Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 1: 97 - 115

DOI: https://doi.org/10.1007/s42967-021-00129-2

ORIGINAL PAPERS

A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations

Expand
  • 1 University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA;
    2 Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA;
    3 University of Rhode Island, Kingston, RI 02881, USA

Received date: 2020-10-02

  Revised date: 2021-01-22

  Online published: 2023-03-08

Supported by

Many of the computations were performed on the MIT/IBM Satori GPU supercomputer supported by the Massachusetts Green High Performance Computing Center (MGHPCC) and Oak Ridge National Laboratory (ORNL)’s Summit leadership-class supercomputer under Director Discretionary allocation AST166. We thank Sebastiano Bernuzzi for helpful discussions. The authors acknowledge support of NSF Grants No. PHY-2010685 (G.K) and No. DMS-1912716 (S.F, S.G, and G.K), AFOSR Grant No. FA9550-18-1-0383 (S.G) and Office of Naval Research/Defense University Research Instrumentation Program (ONR/DURIP) Grant No. N00014181255. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while a subset of the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Advances in Computational Relativity program. A part of this research is sponsored by the Office of Advanced Scientific Computing Research; US Department of Energy, and was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract no. De-AC05-00OR22725. This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

Fold

Abstract

We develop and use a novel mixed-precision weighted essentially non-oscillatory (WENO) method for solving the Teukolsky equation, which arises when modeling perturbations of Kerr black holes. We show that WENO methods outperform higher-order finite-difference methods, standard in the discretization of the Teukolsky equation, due to the need to add dissipation for stability purposes in the latter. In particular, as the WENO scheme uses no additional dissipation, it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio binaries. In the mixed-precision approach, the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of ≈ 3.3. In addition, we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations. Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics. We apply our solver to study the behavior of the Aretakis charge—a conserved quantity, that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.

Key words： Numerical methods; Finite differencing; Hyperbolic; Partial differential equations; Black holesblack holes; WENO

Cite this article

Scott E. Field, Sigal Gottlieb, Zachary J. Grant, Leah F. Isherwood, Gaurav Khanna . A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations[J]. Communications on Applied Mathematics and Computation, 2023 , 5(1) : 97 -115 . DOI: 10.1007/s42967-021-00129-2

References

1. Abbott, B.P., et al.: Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116(6), 061102 (2016)
2. Abbott, B.P., et al.: Tests of general relativity with GW150914. Phys. Rev. Lett. 116(22), 221101 (2016)
3. Abbott, B.P., et al.: GWTC-1: a gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs. Phys. Rev. X9(3), 031040 (2019)
4. Akiyama, K., et al.: First m87 event horizon telescope results. IV. Imaging the central supermassive black hole. Astrophys. J. Lett. 875(1), L4 (2019)
5. Angelopoulos, Y., Aretakis, S., Gajic, D.: Horizon hair of extremal black holes and measurements at null infinity. Phys. Rev. Lett. 121(13), 131102 (2018)
6. Aretakis, S.: A note on instabilities of extremal black holes under scalar perturbations from afar. Class. Quantum Grav. 30(9), 095010 (2013)
7. Burko, L.M., Khanna, G., Zenginoğlu, A.: Cauchy-horizon singularity inside perturbed Kerr black holes. Phys. Rev. D 93(4), 041501 (2016). [Erratum: Phys. Rev. D 96, 129903 (2017)]
8. Burko, L.M., Khanna, G., Sabharwal, S.: (Transient) Scalar hair for (nearly) extreme black holes. Phys. Rev. Res. 1, 033106 (2019)
9. Burko, L.M., Khanna, G., Sabharwal, S.: Scalar and gravitational hair for extreme Kerr black holes. arXiv: 2005. 07294 (2020)
10. Dong, B., Gottlieb, S., Hristova, Y., Jiang, Y,. Wang, H.: The effect of the sensitivity parameter in weighted essentially non-oscillatory methods. In: Brenner, S. (ed.) Topics in Numerical Partial Differential Equations and Scientific Computing. The IMA Volumes in Mathematics and Its Applications, vol. 160, pp. 23–50. Springer, New York, NY (2016)
11. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)
12. Harms, E., Bernuzzi, S., Brügmann, B.: Numerical solution of the 2+1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays. Class. Quantum Grav. 30(11), 115013 (2013)
13. Harms, E., Bernuzzi, S., Nagar, A., Zenginoğlu, A.: A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime. Class. Quantum. Grav. 31(24), 245004 (2014)
14. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)
15. Khanna, G..: High-precision numerical simulations on a CUDA GPU: Kerr black hole tails. J. Sci. Comput. 56(2), 366–380 (2013)
16. Kreiss, H.O., Oliger, J.: Methods for the approximate solution of time dependent problems. Global Atm. Res. Prog. Publ. 10, GARP (1973)
17. Poisson, E.: Absorption of mass and angular momentum by a black hole: time-domain formalisms for gravitational perturbations, and the small-hole or slow-motion approximation. Phys. Rev. D 70(8), 084044 (2004)
18. Regge, T., Wheeler, J.A.: Stability of a Schwarzschild singularity. Phys. Rev. 108, 1063–1069 (1957)
19. Sundararajan, P.A., Khanna, G., Hughes, S.A.: Towards adiabatic waveforms for inspiral into Kerr black holes: a new model of the source for the time domain perturbation equation. Phys. Rev. D 76(10), 104005 (2007)
20. Sundararajan, P.A., Khanna, G., Hughes, S.A., Drasco, S.: Towards adiabatic waveforms for inspiral into Kerr black holes. II. Dynamical sources and generic orbits. Phys. Rev. D 78(2), 024022 (2008)
21. Sundararajan, P.A., Khanna, G., Hughes, S.A.: Binary black hole merger gravitational waves and recoil in the large mass ratio limit. Phys. Rev. D 81(10), 104009 (2010)
22. Teukolsky, S.A.: Perturbations of a rotating black hole. I. Fundamental equations for gravitational electromagnetic and neutrino field perturbations. Astrophys. J. 185, 635–647 (1973)
23. Wysocki, D., Lange, J., O’Shaughnessy, R.: Reconstructing phenomenological distributions of compact binaries via gravitational wave observations. Phys. Rev. D 100(4), 043012 (2019)
24. Zenginoğlu, A.: Hyperboloidal evolution with the Einstein equations. Class. Quantum Grav. 25, 195025 (2008)
25. Zenginoğlu, A.: A hyperboloidal study of tail decay rates for scalar and Yang-Mills fields. Class. Quantum Grav. 25, 175013 (2008)
26. Zenginoğlu, A.: Hyperboloidal foliations and scri-fixing. Class. Quantum Grav. 25, 145002 (2008)
27. Zenginoğlu, A.: Asymptotics of black hole perturbations. Class. Quantum Grav. 27, 045015 (2010)
28. Zenginoğlu, A.: A geometric framework for black hole perturbations. Phys. Rev. D 83, 127502 (2011)
29. Zenginoğlu, A.: Hyperboloidal layers for hyperbolic equations on unbounded domains. J. Comput. Phys. 230, 2286–2302 (2011)
30. Zenginoğlu, A., Khanna, G.: Null infinity waveforms from extreme-mass-ratio inspirals in Kerr spacetime. Phys. Rev. X 1(2), 021017 (2011)
31. Zenginoğlu, A., Khanna, G., Burko, L.M.: Intermediate behavior of Kerr tails. Gen. Relativ. Gravit. 46, 1672 (2014)
32. Zenginoğlu, A., Nunez, D., Husa, S.: Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem. Class. Quantum Grav. 26, 035009 (2009)
33. Zenginoğlu, A., Tiglio, M.: Spacelike matching to null infinity. Phys. Rev. D 80, 024044 (2009)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157