harmonic System . 98 5.2.4 Constraint Evolution System . 99 5.2.5 Relation Between the Generalized Harmonic Constraints and the Standard Hamiltonian and Momentum Constraints . 101 5.2.6 Modified Generalized Harmonic System with Constraint Damping . 101 5.2.7 First Order Generalized Harmonic System . 103 5.3 The Background Metric and Tensor Spherical Decomposition of the Per- turbations . 106 5.4 Extraction of the Regge-Wheeler Function from a Given Geometry . 108 5.5 Construction of Initial Data for Perturbed Black Holes . 110 5.6 Description of the Simulations . 116 5.7 The Standard and Generalized RW Approaches: Numerical Comparisons . 117 5.8 Quasinormal Frequencies . 124 5.9 Summary . 128 6 Conclusions .................................... 131 References ....................................... 135 Appendix A Energy Method .................................. 147 B Notes on Comparing Numerical Results from Chap. 4 with the Semi- Analytic Predictions ................................ 155 C 3+1 Split ...................................... 157 D Vector and Tensor Harmonics ......................... 165 Vita ........................................... 169 v Abstract Numerical simulations of Kerr black holes are presented and the excitation of quasinormal modes is studied in detail. Issues concerning the extraction of gravitational waves from numerical space-times and analyzing them in a systematic way are discussed. A new multi-block infrastructure for solving first order symmetric hyperbolic time de- pendent partial differential equations is developed and implemented in a way that stability is guaranteed for arbitrary high order accurate numerical schemes. Multi-block methods make use of several coordinate patches to cover a computational domain. This provides efficient, flexible and very accurate numerical schemes. Using this code, three dimensional simulations of perturbed Kerr black holes are car- ried out. While the quasinormal frequencies for such sources are well known, until now little attention has been payed to the relative excitation strength of different modes. If an actual perturbed Kerr black hole emits two distinct quasinormal modes that are strong enough to be detected by gravitational wave observatories, these two modes can be used to test the Kerr nature of the source. This would provide a strong test of the so called no hair theorem of general relativity. A systematic method for analyzing ringdown wave- forms is proposed. The so called time shift problem, an ambiguity in the definition of excitation amplitudes, is identified and it is shown that this problem can be avoided by looking at appropriately chosen relative mode amplitudes. Rotational mode coupling, the relative excitation strength of co- and counter rotating modes and overtones for slowly and rapidly spinning Kerr black holes are studied. A method for extracting waves from numerical space-times which generalizes one of the standard methods based on the Regge-Wheeler-Zerilli perturbation formalism is pre- sented. Applying this to evolutions of single perturbed Schwarzschild black holes, the accuracy of the new method is compared to the standard approach and it is found that the errors resulting from the former are one to several orders of magnitude below
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