Skip to main content
Springer Nature Link
Log in

A Note on High-Precision Approximation of Asymptotically Decaying Solution and Orthogonal Decomposition

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In some physical applications, the decaying rate of asymptotically decaying solution is more important than the solution magnitude itself in understanding the physical system such as the late-time behavior of decaying fields in black hole space-time. In Khanna (J Sci Comput 56(2):366–380, 2013), it was emphasized that high-precision arithmetic and high-order methods are required to capture numerically the correct decaying rate of the late-time radiative tails of black-hole system in order to prevent roundoff errors from inducing a wrong power-law decay rate in the numerical approximation. In this paper, we explain how roundoff errors induce a wrong decay mode in the numerical approximation using simple linear differential equations. Then we describe the orthogonal decomposition method as a possible technique to remove wrong decaying modes induced by roundoff errors in the numerical approximation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Antonana, M., Makazaga, J., Murua, A.: Reducing and monitoring round-off error propagation for symplectic implicit Runge–Kutta schemes. Numer. Algorithms (2017). https://doi.org/10.1007/s11075-017-0287-z

  2. Bailey, D.H.: High-precision floating-point arithmetic in scientific computation. Comput. Sci. Eng. 7(3), 54–61 (2005)

    Article  Google Scholar 

  3. Baumgarte, T.W., Shapiro, S.L.: Binary black hole merger. Phys. Today 64, 32–37 (2011)

    Article  Google Scholar 

  4. Burko, L.M., Khanna, G.: Late-time Kerr tails: generic and non-generic initial data sets, “up” modes and superposition. Class. Quant. Gravity 28, 025012 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burko, L.M., Khanna, G.: Mode coupling mechanism for late-time Kerr tails. Phys. Rev. D 89, 044037 (2014)

    Article  Google Scholar 

  6. Campanelli, M., Lousto, C.O., Marronetti, P., Zlochower, Y.: Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett. 96, 111101 (2006)

    Article  Google Scholar 

  7. Canizares, P., Sopuerta, C.F., Jaramillo, J.L.: Pseudospectral collocation methods for the computation of the self-force on a charged particle: generic orbits around a Schwarzschild black hole. Phys. Rev. D 82, 044023 (2010)

    Article  Google Scholar 

  8. Chaitin-Chatelin, F., Gratton, S.: Convergence in finite precision of successive iteration methods under high nonnormality. BIT Numer. Math. 36(3), 455–469 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Donninger, R., Schlag, W., Soffer, A.: A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta. Adv. Math. 226(1), 484–540 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Etienne, Z.B., Paschalidis, V., Shapiro, S.L.: General-relativistic simulations of black-hole-neutron-star mergers: effects of tilted magnetic fields. Phys. Rev. D 86, 084026 (2012)

    Article  Google Scholar 

  11. Field, S., Hesthaven, J., Lau, S.: Discontinuous Galerkin method for computing gravitational waveforms from extreme mass ratio binaries. Class. Quant. Gravity 26, 165010 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gleiser, R.J., Price, R.H., Pullin, J.: Late time tails in the Kerr spacetime. Class. Quant. Gravity 25, 072001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henrici, P.: Error Propagation for Difference Methods. Wiley, New York (1963)

    MATH  Google Scholar 

  14. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  15. Jung, J.-H., Khanna, G., Nagle, I.: A spectral collocation approximation for the radial-infall of a compact object into a Schwarzchild black-hole. Int. J. Mod. Phys. C 20, 1827 (2009)

    Article  MATH  Google Scholar 

  16. Kehlet, B., Logg, A.: A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations. Numer. Algorithms 76, 191–210 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khanna, G.: High-precision numerical simulations on a CUDA GPU: Kerr black hole tails. J. Sci. Comput. 56(2), 366–380 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, 267–293 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lousto, C.O.: A time-domain fourth-order-convergent numerical algorithm to integrate black hole perturbations in the extreme-mass-ratio limit. Class. Quant. Gravity 22, S543–S568 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Price, R.: Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations. Phys. Rev. D 5, 2419 (1972)

    Article  MathSciNet  Google Scholar 

  21. Racz, I., Toth, G.Z.: Numerical investigation of the late-time Kerr tails. Class. Quant. Gravity 28, 195003 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Teukolsky, S.: Perturbations of a rotating black hole. Astrophys. J. 185, 635–647 (1973)

    Article  Google Scholar 

  23. Tiglio, M., Kidder, L., Teukolsky, S.: High accuracy simulations of Kerr tails: coordinate dependence and higher multipoles. Class. Quant. Gravity 25, 105022 (2008)

    Article  MATH  Google Scholar 

  24. Valdettaro, L., Rieutord, M., Braconnier, T., Frayssè, V.: Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi–Chebyshev algorithm. J. Comput. Appl. Math. 205, 382–393 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zenginoğlu, A., Khanna, G., Burko, L.M.: Intermediate behavior of Kerr tails. Gen. Rel. Gravit. 46, 1672 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the two anonymous reviewers whose comments helped them to understand the problem better and improve their paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University at Buffalo, The State University of New York, Buffalo, Buffalo, NY, 14260-2900, USA

    John Nicponski & Jae-Hun Jung

  2. Department of Data Science, Ajou University, Yeongtong-gu, Suwon, 16499, Korea

    Jae-Hun Jung

Authors
  1. John Nicponski
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jae-Hun Jung
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jae-Hun Jung.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nicponski, J., Jung, JH. A Note on High-Precision Approximation of Asymptotically Decaying Solution and Orthogonal Decomposition. J Sci Comput 76, 189–215 (2018). https://doi.org/10.1007/s10915-017-0619-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0619-0

Keywords