Journal of Scientific Computing

, Volume 56, Issue 2, pp 366–380 | Cite as

High-Precision Numerical Simulations on a CUDA GPU: Kerr Black Hole Tails

Article

Abstract

Computational science has advanced significantly over the past decade and has impacted almost every area of science and engineering. Most numerical scientific computation today is performed with double-precision floating-point accuracy (64-bit or \(\sim \)15 decimal digits); however, there are a number of applications that benefit from a higher level of numerical precision. In this paper, we describe such an application in the research area of black hole physics: studying the late-time behavior of decaying fields in Kerr black hole space-time. More specifically, this application involves a hyperbolic partial-differential-equation solver that uses high-order finite-differencing and quadruple (128-bit or \(\sim \)30 decimal digits) or octal (256-bit or \(\sim \)60 decimal digits) floating-point precision. Given the computational demands of this high-order and high-precision solver, in addition to the rather long evolutions required for these studies, we accelerate the solver using a many-core Nvidia graphics-processing-unit and obtain an order-of-magnitude speed-up over a high-end multi-core processor. We thus demonstrate a practical solution for demanding problems that utilize high-precision numerics today.

Keywords

CUDA GPU Precision Tails Quadruple  Octal 

References

  1. 1.
    Bailey, D.H.: High-precision arithmetic in scientific computation. Comput. Sci. Eng. 7, 54 (2005)CrossRefGoogle Scholar
  2. 2.
    Bailey, D.H., Barrio, R., Borwein, J.M.: High-precision computation: Mathematical physics and dynamics. Appl. Math. Comput. 218, 10106 (2012)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Higham, N..: Accuracy and Stability of Numerical Analysis, pp 506–507. SIAM, (1996).Google Scholar
  4. 4.
    Valdettaro, L., Rieutord, M., Braconnier, T., Fraysse, V.: Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and ArnoldiChebyshev algorithm. J. Comput. Appl. Math. 205, 382 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Sarra, S.: Radial basis function approximation methods with extended precision floating point arithmetic. Eng. Anal. Boundary Elements 35, 68 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gottieb, S., Fischer, Paul F.: Modified Conjugate Gradient Method for the Solution of \(Ax = b\), J. Sci. Comp. 13, 173 (1998)CrossRefGoogle Scholar
  7. 7.
    Nvidia’s CUDA, http://www.nvidia.com/cuda/. Accessed 13 Aug 2012
  8. 8.
    Burko, L., Khanna, G.: Radiative falloff in the background of rotating black hole. Phys. Rev. D 67, 081502 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Burko, L., Khanna, G.: Universality of massive scalar field late-time tails in black-hole spacetimes. Phys. Rev. D 70, 044018 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burko, L., Khanna, G.: Late-time Kerr tails revisited. Class. Quant. Grav. 26, 015014 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Burko, L., Khanna, G.: Late-time Kerr tails: generic and non-generic initial data sets, “up” modes, and superposition. Class. Quant. Grav. 28, 025012 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zenginoglu, A., Khanna, G., and Burko, L., Mode coupling and intermediate behavior of Kerr tails, Class. Quant. Grav., 2012.Google Scholar
  13. 13.
    Baumgarte, T., Shapiro, S.: Numerical Relativity. Cambridge University Press,   (2010)Google Scholar
  14. 14.
    Alcubierre, M.: Introduction to 3+1 Numerical Relativity. Oxford University Press,   (2008)Google Scholar
  15. 15.
    Cactus Code, http://www.cactuscode.org/. Accessed 13 Aug 2012
  16. 16.
    Gundlach, C.: Critical Phenomena in Gravitational Collapse. Living Rev. Relat. 2, 4 (1999)MathSciNetGoogle Scholar
  17. 17.
    Amaro-Seoane, P., et al.: Intermediate and Extreme Mass-Ratio Inspirals - Astrophysics, Science Applications and Detection using LISA. Class. Quant. Grav. 24, R113 (2007)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Teukolsky, S.: Perturbations of a rotating black hole. Astrophys. J. 185, 635 (1973)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lax, P., Richtmyer, R.: Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, 267 (1956)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Krivan, W., Laguna, P., Papadopoulos, P., Andersson, N.: Dynamics of perturbations of rotating black holes. Phys. Rev. D 56, 3395 (1997)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Price, R.: Nonspherical Perturbations of Relativistic Gravitational Collapse. Phys. Rev. D 5, 2419 (1972)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tiglio, M., Kidder, L., Teukolsky, S.: High accuracy simulations of Kerr tails: coordinate dependence and higher multipoles. Class. Quant. Grav. 25, 105022 (2008)CrossRefGoogle Scholar
  23. 23.
    Khanna, G., McKennon, J.: Numerical modeling of gravitational wave sources accelerated by OpenCL. Comput. Phys. Commun. 181, 1605 (2010)MATHCrossRefGoogle Scholar
  24. 24.
    McKennon, J., Forrester, G. and Khanna, G.,: High accuracy gravitational waveforms from black hole binary inspirals using OpenCL. Proceedings of the NSF XSEDE12 Conference, Chicago IL, 2012.Google Scholar
  25. 25.
    LBNL QD Library, http://crd.lbl.gov/~dhbailey/mpdist/. Accessed 13 Aug 2012
  26. 26.
    GQD Library, http://code.google.com/p/gpuprec/. Accessed 13 Aug 2012
  27. 27.
    Ginjupalli, R., Khanna, G.: High-precision numerical simulations of rotating black holes accelerated by CUDA, Proceedings of the International Conference on High Performance Computing Systems (HPCS). Orlando, FL 2010.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of Massachusetts DartmouthNorth DartmouthUSA

Personalised recommendations