Skip to main content

Aliasing instabilities in the numerical evolution of the Einstein field equations

Abstract

The Einstein field equations of gravitation are characterized by cross-scale, high-order nonlinear terms, representing a challenge for numerical modeling. In an exact spectral decomposition, high-order nonlinearities correspond to a convolution that numerically might lead to aliasing instabilities. We present a study of this problem, in vacuum conditions, based on the \(3+1\) Baumgarte–Shibata–Shapiro–Nakamura (BSSN) formalism. We inspect the emergence of numerical artifacts, in a variety of conditions, by using the Spectral-FIltered Numerical Gravity codE (SFINGE)—a pseudo-spectral algorithm, based on a classical (Cartesian) Fourier decomposition. By monitoring the highest \(k-\)modes of the dynamical fields, we identify the culprits of the aliasing and propose procedures that cure such instabilities, based on the suppression of the aliased wavelengths. This simple algorithm, together with appropriate treatment of the boundary conditions, can be applied to a variety of gravitational problems, including those related to massive objects dynamics.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. 1.

    Diener, P., Herrmann, F., Pollney, D., Schnetter, E., Seidel, E., Takahashi, R., Thornburg, J., Ventrella, J.: Accurate evolution of orbiting binary black holes. Phys. Rev. Lett. 96, 121101 (2006)

  2. 2.

    Baker, J.G., Centrella, J., Choi, D., Koppitz, M., Meter, J.: Binary black hole merger dynamics and waveforms. Phys. Rev. D 73, 104002 (2006)

  3. 3.

    Scheel, M.A., Pfeiffer, H.P., Lindblom, L., Kidder, L.E., Rinne, O., Teukolsky, S.A.: Solving Einstein’s equations with dual coordinate frames. Phys. Rev. D 74, 104006 (2006)

  4. 4.

    Herrmann, F., Hinder, I., Shoemaker, D., Laguna, P.: Binary black holes and recoil velocities. In: AIP Conference Proceedings, vol. 873, pp. 89–93. American Institute of Physics (2006)

  5. 5.

    Schnetter, E., Hawley, S.H., Hawke, I.: Evolutions in 3d numerical relativity using fixed mesh refinement. Class. Quant. Gravity 21, 1465 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Ajith, et al.: Inspiral-merger-ringdown waveforms for black-hole binaries with nonprecessing spins. Phys. Rev. L 106, 241101 (2011)

  7. 7.

    Akbarian, A., Choptuik, M.W.: Black hole critical behavior with the generalized BSSN formulation. Phys. Rev. D 92, 084037 (2015)

  8. 8.

    Shibata, M.: Numerical Relativity. World Scientific, Singapore (2015)

    Book  Google Scholar 

  9. 9.

    Mewes, V., Font, J.A., Galeazzi, F., Montero, P.J., Stergioulas, N.: Numerical relativity simulations of thick accretion disks around tilted Kerr black holes. Phys. Rev. D 93, 064055 (2016)

  10. 10.

    Lovelace, et al.: Modeling the source of GW150914 with targeted numerical-relativity simulations. Class. Quant. Gravity 33, 244002 (2016)

  11. 11.

    Porth, O., et al.: The black hole accretion code. Comput. Astrophys. Cosmol. 4, 1–42 (2017)

    ADS  Article  Google Scholar 

  12. 12.

    Cao, Z., Han, W.B.: Waveform model for an eccentric binary black hole based on the effective-one-body-numerical-relativity formalism. Phys. Rev. D 96, 044028 (2017)

  13. 13.

    Okounkova, M., Stein, L.C., Moxon, J., Scheel, M.A., Teukolsky, S.A.: Numerical relativity simulation of GW150914 beyond general relativity. Phys. Rev. D 101, 104016 (2020)

  14. 14.

    Most, E.R., Papenfort, L.J., Weih, L.R., Rezzolla, L.: A lower bound on the maximum mass if the secondary in GW190814 was once a rapidly spinning neutron star. MNRAS 499, L82–L86 (2020)

    ADS  Article  Google Scholar 

  15. 15.

    Richards, C.B., Baumgarte, T.W., Shapiro, S.L.: Accretion onto a small black hole at the center of a neutron star. Phys. Rev. D 103, 104009 (2021)

  16. 16.

    Laguna, P., Shoemaker, D.: Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation. Class. Quant. Gravity 19, 3679 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Witek, H., Hilditch, D., Sperhake, U.: Stability of the puncture method with a generalized Baumgarte–Shapiro–Shibata–Nakamura formulation. Phys. Rev. D 10, 104041 (2011)

  18. 18.

    Bernuzzi, B., Hilditch, D.: Constraint violation in free evolution schemes: comparing the BSSNOK formulation with a conformal decomposition of the Z4 formulation. Phys. Rev. D 81, 084003 (2010)

  19. 19.

    Calabrese, G., Pullin, J., Sarbach, O., Tiglio, M.: Stability properties of a formulation of Einstein’s equations. Phys. Rev. D 66, 064011 (2002)

  20. 20.

    Zlochower, Y., Baker, J., G., Campanelli, M., Lousto, C., O.: Accurate black hole evolutions by fourth-order numerical relativity. Phys. Rev. D 72, 024021 (2005)

  21. 21.

    Jansen, N., Bruegmann, B., Tichy, W.: Numerical stability of the Alekseenko–Arnold evolution system compared to the ADM and BSSN systems. Phys. Rev. D 74, 084022 (2006)

  22. 22.

    Pretorius, F.: Simulation of binary black hole spacetimes with a harmonic evolution scheme. Class. Quant. Gravity 23, S529 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Baumgarte, T.W., Shapiro, S.L.: Numerical integration of Einstein’s field equations. Phys. Rev. D 59, 024007 (1998)

  24. 24.

    Shibata, M., Nakamura, T.: Evolution of three-dimensional gravitational waves: harmonic slicing case. Phys. Rev. D 52, 5428 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Beyer, H., Sarbach, O.: Well-posedness of the Baumgarte–Shapiro–Shibata–Nakamura formulation of Einstein’s field equations. Phys. Rev. D 70), 104004 (2004)

  26. 26.

    Brown, J., D.: BSSN in spherical symmetry. Class. Quant. Gravity 25, 205004 (2008)

  27. 27.

    Sarbach, O., Calabrese, G., Pullin, J., Tiglio, M.: Hyperbolicity of the Baumgarte–Shapiro–Shibata–Nakamura system of Einstein evolution equations. Phys. Rev. D 66, 064002 (2002)

  28. 28.

    Boyd, J.: Chebyshev and Fourier Spectral Methods. Dover Publications, New York (2001)

    MATH  Google Scholar 

  29. 29.

    Orszag, S.A.: On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. J. Atmos. Sci. 28, 1074 (1971)

    ADS  Article  Google Scholar 

  30. 30.

    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)

    Book  Google Scholar 

  31. 31.

    Meringolo, C., Servidio, S., Veltri, P.: A spectral method algorithm for numerical simulations of gravitational fields. Class. Quant. Gravity 28, 075027 (2021)

  32. 32.

    Orszag, S.A.:Comparison of pseudospectral and spectral approximation. Stud. Appl. Math. 51, 253–259 (1972)

  33. 33.

    Orszag, S.A.: Spectral methods for problems in complex geometrics. Numerical methods for partial differential equations, pp. 273–305 (1979)

  34. 34.

    Canuto, C., et al.: Spectral Methods in Fluid Dynamics. Springer, Berlin (2012)

    Google Scholar 

  35. 35.

    Dutykh, D.: A brief introduction to pseudo-spectral methods: application to diffusion problems (2016). preprint arXiv:1606.05432

  36. 36.

    Grandcleement, P., Novak, J.: Spectral methods for numerical relativity. Liv. Rev. Relat. 12, 1–103 (2009)

    Article  Google Scholar 

  37. 37.

    Hou, T.Y., Li, R.: Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  38. 38.

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

  39. 39.

    Bona, C., Masso, J.: Einstein’s evolution equations as a system of balance laws. Phys. Rev. D 4, 1022 (1989)

  40. 40.

    Bona, C., Masso, J., Seidel, E., Stela, J.: New formalism for numerical relativity. Phys. Rev. Lett. 75, 600 (1995)

    ADS  Article  Google Scholar 

  41. 41.

    Alcubierre, M., et al.: Towards standard testbeds for numerical relativity. Class. Quant. Gravity 21, 589 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  42. 42.

    Alekseenko, A.M.: Constraint preserving boundary conditions for the linearized BSSN formulation. preprint arXiv gr-qc/0405080 (2004)

  43. 43.

    Gundlach, C., Martin-Garcia, J.M.: Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions. Phys. Rev. D 74, 024016 (2006)

  44. 44.

    Brown, J.D.: Covariant formulations of Baumgarte, Shapiro, Shibata, and Nakamura and the standard gauge. Phys. Rev. D 79, 104029 (2009)

  45. 45.

    Cooley, J.W., Lewis, P.A., Welch, P.D.: The fast Fourier transform and its applications. IEEE Trans. Educ. 12, 27–34 (1969)

    Article  Google Scholar 

  46. 46.

    Boyle, M., Lindblom, L., Pfeiffer, H.P., Scheel, M.A., Kidder, L.E.: Testing the accuracy and stability of spectral methods in numerical relativity. Phys. Rev. D 72, 024006 (2007)

  47. 47.

    Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Hossain, M., Matthaeus, W.H., Ghosh, S.: On computing high order Galerkin products. Comput. Phys. Commun. 69, 1–6 (1992)

    ADS  Article  Google Scholar 

  49. 49.

    Frisch, U., Kurien, S., Pandit, R., Pauls, W., SankarRay, S., Wirth, A., Zhu, J.Z.: Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence. Phys. Rev. Lett. 101, 144501 (2008)

  50. 50.

    Shu, C.W., Don, W.S., Gottlieb, D., Schilling, O., Jameson, L.: Numerical convergence study of nearly incompressible, inviscid Taylor–Green vortex flow. J. Sci. Comput. 24, 1–27 (2005)

  51. 51.

    Schneider, K., Farge, M.: Decaying two-dimensional turbulence in a circular container. Phys. Rev. Lett. 95, 244502 (2005)

  52. 52.

    Dobler, W., Stix, M., Brandenburg, A.: Magnetic field generation in fully convective rotating spheres. Astrophys. J. 638, 336 (2006)

    ADS  Article  Google Scholar 

  53. 53.

    Servidio, S., Carbone, V., Primavera, L., Veltri, P., Stasiewicz, K.: Compressible turbulence in hall magnetohydrodynamics. Planet. Space Sci. 55, 2239–2243 (2007)

    ADS  Article  Google Scholar 

  54. 54.

    Dumbser, M., Guercilena, F., Kooppel, S., Rezzolla, L., Zanotti, O.: Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes. Phys. Rev. D 97, 084053 (2018)

  55. 55.

    Cao, Z., Yo, H.J., Yu, J.P.: Reinvestigation of moving punctured black holes with a new code. Phys. Rev. D 78, 124011 (2008)

  56. 56.

    Bona, C., Bona-Casas, C.: Gowdy waves as a testbed for constraint-preserving boundary conditions. J. Phys. Conf. Ser. 229, 012022 (2010)

  57. 57.

    New, K., Watt, K., Misner, C.W., Centrella, J.M.: Stable 3-level leapfrog integration in numerical relativity. Phys. Rev. D 58, 064022 (1998)

  58. 58.

    Babiuc, M., et al.: Implementation of standard testbeds for numerical relativity. Class. Quant. Gravity 25, 125012 (2008)

  59. 59.

    Daverio, D., Dirian, Y., Mitsou, E.: Apples with apples comparison of 3+1 conformal numerical relativity schemes (2018). preprint arXiv:1810.12346

  60. 60.

    Clough, K., Figueras, P., Finkel, H., Kunesch, M., Lim, E.A., Tunyasuvunakool, S.: Grchombo: numerical relativity with adaptive mesh refinement. Class. Quant. Gravity 32, 245011 (2015)

  61. 61.

    Brill, D.R., Lindquist, R.W.: Interaction energy in geometrostatics. Phys. Rev. D 131, 471 (1963)

  62. 62.

    Misner, C.W., Wheeler, J.A.: Classical physics as geometry. Ann. Phys. 2, 525–603 (1957)

    ADS  Article  Google Scholar 

  63. 63.

    Matzner, R.A., Huq, M.F., Shoemaker, D.: Initial data and coordinates for multiple black hole systems. Phys. Rev. D 59, 024015 (1998)

  64. 64.

    Lousto, C.O., Zlochower, Y.: Foundations of multiple-black-hole evolutions. Phys. Rev. D 77, 024034 (2008)

Download references

Acknowledgements

The simulations have been performed at the Newton HPCC Computing Facility, at the University of Calabria.

Author information

Affiliations

Authors

Corresponding author

Correspondence to C. Meringolo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Meringolo, C., Servidio, S. Aliasing instabilities in the numerical evolution of the Einstein field equations. Gen Relativ Gravit 53, 95 (2021). https://doi.org/10.1007/s10714-021-02865-5

Download citation

Keywords

  • Numerical relativity
  • Gravitational testbeds
  • BSSN decomposition
  • Black hole dynamics