Introduction
The Spectral Einstein Code (SpEC) is a flexible infrastructure for solving partial differential equations using multi-domain spectral methods. While SpEC was primarily designed for fully general-relativistic compact object simulations, it can be applied to a wide range of hyperbolic and elliptic equations. Some of its features are:
- A flexible domain decomposition supporting individual subdomains (cells, elements) of various topologies, e.g. blocks, spheres, cylinders, and any combination thereof. Subdomains can touch or overlap.
- Solves hyperbolic and elliptic PDEs.
- Interfaces to the visualization software ParaView and VisIt.
SpEC simulation of inspiral and merger of two black holes
The main application area of SpEC lies in simulating compact binary objects. Specifically, it is one of the most accurate and efficient codes to compute the gravitational waveforms for inspiraling and coalescing binary black holes.
Contributors
SpEC was originally developed by Lawrence Kidder, Harald Pfeiffer, and Mark Scheel, who remain the principal maintainers of the code. Since then, many further individuals have contributed to SpEC. Most especially, Matt Duez and Francois Foucart have developed the hydrodynamics module, and Béla Szilágyi has made numerous valuable additions throughout the code. In addition to these people, Lee Lindblom has contributed significantly to the algorithms used in SpEC.
Further contributions to SpEC were made by Thomas Baumgarte, Andy Bohn, Michael Boyle, Jeandrew Brink, Luisa Buchman, Tony Chu, Michael Cohen, Gregory Cook, Jason Grigsby, Francois Hebert, Dan Hemberger, Kate Henriksson, Frank Herrmann, Jeff Kaplan, Stephen Lau, François Limousin, Geoffrey Lovelace, Keith Matthews, Abdul Mroué, Curran Muhlberger, Rob Owen, Oliver Rinne, Olivier Sarbach, Deirdre Shoemaker, Nick Taylor, Saul Teukolsky, Will Throwe, Manuel Tiglio, Anil Zenginoglu, and Fan Zhang.
Finally, we thank the following undergraduate students for assisting with visualization and improving visualization capabilities: Adam Bartnik, Deshpreet Bedi, Patrick Calhoun, Cameron Cogburn, Bryant Garcia, Daniel Jones, Dave Kotfis, Yor Limkumnerd, Ian MacCormack, Robert McGehee, Dmitry Meyerson, Adam Neumann, Amin Nikbin, Hiroaki Oyaizu, Jennifer Seiler, Keara Soloway, Alexandre Streicher, Allen Sussman.
Availability
Because of the steep learning curve and complexity of SpEC, new users are typically introduced to SpEC through a collaboration with experienced SpEC users. If interested in using SpEC, please contact Lawrence Kidder, Harald Pfeiffer, or Mark Scheel.
Publications
These publications have made use of SpEC:
2000
- Black hole evolution by spectral methods Phys. Rev. D 62, 084032 (2000) [arXiv:gr-qc/0005056]
2001
- Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations Phys. Rev. D 64, 064017 (2001) [arXiv:gr-qc/0105031]
2002
- Comparing initial-data sets for binary black holes Phys. Rev. D 66, 024047 (2002) [arXiv:gr-qc/0203085]
- Energy norms and the stability of the Einstein evolution equations Phys. Rev. D 66, 084014 (2002) [arXiv:gr-qc/0206035]
- Toward stable 3D numerical evolutions of black-hole spacetimes Phys. Rev. D 66, 124005 (2002) [arXiv:gr-qc/0209115]
2003
- A multidomain spectral method for solving elliptic equations Computer Physics Communications 152, 253-273 (2003) [arXiv:gr-qc/0202096]
- Extrinsic curvature and the Einstein constraints Phys. Rev. D 67, 044022 (2003) [arXiv:gr-qc/0207095]
- Dynamical gauge conditions for the Einstein evolution equations Phys. Rev. D 67, 124005 (2003) [arXiv:gr-qc/0301120]
2004
- 3D simulations of linearized scalar fields in Kerr spacetime Phys. Rev. D 69, 104006 (2004) [arXiv:gr-qc/0305027]
- Controlling the growth of constraints in hyperbolic evolution systems Phys. Rev. D 69, 124025 (2004) [arXiv:gr-qc/0402027]
- Optimal constraint projection for hyperbolic evolution systems Phys. Rev. D 70, 084017 (2004) [arXiv:gr-qc/0407011]
- Excision boundary conditions for black-hole initial data Phys. Rev. D 70, 104016 (2004) [arXiv:gr-qc/0407078]
2005
- Initial data for Einstein's equations with superposed gravitational waves Phys. Rev. D 71, 024020 (2005) [arXiv:gr-qc/0410016]
- Boundary conditions for the Einstein evolution system Phys. Rev. D 71, 064020 (2005) [arXiv:gr-qc/0412116]
- Uniqueness and Nonuniqueness in the Einstein Constraints Physical Review Letters 95, 091101 (2005) [arXiv:gr-qc/0504142]
2006
- A new generalized harmonic evolution system Classical and Quantum Gravity 23, 447 (2006) [arXiv:gr-qc/0512093]
- Circular orbits and spin in black-hole initial data Phys. Rev. D 74, 064011 (2006) [arXiv:gr-qc/0605053]
- Approximate initial data for binary black holes Phys. Rev. D 74, 064016 (2006) [arXiv:gr-qc/0606037]
- Solving Einstein's equations with dual coordinate frames Phys. Rev. D 74, 104006 (2006) [arXiv:gr-qc/0607056]
2007
- Testing the accuracy and stability of spectral methods in numerical relativity Phys. Rev. D 75, 024006 (2007) [arXiv:gr-qc/0609047]
- Einstein constraints: Uniqueness and nonuniqueness in the conformal thin sandwich approach Phys. Rev. D 75, 044009 (2007) [arXiv:gr-qc/0610120]
- Reducing orbital eccentricity in binary black hole simulations Classical and Quantum Gravity 24, 59 (2007) [arXiv:gr-qc/0702106]
- Constraint damping in first-order evolution systems for numerical relativity Phys. Rev. D 76, 044019 (2007) [arXiv:gr-qc/0703145]
- Testing outer boundary treatments for the Einstein equations Classical and Quantum Gravity 24, 4053-4078 (2007) [arXiv:0704.0782]
- High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions Phys. Rev. D 76, 124038 (2007) [arXiv:0710.0158]
2008
- Gauge drivers for the generalized harmonic Einstein equations Phys. Rev. D 77, 084001 (2008) [arXiv:0711.2084]
- Initial data for black hole neutron star binaries: A flexible, high-accuracy spectral method Phys. Rev. D 77, 124051 (2008) [arXiv:0804.3787]
- High-accuracy numerical simulation of black-hole binaries: Computation of the gravitational-wave energy flux and comparisons with post-Newtonian approximants Phys. Rev. D 78, 104020 (2008) [arXiv:0804.4184]
- Ineffectiveness of Padé resummation techniques in post-Newtonian approximations Phys. Rev. D 78, 044004 (2008) [arXiv:0805.2390]
- Binary-black-hole initial data with nearly extremal spins Phys. Rev. D 78, 084017 (2008) [arXiv:0805.4192]
- Evolving black hole-neutron star binaries in general relativity using pseudospectral and finite difference methods Phys. Rev. D 78, 104015 (2008) [arXiv:0809.0002]
2009
- IMEX Evolution of Scalar Fields on Curved Backgrounds Communications in Computational Physics 6, 1063-1094 (2009) [arXiv:0808.2597]
- Revisiting event horizon finders Classical and Quantum Gravity 26, 035005 (2009) [arXiv:0809.2628]
- High-accuracy waveforms for binary black hole inspiral, merger, and ringdown Phys. Rev. D 79, 024003 (2009) [arXiv:0810.1767]
- Implementation of higher-order absorbing boundary conditions for the Einstein equations Classical and Quantum Gravity 26, 075009 (2009) [arXiv:0811.3593]
- Reducing spurious gravitational radiation in binary-black-hole simulations by using conformally curved initial data Classical and Quantum Gravity 26, 114002 (2009) [arXiv:0812.3132]
- Comparison of high-accuracy numerical simulations of black-hole binaries with stationary-phase post-Newtonian template waveforms for initial and advanced LIGO Classical and Quantum Gravity 26, 114006 (2009) [arXiv:0901.1628]
- Samurai project: Verifying the consistency of black-hole-binary waveforms for gravitational-wave detection Phys. Rev. D 79, 084025 (2009) [arXiv:0901.2437]
- Testing gravitational-wave searches with numerical relativity waveforms: results from the first Numerical INJection Analysis (NINJA) project Classical and Quantum Gravity 26, 165008 (2009) [arXiv:0901.4399]
- Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of nonspinning, equal-mass black holes Phys. Rev. D 79, 124028 (2009) [arXiv:0902.0790]
- Orbiting binary black hole evolutions with a multipatch high order finite-difference approach Phys. Rev. D 80, 024027 (2009) [arXiv:0904.0493]
- Improved gauge driver for the generalized harmonic Einstein system Phys. Rev. D 80, 084019 (2009) [arXiv:0904.4873]
- Extrapolating gravitational-wave data from numerical simulations Phys. Rev. D 80, 124045 (2009) [arXiv:0905.3177]
- Status of NINJA: the Numerical INJection Analysis project Classical and Quantum Gravity 26, 114008 (2009) [arXiv:0905.4227]
- Final remnant of binary black hole mergers: Multipolar analysis Phys. Rev. D 80, 084012 (2009) [arXiv:0907.0280]
- Black hole initial data on hyperboloidal slices Phys. Rev. D 80, 084024 (2009) [arXiv:0907.3163]
- High accuracy simulations of black hole binaries: Spins anti-aligned with the orbital angular momentum Phys. Rev. D 80, 124051 (2009) [arXiv:0909.1313]
- Simulations of binary black hole mergers using spectral methods Phys. Rev. D 80, 124010 (2009) [arXiv:0909.3557]
2010
- Momentum flow in black-hole binaries. II. Numerical simulations of equal-mass, head-on mergers with antiparallel spins Phys. Rev. D 82, 064031 (2010) [arXiv:0907.0869]
- Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of nonprecessing, spinning, equal-mass black holes Phys. Rev. D 81, 084041 (2010) [arXiv:0912.3466]
- Equation of state effects in black hole\ndashneutron star mergers Classical and Quantum Gravity 27, 114106 (2010) [arXiv:0912.3528]
- Hyperboloidal evolution of test fields in three spatial dimensions Phys. Rev. D 81, 124010 (2010) [arXiv:1004.0760]
- Degeneracy measures for the algebraic classification of numerical spacetimes Phys. Rev. D 81, 124042 (2010) [arXiv:1004.3768]
- Measuring orbital eccentricity and periastron advance in quasicircular black hole simulations Phys. Rev. D 82, 124016 (2010) [arXiv:1004.4697]
- Spectral methods for the wave equation in second-order form Phys. Rev. D 82, 024037 (2010) [arXiv:1005.2922]
2011
- Black hole-neutron star mergers: Effects of the orientation of the black hole spin Phys. Rev. D 83, 024005 (2011) [arXiv:1007.4203]
- Hyperboloidal layers for hyperbolic equations on unbounded domains Journal of Computational Physics 230, 2286-2302 (2011) [arXiv:1008.3809]
- Simulating merging binary black holes with nearly extremal spins Phys. Rev. D 83, 024010 (2011) [arXiv:1010.2777]
- Horizon dynamics of distorted rotating black holes Phys. Rev. D 83, 104018 (2011) [arXiv:1011.2601]
- Reducing orbital eccentricity of precessing black-hole binaries Phys. Rev. D 83, 104034 (2011) [arXiv:1012.1549]
- Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime Physical Review Letters 106, 151101 (2011) [arXiv:1012.4869]
- Suitability of post-Newtonian/numerical-relativity hybrid waveforms for gravitational wave detectors Classical and Quantum Gravity 28, 134002 (2011) [arXiv:1102.5128]
- Implicit-explicit evolution of single black holes Phys. Rev. D 84, 084023 (2011) [arXiv:1105.3922]
- Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism Phys. Rev. D 84, 124052 (2011) [arXiv:1106.1021]
- Periastron Advance in Black-Hole Binaries Physical Review Letters 107, 141101 (2011) [arXiv:1106.3278]
- Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: General theory and weak-gravity applications Phys. Rev. D 84, 124014 (2011) [arXiv:1108.5486]
- Geometric approach to the precession of compact binaries Phys. Rev. D 84, 124011 (2011) [arXiv:1110.2965]
2012
- Toroidal horizons in binary black hole inspirals Phys. Rev. D 85, 024031 (2012) [arXiv:1110.1668]
- High-accuracy gravitational waveforms for binary black hole mergers with nearly extremal spins Classical and Quantum Gravity 29, 045003 (2012) [arXiv:1110.2229]
- Black hole-neutron star mergers for 10M$_&sun;$ black holes Phys. Rev. D 85, 044015 (2012) [arXiv:1111.1677]
- The NINJA-2 catalog of hybrid post-Newtonian/numerical-relativity waveforms for non-precessing black-hole binaries Classical and Quantum Gravity 29, 124001 (2012) [arXiv:1201.5319]
- Prototype effective-one-body model for nonprecessing spinning inspiral-merger-ringdown waveforms Phys. Rev. D 86, 024011 (2012) [arXiv:1202.0790]
- Numerical simulations of compact object binaries Classical and Quantum Gravity 29, 124004 (2012) [arXiv:1203.5166]
- Are different approaches to constructing initial data for binary black hole simulations of the same astrophysical situation equivalent? Phys. Rev. D 86, 084054 (2012) [arXiv:1206.2943]
- Simulations of unequal-mass black hole binaries with spectral methods Phys. Rev. D 86, 084033 (2012) [arXiv:1206.3015]
- Black-hole-neutron-star mergers: Disk mass predictions Phys. Rev. D 86, 124007 (2012) [arXiv:1207.6304]
- Geometrically motivated coordinate system for exploring spacetime dynamics in numerical-relativity simulations using a quasi-Kinnersley tetrad Phys. Rev. D 86, 084020 (2012) [arXiv:1208.0630]
- Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. II. Stationary black holes Phys. Rev. D 86, 084049 (2012) [arXiv:1208.3034]
- Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. III. Quasinormal pulsations of Schwarzschild and Kerr black holes Phys. Rev. D 86, 104028 (2012) [arXiv:1208.3038]
- Precessing Binary Black Holes Simulations: Quasicircular Initial Data [arXiv:1210.2958]
2013
- Suitability of hybrid gravitational waveforms for unequal-mass binaries Phys. Rev. D 87, 024009 (2013) [arXiv:1210.3007]
- Solving partial differential equations numerically on manifolds with arbitrary spatial topologies Journal of Computational Physics 243, 151-175 (2013) [arXiv:1210.5016]
- Dynamical excision boundaries in spectral evolutions of binary black hole spacetimes Classical and Quantum Gravity 30, 115001 (2013) [arXiv:1211.6079]
- Black-hole-neutron-star mergers at realistic mass ratios: Equation of state and spin orientation effects Phys. Rev. D 87, 084006 (2013) [arXiv:1212.4810]
- Massive disc formation in the tidal disruption of a neutron star by a nearly extremal black hole Classical and Quantum Gravity 30, 135004 (2013) [arXiv:1302.6297]
- Precession-tracking coordinates for simulations of compact-object binaries Phys. Rev. D 88, 084031 (2013) [arXiv:1304.3067]
- Black Hole-Neutron Star Mergers with a Hot Nuclear Equation of State: Outflow and Neutrino-cooled Disk for a Low-mass, High-spin Case Astrophys. J. 776, 47 (2013) [arXiv:1304.3384]
- Catalog of 174 Binary Black Hole Simulations for Gravitational Wave Astronomy Physical Review Letters 111, 241104 (2013) [arXiv:1304.6077]
- Final spin and radiated energy in numerical simulations of binary black holes with equal masses and equal, aligned or antialigned spins Phys. Rev. D 88, 064014 (2013) [arXiv:1305.5991]
- Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR Collaboration Classical and Quantum Gravity 31, 025012 (2013) [arXiv:1307.5307]
- First direct comparison of nondisrupting neutron star-black hole and binary black hole merger simulations Phys. Rev. D 88, 064017 (2013) [arXiv:1307.7685]
- Periastron advance in spinning black hole binaries: Gravitational self-force from numerical relativity Phys. Rev. D 88, 124027 (2013) [arXiv:1309.0541]
- Periastron advance in spinning black hole binaries: comparing effective-one-body and numerical relativity Phys. Rev. D 88, 084005 (2013) [arXiv:1309.0544]
- Joint approach for reducing eccentricity and spurious gravitational radiation in binary black hole initial data construction Phys. Rev. D 88, 084033 (2013) [arXiv:1309.1141]
- Comparing gravitational waveform extrapolation to Cauchy-characteristic extraction in binary black hole simulations Phys. Rev. D 88, 124010 (2013) [arXiv:1309.3605]
2014
- Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism Phys. Rev. D 89, 084006 (2014) [arXiv:1307.6232]
- Including realistic tidal deformations in binary black-hole initial data Phys. Rev. D 89, 064062 (2014) [arXiv:1310.7900]
- Template banks for binary black hole searches with numerical relativity waveforms Phys. Rev. D 89, 042002 (2014) [arXiv:1310.7949]
- Effective-one-body model for black-hole binaries with generic mass ratios and spins Phys. Rev. D 89, 061502 (2014) [arXiv:1311.2544]
- Stability of nonspinning effective-one-body model in approximating two-body dynamics and gravitational-wave emission Phys. Rev. D 89, 061501 (2014) [arXiv:1311.2565]
- Accretion disks around binary black holes of unequal mass: General relativistic magnetohydrodynamic simulations near decoupling Phys. Rev. D 89, 064060 (2014) [arXiv:1312.0600]
- Solving Einstein's equation numerically on manifolds with arbitrary spatial topologies Phys. Rev. D 89, 044044 (2014) [arXiv:1312.0701]
- The NINJA-2 project: detecting and characterizing gravitational waveforms modelled using numerical binary black hole simulations Classical and Quantum Gravity 31, 115004 (2014) [arXiv:1401.0939]
- Decline of the current quadrupole moment during the merger phase of binary black hole coalescence [arXiv:1403.0512]
- Neutron star-black hole mergers with a nuclear equation of state and neutrino cooling: Dependence in the binary parameters Phys. Rev. D 90, 024026 (2014) [arXiv:1405.1121]
- Magnetic effects on the low-T /|W | instability in differentially rotating neutron stars Phys. Rev. D 90, 104014 (2014) [arXiv:1405.2144]
- Key elements of robustness in binary black hole evolutions using spectral methods International Journal of Modern Physics D 23, 30014 (2014) [arXiv:1405.3693]
- Gravitational-wave modes from precessing black-hole binaries [arXiv:1409.4431]
- Initial data for high-compactness black hole-neutron star binaries [arXiv:1409.7159]
- Accretion disks around binary black holes of unequal mass: General relativistic MHD simulations of postdecoupling and merger Phys. Rev. D 90, 104030 (2014) [arXiv:1410.1543]
2015
- Spectral characteristic evolution: a new algorithm for gravitational wave propagation Classical and Quantum Gravity 32, 025008 (2015) [arXiv:1406.7029]
- What does a binary black hole merger look like? Classical and Quantum Gravity 32, 065002 (2015) [arXiv:1410.7775]
- Nearly extremal apparent horizons in simulations of merging black holes Classical and Quantum Gravity 32, 065007 (2015) [arXiv:1411.7297]
- Improved methods for simulating nearly extremal binary black holes Classical and Quantum Gravity 32, 105009 (2015) [arXiv:1412.1803]
- Comparing Post-Newtonian and Numerical-Relativity Precession Dynamics [arXiv:1502.01747]
- Post-merger evolution of a neutron star-black hole binary with neutrino transport Phys. Rev. D 91, 124021 (2015) [arXiv:1502.04146]
- Approaching the Post-Newtonian Regime with Numerical Relativity: A Compact-Object Binary Simulation Spanning 350 Gravitational-Wave Cycles Physical Review Letters 115, 031102 (2015) [arXiv:1502.04953]
- Gauge Invariant Spectral Cauchy Characteristic Extraction [arXiv:1502.06987]
- Fast and accurate prediction of numerical relativity waveforms from binary black hole mergers using surrogate models Phys. Rev. Lett. 115, 121102 (2015) [arXiv:1502.07758]
- Improvements to the construction of binary black hole initial data [arXiv:1506.01689]
- Accuracy and precision of gravitational-wave models of inspiraling neutron star -- black hole binaries with spin: comparison with numerical relativity in the low-frequency regime [arXiv:1507.00103]
- Binary Neutron Stars with Arbitrary Spins in Numerical Relativity [arXiv:1508.06986]
- Transformations of asymptotic gravitational-wave data [arXiv:1509.00862]
- Gravitational waveforms for neutron star binaries from binary black hole simulations [arXiv:1509.05782]
External Software
SpEC benefits from the years of work put into the following general-purpose scientific software packages:
- PETSc: Parallel linear and nonlinear equation solvers
- SPHEREPACK: Spherical harmonic transforms
- FFTW: Fast Fourier transforms
- DFFTPACK: Double-precision fast Fourier transforms
- GNU Scientific Library: Mathematical routines for scientific applications
- Numerical Recipes: Numerical routines for scientific computing