Project A5;     (2007 - 2014)

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Numerical Methods for General Relativity

    Principal Investigators: Bernd Brügmann, Sascha Husa, Christian Lubich, Gerhard Zumbusch

Goal of project A5 is the development of robust and efficient methods for the numerical solution of the Einstein equations. Gravitational wave astronomy requires numerical solutions of high reliability and accuracy for the theoretical prediction and analysis of gravitational waves. Focus areas are reformulations of the Einstein equations as a first order in time and second order in space system, numerical methods for the Einstein evolution problem including structure preservation, efficient elliptic solvers for adaptive meshes and black hole geometries, and the adaptation of these numerical algorithms to wave problems.

Current work in the gravitational wave community is focused both on getting first results from the detectors that are currently achieving design sensitivity, and also on developing the next generation of ground based and the first generation of space based detectors, which are expected to reach the sensitivities required by gravitational wave astronomy.

The task for numerical relativity (NR) is twofold as well: First, to produce results with state-of-the-art methods and second, the development of a new generation of numerical codes to bridge the wide gap from current technology to the demand in terms of robustness and efficiency, to perform the type of parameter studies called for by gravitational wave astronomy.
The latter is the aim of project A5.

NR has become a very rapidly developing field: in the last years several breakthroughs have been obtained both regarding the mathematical basis of the field, but also several groups have obtained a first orbits of two black holes.

The German numerical relativity community has traditionally been very strong at performing pioneering large scale simulations of black holes, which have produced first glimpses into uncharted territory of black hole dynamics. However, to develop a counterpart to the strong US efforts at developing entirely new techniques, often with a direct application to the binary black hole problem in mind, will require a major effort involving scientists with expertise ranging from astrophysical simulations to applied mathematics. The SFB/TR7 provides an ideal platform to work on such a project and integrate it with existing efforts.

The basis of this project is a clear view on the key methodological problems in NR, all of which will be addressed to some degree in at least one of the SFB projects, and will be relevant to the work programme of project A5. Naturally, we will define some focus areas, which we outline briefly along with our general view of the problem.

  • A central problem of computational physics is to make the physical features manifest in the discrete system, e.g. to preserve structure during a time integration. This is particularly hard in NR, essentially due to diffeomorphism invariance. Writing the Einstein equations in the form of an initial value problem, the number of computational degrees of freedom (d.o.f.) is much larger than the physical d.o.f., since it is not known in general how to separate physical from gauge and constraint violating d.o.f. Due to the complicated nonlinear structure of the Einstein equations it has not yet been possible to directly carry over techniques that solve conceptually related issues in other gauge theories such as the Maxwell equations. Consequently, numerical relativity simulations are typically plagued by instabilities, which are often rooted in the continuum formulation of the problem.
    This project approaches this problem along several lines: studies of the continuum equations, alternative discretization procedures to preserve the constraints, and investigation of constrained evolution schemes.

  • The main study of wave emission from compact objects requires resolution at different scales: a code needs to resolve the compact objects, their orbits, emitted waves and a slowly varying background. In order to obtain accurate results both the use of mesh refinement techniques and a good choice of coordinate gauges is essential. Here the focus of this project will be on efficient techniques for mesh refinement, in particular also concerning efficient solvers for elliptic equations.

  • Given that at least from the observational point of view the most valuable result of a GW source simulation is the GW signal, it seems natural to design codes with wave extraction in mind. This is difficult for various reasons. First, even though some gravitational wave sources will be among the brightest objects in the universe, gravitational waves are typically only a small effect in the energy balance of such systems.

    Second, in GR such fundamental quantities as energy, momentum, or emitted gravitational radiation can only be defined unambiguously in terms of asymptotic limits. Consequently, it also becomes particularly difficult to formulate `outgoing radiation boundary conditions' at finite distance from the sources. For black hole simulations the domain of interest typically has two types of boundary, those through which "radiation leaves the system", and those which isolate the computational domain from the singularity inside the black holes. Both can be realized in many ways, in certain approaches not as actual boundaries in the immediate sense, but as regions that act as "dissipative sponges", sometimes containing formally singular points. Many approaches have been used with some success, however, none is sufficiently understood, e.g. to formulate "best practices".

    In this context we will mainly work along two directions: First, to continue work roughly along the lines of projects A1 and A2, and second by trying to adapt techniques from numerical analysis to handle radiation problems.

  • Given the abundance of options and approaches, and the subtleties of diffeomorphism invariance, setting up quality criteria for numerical codes is as important as difficult. A major effort in this direction is the `Apples with Apples´ collaboration to compare codes, and this project would draw major benefit and make major contributions to this project.

Integration of new methods into the existing computational infrastructure within the SFB poses a significant software engineering problem, which we will address from the start. Our work will be placed into the context of the development of both the Cactus and BAM frameworks which are used within the SFB, and utilizing our tools of automatized (and at least partially cross-platform) code generation.

Researchers

  Bernd Brügmann   Professor, PI 2007
  Nathan Johnson-McDaniel   Postdoc, 2011-2014
  Georgios Loukes-Gerakopoulos   Postdoc, 2011-2014
  Christian Lubich   Professor, PI 2007
  Charalampos Markakis   Postdoc, 2011-2014
  Gerhard Zumbusch   Professor, PI 2007-2014

Former Associates
  Thomas Fischer   PhD Student, 2007
  Julia Gundermann   Student, 2009-2010
  Sascha Husa   Postdoc, PI 2007
  Ronald Laesker   PhD Student, 2007
  Norbert Lages   PhD Student, 2007-2010
  Jonathan McDonald   Postdoc, 2009-2010
  Doreen Mueller   PhD Student, 2010
  Andrea Nerozzi   Postdoc, 2007-2008
  Hans-Peter Nollert   Staff, 2003
  Rodrigo Panosso Macedo   Postdoc, 2010
  Frank Peuker   PhD Student, 2007-2009
  Ronny Richter   Postdoc, 2007-2010
  Andreas Weyhausen   PhD Student, 2009-2013

Publications

[1] Spinning black hole in the puncture method: Numerical experiments
T.Dietrich, B. Brügmann, J.Phys.Conf.Ser.490 012155

[2] Solving the Hamiltonian constraint for 1+log trumpets
T.Dietrich, B. Brügmann, Phys. Rev. D, 89 024014

[3] Vectorized Higher Order Finite Difference Kernels
G. Zumbusch, PARA 2012, State-of-the-Art in Scientific and Parallel Computing

[4] Collapse of Nonlinear Gravitational Waves in Moving-Puncture Coordinates
D. Hilditch, T. Baumgarte, A. Weyhausen, T. Dietrich, B. Brügmann, P. Montero, E. Müller, Phys. Rev. D, 88 103009

[5] An Introduction to Well-posedness and Free-evolution
D. Hilditch, Int.J.Mod.Phys. A28 1340015

[6] Hyperbolicity of Physical Theories with Application to General Relativity
D. Hilditch, R. Richter, preprint

[7] A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
B. Brügmann, J.Comput.Phys. 235 216-240

[8] Tuning a Finite Difference Computation for Parallel Vector Processors
G. Zumbusch, 2012 11th International Symposium on Parallel and Distributed Computing

[9] Tuning a Finite Difference Stencil
G. Zumbusch, Poster, GPU Technology Conference 2012 (GTC), San Jose, CA

[10] Hyperbolic formulations of General Relativity with Hamiltonian structure
D. Hilditch and R. Richter, Phys.Rev. D86 123017

[11] Numerical stability of the Z4c formulation of general relativity
Z. Cao, D. Hilditch, Phys. Rev. D, 85 124032

[12] Constraint damping for the Z4c formulation of general relativity
A. Weyhausen, S. Bernuzzi, D. Hilditch, Phys. Rev. D, 85 024038

[13] Stability of the puncture method with a generalized BSSN formulation
H. Witek, D. Hilditch, U. Sperhake, Phys. Rev. D, 83 104041

[14] Hyperbolicity of Hamiltonian formulations in General Relativity
R. Richter, D. Hilditch, J.Phys.Conf.Ser. 314 012102

[15] The trumpet solution from spherical gravitational collapse with puncture gauges
M. Thierfelder, S. Bernuzzi, D. Hilditch, B. Brügmann, L. Rezzolla, Phys. Rev. D, 83 064022

[16] Symplectic Integration of Post-Newtonian Equation of Motion with Spin
C. Lubich, B. Walther and B. Bruegmann, Phys. Rev. D, 81 104025

[17] Constraint preserving boundary conditions for the Z4c formulation of general relativity
M. Ruiz, D. Hilditch, S. Bernuzzi, Phys. Rev. D, 83 024025

[18] Portable Multi-Level Parallel Programming for Cell processor, GPU, and Clusters
G. Zumbusch, accepted for Proc. Para08, LNCS, Springer (11/2009), A. Elster (ed)

[19] Coupling Non-Gravitational Fields with Simplicial Spacetimes
R. McDonald, W. A. Miller, Class. Quantum Grav., 27 095011 (2010)

[20] Discretization of the Cauchy problem for second order in space, first order in time systems using high order finite difference operators
M. Chirvasa, S. Husa, J. Comput. Phys., 229 2675–2696 (2010)

[21] Discrete differential forms for 1+1 dimensional cosmological space-times
R. Richter, and J. Frauendiener, accepted by SIAM J. Sci. Comp. (2009)

[22] Bowen-York trumpet data and black-hole simulations
M. Hannam, S. Husa, and N. O Murchadha, Phys. Rev. D, 80 124007 (2009)

[23] Schwarzschild black hole as moving puncture in isotropic coordinates
B. Bruegmann, Gen. Rel. Grav., 41 2131–2151 (2009)

[24] Strongly hyperbolic Hamiltonian systems in numerical relativity Formulation and symplectic integration
R. Richter, Class. Quant. Grav., 26 145017 (2009)

[25] Finite element, discontinuous Galerkin, and finite difference evolution schemes in spacetime
G. Zumbusch, Class. Quantum Grav., 26 175011 (2009)

[26] Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem
A. Zenginoglu, D. Nunez, and S. Husa, Class. Quantum Grav., 26 035009 (2009)

[27] Stationary hyperboloidal slicings with evolved gauge conditions
F. Ohme, M. Hannam, S. Husa, and N. O Murchadha, Class. Quantum Grav., 26 175014 (2009)

[28] Gravitational-wave detectability of equal-mass black-hole binaries with aligned spins
C. Reisswig, S. Husa, L. Rezzolla, E. N. Dorband, D. Pollney and J. Seiler, Phys. Rev. D, 80 124026 (2009)

[29] Searching for numerically-simulated signals of black hole binaries with a phenomenological template family
L. Santamaria, B. Krishnan and J. T. Whelan, Class. Quantum Grav., 26 114010 (2009)

[30] Symplectic Time Integrators for Numerical General Relativity
R. Richter, in K. E. Kunze, M. Mars, and M. A. Vazquez-Mozo (eds), ”Physics and Mathematics of Gravitation”, AIP Conference Proceedings, 1122 376–379 (2009)

[31] Strongly hyperbolic Hamiltonian systems in numerical relativity: Formulation and symplectic integration
Ronny Richter, Class.Quant.Grav.26:145017

[32] Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem
A. Zenginoglu, D. Nunez, S. Husa, Class. Quantum Grav. 26 (2009) 035009

[33] Stationary hyperboloidal slicings with evolved gauge conditions
F. Ohme, M. Hannam, S. Husa, N. O'Murchadha, Class. Quantum Grav. 26 (2009) 175014

[34] Numerical black hole initial data with low eccentricity based on post-Newtonian orbital parameters
B. Walther, B. Brügmann and D. Müller, Phys.Rev.D79 (2009), 124040

[35] Discretization of the Cauchy problem for second order in space, first order in time systems using high order finite difference operators.
M. Chirvasa and S. Husa, submitted to J. Comp. Phys.

[36] Schwarzschild black hole as moving puncture in isotropic coordinates
B. Bruegmann, (2009)

[37] Application of Discrete Differential Forms in numerical General Relativity
R. Richter, J. Frauendiener, in A. Oscoz, E. Mediavilla and M. Serra-Ricart (eds), "Spanish Relativity Meeting - Encuentros Relativistas Espanoles ERE2007 Relativistic Astrophysics and Cosmology", EAS Publications Series, 30 (2008) 219-222

[38] Using curvature invariants for wave extraction in numerical relativity
A. Nerozzi, and O. Elbracht, preprint

[39] Resolving Super Massive Black Holes with LISA
S. Babak, M. Hannam, S. Husa, and B. Schutz, preprint

[40] Wormholes and trumpets: The Schwarzschild spacetime for the moving-puncture generation
Mark Hannam, Sascha Husa, Frank Ohme, Bernd Brügmann , Niall O'Murchadha, Phys. Rev. D 78, 064020 (2008)

[41] Reducing eccentricity in black-hole binary evolutions with initial parameters from post-Newtonian inspiral
S. Husa, M. Hannam, J. A. Gonzalez, U. Sperhake, and B. Bruegmann, Phys. Rev. D, 77 044037 (2008)

[42] Implementation of standard testbeds for numerical relativity
M. C. Babiuc, S. Husa, D. Alic, I. Hinder, C. Lechner, E. Schnetter, B. Szilagyi, Y. Zlochower, N. Dorband, D. Pollney, and J. Winicour, Class. Quant. Grav., 25 125012 (2008)

[43] Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing
S. Husa, J. A, Gonzalez, M. Hannam, B. Br ̈gmann, and U. Sperhake, Class. Quant. Grav., 25 105006 (2008)

[44] Application of Discrete Differential Forms in numerical General Relativity
R. Richter and J. Frauendiener, in A. Oscoz, E. Mediavilla, and M. Serra-Ricart (eds), ”Spanish Relativity Meeting - Encuentros Relativistas Espanoles ERE2007 Relativistic Astrophysics and Cosmology”, EAS Publications Series, 30 219-222 (2008)

[45] A container-iterator parallel programming model
Gerhard Zumbusch, Parallel Processing and Applied Mathematics, volume 4967 of LNCS, pages 1130-1139, Heidelberg, 2008. Springer., R. Wyrzykowskii, J. Dongarra, K. Karczewski, and J. Wasniewski, editors

[46] Free and constrained symplectic integrators for numerical general relativity
Ronny Richter, Christian Lubich, Class.Quant.Grav.25:225018

[47] Numerical modeling of black holes as sources of gravitational waves in a nutshell
S. Husa, Eur. Phys. J., Special Topics 152 183–207 (2007)

[48] Phenomenological template family for black-hole coalescence waveforms
P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, J. T. Whelan, B. Bruegmann ,P. Diener, J. Gonzalez, M. Hannam, S. Husa, M. Koppitz, D. Pollney, L. Rezzolla, L. Santamaria, A. M. Sintes, U. Sperhake, and J. Thornburg, Class. Quant. Grav., 24 S689–S700 (2007)

[49] Numerical calculations near spatial infinity
A. Zenginoglu, Journal of Physics Conference Series 66(1), (2007)

[50] Inspiral, merger and ringdown of unequal mass black hole binaries: a multipolar analysis
E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. Hannam, S. Husa, B. Brügmann, Phys.Rev.D76, 064034 (2007)

[51] Implementation of standard testbeds for numerical relativity
M. C. Babiuc, S. Husa, I. Hinder, C. Lechner, E. Schnetter, B. Szilagyi, Y. Zlochower, N. Dorband, D. Pollney, J. Winicour, Class.Quant.Grav.25:125012,(2008)

[52] Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing
S. Husa, J. A, Gonzalez, M. Hannam, B. Brügmann, U. Sperhake, Class.Quant.Grav.25:105006 (2008)

[53] Numerical modeling of black holes as sources of gravitational waves in a nutshell
S. Husa, Eur. Phys. J. Special Topics 152, 183-207 (2007)

[54] Data dependence analysis for parallel tree codes
Gerhard Zumbusch, Applied Parallel Computing. State of the Art in Scientific Computing, volume 4699 of LNCS, pages 890-899, Heidelberg, 2007. Springer. DOI 10.1007/978-3-540-75755-9_106., B. Kagström, E. Elmroth, J. Dongarra, and J. Wasniewski, editors

[55] Binary black holes on a budget: Simulations using workstations
P. Marronetti, W. Tichy, B. Brügmann, J. A. Gonzalez, M. Hannam, S. Husa, U. Sperhake, Class.Quant.Grav.24, S43-S58 (2007)

[56] Geometry and Regularity of Moving Punctures
M. Hannam, S. Husa, D. Pollney, B. Brügmann, and N. 'O Murchadha, Phys.Rev.Lett.99, 241102 (2007)

[57] Beyond the Bowen-York extrinsic curvature for spinning black holes
M. Hannam, S. Husa, B. Brügmann, J. A. Gonzalez, U. Sperhake, Class.Quant.Grav.24, S15-S24 (2007)

[58] Where do moving punctures go?
M. Hannam, S. Husa, N. O Murchadha, B. Brügmann, J. A. Gonzalez, U. Sperhake, J.Phys.Conf.Ser.66, 012047 (2007) Proceedings for 29th Spanish Relativity Meeting

Theses

[59] Spektralverfahren auf GPUs
Jörg Feierabend, Diplomarbeit, University of Jena

[60] Konvergenzuntersuchungen bei Lösungen der Wellengleichung mittels finiter Differenzen
Roland Wickles, Bachelor Thesis, University of Jena

[61] Finite Difference Methods for 1st order in time, 2nd order in space, Hyperbolic Systems used in Numerical Relativity
Mihaela Chirvasa, PhD Thesis, University of Potsdam

[62] Apparent Horizons and Marginally Trapped Surfaces in Numerical General Relativity
Norbert Lages, PhD Thesis, University of Jena

[63] Simplicial Methods for Solving Selected Problems in General Relativity Numerically: Regge Calculus and the Finite-Element Method
Frank Peuker, PhD Thesis, University of Jena

[64] Loesung elliptischer Randwertprobleme mit Hilfe der CUDA Technologie
C. Reibiger, Diplomarbeit, University of Jena

[65] Schurkomplement-Vorkonditionierer fur pseudospektrale Diskretisierungen
A. Boos, Diplomarbeit, University of Jena

[66] The Hamiltonian constraint for puncture initial data
J. Gundermann, Diplomarbeit, University of Jena

[67] Numerical Algorithms of General Relativity for Heterogeneous Computing Environments
A. Weyhausen, Diplomarbeit, University of Jena