TESTING BINARY BLACK HOLE CODES IN STRONG FIELD REGIMES

Transcription

1 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

2 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

3 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.

4 iv Table of Contents List of Tables viii List of Figures ix Acknowledgments xv Preface xvi Chapter 1. Introduction Chapter 2. Computing Environment Software Cactus Maya Hardware Chapter 3. ADM Derivation Gauge: Lapse Function and Shift Vector The 3-metric evolution equation The extrinsic curvature evolution equation The Hamiltonian Constraint The Momentum Constraint

5 v 3.2 Gauge Choices Periodic Boundary Conditions Finite Differencing Chapter 4. Issues of Stability Definitions L2 norm Form of Equations Well-posedness Defining an unstable evolution Consistency, Convergence, Dispersion and Dissipation Convergence Tests Modes Unanswered Questions The Test Chapter 5. Nearly Trivial Solutions: Perturbed Flat-space & Kasner Perturbed Flat-space Error growth for different resolutions Error growth for different amplitudes Testing the stability of Iterated Crank-Nicholson Kasner Introduction Convergence Tests

6 vi Errors in the Kasner spacetime Type I Stability Tests Conclusion Chapter 6. Cosmological Spacetimes: Gowdy & Bondi Bondi Waves Non-linear Periodic Plane Waves Introduction to Bondi Waves Convergence Tests for Bondi Type I Stability Analysis Gowdy Introduction Convergence Tests for Gowdy Type I Stability Analysis Conclusion Chapter 7. The Gauge Wave Solution & Stability Gauge Wave Derivation Convergence Tests Type I Stability Analysis Why Gauge Wave is unstable for large A Type II Stability in all the spacetimes Conclusion

7 vii Chapter 8. Results Summary The effect of constraint violations on Stability Conclusion Appendix A. Additional plots for the Bondi System Appendix B. Additional plots for the Gowdy System Appendix C. Additional plots for the Gauge Wave System References

8 viii List of Tables 5.1 Error growth rate of perturbed flat-space for different amplitudes Convergence test specifications for Kasner Convergence test specifications for Bondi Convergence test specifications for Bondi Convergence test specifications for Gowdy Convergence test specifications for Gauge Wave

9 ix List of Figures 3.1 The above drawing shows how four dimensional spacetime is sliced into 3D space-like hypersurfaces Noise 1: Above shows the value of the L2 norm of the g xx 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 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 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 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 Kasner 2: This is a close-up how the Kasner g yy metric component grows with time. Notice that the numerical solution lags the analytic solution

10 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 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 analytic solutions (ex. neg yy = g yy agyy gyy + agyy ). The normalized constraint values are defined by dividing the constraint values by the sum the the absolute values of their components (ex. nham = R+K 2 K ij K ij R + K 2 + K ij K ij ). Notice that the constraints begin to explode before the numerical errors Kasner 5: The above shows how the errors in both g xx and g yy 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 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

11 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 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 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 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 g xx are shown in Fig B 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 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 Gauge Wave 1: These are the results of the conversion tests for the g xx component of the gauge wave spacetime

12 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 Gauge Wave 3: Error in the gauge wave metric versus time with an amplitude A = 1.0 using a Courant factor of 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 = Notice the loss of convergence around t = 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 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 Consistency 3: The consistency of the Gowdy system. Notice the fine solution explodes first proving that the system is type II unstable Consistency 4: The consistency of the Gauge wave system. Here the medium solution explodes first followed by the fine and coarse solutions 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 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

13 xiii 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 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 A.1 Bondi Wave 1: Bondi metric at t = 1. These next two plots demonstrate the dispersion seen in the Bondi system A.2 Bondi Wave 2: Bondi metric at t = 9. Notice the growing phase shift in the solutions A.3 Bondi Wave 2: Bondi metric at t = 17. As time increases the phase error continues to grow 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 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

14 xiv B.2 Gowdy 2: Shows the value of eg yy at different times immediately before during and after the hyperconvergence peak at t = Notice how the error oscillates through a value near zero 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 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 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 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 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 C.5 Gauge Wave 5: Error in the gauge wave metric versus time with an amplitude A = 1.0 using a Courant factor of This is the result mentioned in section

15 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.

16 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,

17 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

18 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.

19 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.

20 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 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.

21 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.

22 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 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 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

23 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.

24 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 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 K ij 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

25 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 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

26 gnuplot are used to visualize the output and MapleV was used to calculate the initial data. Additional analysis software was used on other computers. 8

27 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-Deser- Misner [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) g µν. The spacetime (M, (4) g µν ) 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 into R τ where τ is a set of spacelike hypersurfaces that are parameterized by τ. The collection of all τ form the foliation of the spacetime manifold.

28 10 Σ dt αn β t(x α ) = dt n t Σ 0 t(x α ) = 0 Fig The above drawing shows how four dimensional spacetime is sliced into 3D space-like hypersurfaces 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, n µ = α µ τ (3.2) Because coordinates may shift between hypersurfaces, a timelike vector field can then be defined as, t i = αn i + β i. (3.3)

29 β i is a spacelike vector which is referred to as the shift vector. It is always tangent to the hypersurface. t i are then tangent vectors to the worldlines of coordinate observers on τ and α dt is the proper time between hypersurfaces. α and βi 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 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 ν δµ ν + nµ n ν is the projection tensor that projects v µ M onto τ. Nµ ν nµ n ν projects vectors in M along n µ orthogonal to τ. The 3-metric g ij then describes the geometry of τ and is found by projecting (4) gµν onto τ. g ij = µ i ν (4) gµν = (4) g j ij (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. The covariant derivative on τ is then defined as D i so that D k g ij = 0. The covariant derivative of a tensor can be obtained by projecting all free components of the 4D covariant derivative onto τ. Example, D k T i 1 i 2...i N j 1 j 2...j N = i 1 µ1... i N µn ν 1 j 1... ν N jn η k η T µ 1 µ 2...µ N ν 1 ν 2...ν N (3.5)

30 12 The extrinsic curvature, K ij, describes the curvature of τ relative to the manifold M. It is often thought of as the rate of change of g ij. It is defined as K ij (i n j). (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 n j = δ µ i δν j µ n ν... = ( µ i nµ n i )( ν j nν n j ) µ n ν... = i n j n i n k k n j (3.7) Here n k k n j is the acceleration of the normal a j. By using the definition of the normal, n ν = (4) g νη n η, 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, K ij = 1 2 L n g ij. (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, L t g ij = L αn g ij + L β g ij. (3.9) Using the definition of the extrinsic curvature the evolution equation for the metric becomes: t g ij = 2αK ij + L β g ij. (3.10)

31 The extrinsic curvature evolution equation Starting with the Einstein tensor, we can derive the constraint equations and evolution equations for the extrinsic curvature. G µν (4) R µν 1 2 (4) gµν (4) R (3.11) The evolution equation for the extrinsic curvature is given by G ij while the Hamiltonian constraint is given by G ij n i n j and the momentum constraints are given by G ij n i. 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) R ijkl = R ijkl + K ik K jl K il K jk (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) R ijkl n l = D j K ik D i K jk. (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) R ikjl n k n l = L n K ij + K ik K k j + a i a j + D i a j. (3.14)

32 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: L t K ij = αl n K ij + L β K ij. (3.15) Using Ricci s equation and the identity a i = D i lnα, L t K ij = α( (4) R ikjl n k n l K ik K k j ) D i D j α + L β K ij. (3.16) The relationship, (4) g µν η σ π ρ ξ µ ζ Rηξπζ = ν(4) η σ π (4) Rηπ + n ξ n ζ η ρ σ π (4) Rηπξζ (3.17) ρ along with Gauss equations we can simplify the Riemann tensor term so that (4) R ikjl n k n l = R ij + KK ij K ki K k j µ i ν (4) Rµν. (3.18) j We now replace the expression µ i ν j (4) R µν with 8π(S ij 1 2 g ij (S ρ)), where S ij = µ i ν j T µν 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 K ij = α(r ij + KK ij 2K ki K k j ) D i D j α + L β K ij. (3.19)

33 The Hamiltonian Constraint The Hamiltonian constraint is obtained by projecting G ij along the normals. G ij n i n j = 1 2 ((4) R ij n i n j (4) R) (3.20) Using the expression, (4) g µν g σρ η σ π ρ ξ µ ζ Rηπξζ = ν(4) (4) R ηπ n η n π + 1 (4) R (3.21) 2 we see that the Hamiltonian constraint can be simplified to, G ij n i n j = 1 2 (R + K2 K ij K ij ). (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. i k nj G ij = i k nj ( (4) R ij 1 2 (4) gij (4) R) (3.23) Because i k n i = 0 this equation becomes, i k nj G ij = i k nj(4) R ij. (3.24)

34 By using Codazzi s equation we get the final form of the expression for the momentum constraint, 16 i k nj G ij = D j K j k D kk. (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, ds 2 = α 2 dt 2 + g ij (dx i + β i dt)(dx j + β j dt). (3.26) By substituting the L β terms for their explicit forms, the evolution and constraint equations become, t g ij 2αK ij + 2D (i β j) (3.27) t K ij α(r ij + KK ij 2K ki K k j ) D i D j α + βk D k K ij 2K k(i D k β j) (3.28) H R + K 2 K ij K ij = 0 (3.29)

35 17 M k D j K j k D kk = 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

36 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) 2 x + O( x 2 ) (3.31) and 2 u(x + x, t) 2u(x, t) + u(x x, t) u(x, t) = x x 2 + O( x 2 ) (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, t u(x, t) = A(u, x u, 2 u, t). (3.33) x 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

37 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 ). u n+1 j (2)ũn+1 j (1)ũn+1 j = u n j + ta(1 2 ((2) ũ n+1 j = u n j + ta(1 2 ((1) ũ n+1 j + u n )) (3.34) j + u n )) (3.35) j = u n j + ta(un 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 x 2 we expect the total numerical errors to vary as x 2 or second order, this will be important when we discuss convergence.

38 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 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: L2 norm = 1 N N (u k ) 2. (4.1) k=1

39 Form of Equations We begin with a linear second order PDE in terms of two variables [30], au xx + 2bu xy + cu yy + du x + eu y + fu = g (4.2) By replacing the differentials with greek indices (u xx = α 2, u xy = αβ, u yy = β 2, u x = α and u y = β) we get the second degree polynomial: P (α, β) aα 2 + 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 t 2 u + 2 u = RHS. An example of this is the x Poisson equation, 2 x u + 2 u = ρ(x, y). (4.4) y Parabolic PDEs have the form t u - x 2 u = RHS like in the diffusion equation, t u = x (D x u). (4.5) Hyperbolic PDEs have the form t 2 u - 2 u = RHS. A common example of this is the x one dimensional wave equation, 2 t u = v2 2 u. (4.6) x

40 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 u ij is broken down into t u ij - L β u ij. Here L β u ij contains the advective term β k k u ij 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

41 23 numerical solutions satisfy u(, t) Ke αt f( ). (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 Defining an unstable evolution If not for years of computational experience, no one would expect unstable evolutions 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:

42 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.

43 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. u n k fn 0 x, t 0 (4.9) k Here u n k is the numerical solution and f n k is the analytic solution. According to the 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