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Local Characteristic Algorithms for Relativistic Hydrodynamics

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The Conformal Structure of Space-Time

Part of the book series: Lecture Notes in Physics ((LNP,volume 604))

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Abstract

Numerical schemes for the general relativistic hydrodynamic equations are discussed. The use of conservative algorithms based upon the characteristic structure of those equations, developed during the last decade building on ideas first applied in Newtonian hydrodynamics, provides a robust methodology to obtain stable and accurate solutions even in the presence of discontinuities. The knowledge of the wave structure of the above system is essential in the construction of the so-called linearized Riemann solvers, a class of numerical schemes specifically designed to solve nonlinear hyperbolic systems of conservation laws. In the last part of the review some astrophysical applications of such schemes, using the coupled system of the (characteristic) Einstein and hydrodynamic equations, are also briefly presented.

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References

  1. M.W. Choptuik: Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett. 70, 9–12 (1993)

    Article  ADS  Google Scholar 

  2. R. Gómez et al: Stable characteristic evolution of generic 3-dimensional singleblack-hole spacetimes, Phys. Rev. Lett. 80, 3915–3918 (1998)

    Article  ADS  Google Scholar 

  3. A. Lichnerowicz: J. Math. Pure Appl. 23, 37 (1944)

    MATH  MathSciNet  Google Scholar 

  4. Y. Choquet-Bruhat: In: Gravitation: An introduction to current research (John Wiley, New York 1962)

    Google Scholar 

  5. R. Arnowitt, S. Deser, C.W. Misner: ‘The Dynamics of General Relativity’. In: Gravitation: An Introduction to Current Research (John Wiley, New York 1962) pp. 227–265

    Google Scholar 

  6. H. Friedrich, A.D. Rendall: The Cauchy Problem for the Einstein Equations, Lect. Notes Phys. 540, 127–224 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  7. O. Reula: Hyperbolic Methods for Einstein’s Equations, Liv. Rev. Relativ. 1, 3 (1998)

    ADS  MathSciNet  Google Scholar 

  8. M. Shibata, T. Nakamura: Evolution of three-dimensional gravitational waves: Harmonic slicing case, Phys. Rev. D 52, 5428–5444 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  9. T.W. Baumgarte, S.L. Shapiro: On the numerical integration of Einstein’s field equations, Phys. Rev. D 59, 024007 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  10. E. Seidel, W.-M. Suen: Numerical Relativity As A Tool For Computational Astrophysics, J. Comput. Appl. Math. 109, 493–525 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. H. Bondi, M.J.G. van der Burg, A.W.K. Metzner: Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems, Proc. R. Soc. London, Sect. A 269, 21–52 (1962)

    Article  MATH  ADS  Google Scholar 

  12. R.K. Sachs: Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time, Proc. R. Soc. London, Sect. A 270, 103–126 (1962)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. R. Penrose: Asymptotic Properties of Fields and Space-Times, Phys. Rev. Lett. 10, 66–68 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Winicour: Characteristic evolution and matching, Liv. Rev. Relativ. 4, 3 (2001)

    ADS  MathSciNet  Google Scholar 

  15. H. Friedrich: Proc. R. Soc. London A 375, 169 (1981)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. P. Hübner: A Method for Calculating the Structure of (Singular) Spacetimes in the Large, Phys. Rev. D 53, 701 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  17. J. Frauendiener: Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. II. The evolution equations, Phys. Rev. D 58, 064003 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  18. P. Hübner: Numerical Calculation of Conformally Smooth Hyperboloidal Data, Class. Quant. Grav. 18, 1421 (2001)

    Article  MATH  ADS  Google Scholar 

  19. M.R. Dubal, R.A. d’Inverno, J.A. Vickers: Combining Cauchy and characteristic codes. V. CCM for a spherical spacetime containing a perfect fluid, Phys. Rev. D 58, 044019 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  20. N. Bishop, R. Gómez, L. Lehner, M. Maharaj, J. Winicour: The incorporation of matter into characteristic numerical relativity, Phys. Rev. D 60, 024005 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  21. P. Papadopoulos, J.A. Font: Relativistic hydrodynamics on spacelike and null surfaces: Formalism and computations of spherically symmetric spacetimes, Phys. Rev. D 61, 024015 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  22. P. Papadopoulos, J.A. Font: Imprints of accretion on gravitational waves from black holes, Phys. Rev. D 63, 044016 (2001)

    Article  ADS  Google Scholar 

  23. F. Linke, J.A. Font, H.-Th. Janka, E. Müller, P. Papadopoulos: Spherical collapse of supermassive stars: neutrino emission and gamma-ray bursts, Astron. Astrophys. 376, 568–579 (2001)

    Article  ADS  Google Scholar 

  24. F. Siebel, J.A. Font, E. Müller, P. Papadopoulos: Characteristic numerical relativity applied to hydrodynamic studies of neutron stars. Proceedings of the 9th Marcel Grossmann Meeting. Ed. R. Ruffini. World Scientific (2001); gr-qc/0011096

    Google Scholar 

  25. F. Siebel, J.A. Font, P. Papadopoulos: Scalar field induced oscillations of relativistic stars and gravitational collapse, Phys. Rev. D 65, 024021 (2002)

    Article  ADS  Google Scholar 

  26. F. Siebel, J.A. Font, E. Müller, P. Papadopoulos: Simulating the dynamics of relativistic stars via a light-cone approach, Phys. Rev. D 65, 064038 (2002)

    Article  ADS  Google Scholar 

  27. J.A. Font: Numerical hydrodynamics in general relativity, Liv. Rev. Relativ. 3, 2 (2000)

    ADS  MathSciNet  Google Scholar 

  28. J.R. Wilson: Numerical study of fluid flow in a Kerr space, Astrophys. J. 173, 431–438 (1972)

    Article  ADS  Google Scholar 

  29. J.M. Martí, J.M. Ibáñez, J.A. Miralles: Numerical relativistic hydrodynamics: local characteristic approach, Phys. Rev. D 43 3794–3801 (1991)

    Article  ADS  Google Scholar 

  30. M.M May, R.H. White: Hydrodynamic calculations of general relativistic collapse, Phys. Rev. D 141, 1232–1241 (1966)

    Article  ADS  MathSciNet  Google Scholar 

  31. T. Nakamura: General relativistic collapse of axially symmetric stars leading to the formation of rotating black holes, Prog. Theor. Phys. 65, 1876–1890 (1981)

    Article  ADS  Google Scholar 

  32. R.F. Stark, T. Piran: Gravitational-Wave Emission from Rotating Gravitational Collapse, Phys. Rev. Lett. 55, 891–894 (1985)

    Article  ADS  Google Scholar 

  33. H. Dimmelmeier, J.A. Font, E. Müller: Gravitational waves from relativistic rotational core collapse, Astrophys. J 560, L163–L166 (2001)

    Article  ADS  Google Scholar 

  34. T.W. Baumgarte, S.A. Hughes, S.L. Shapiro: Evolving Einstein’s field equations with matter: The “hydro without hydro” test, Phys. Rev. D 60, 087501 (1999)

    Article  ADS  Google Scholar 

  35. J.A. Font, M. Miller, W.-M. Suen, M. Tobias: Three Dimensional Numerical General Relativistic Hydrodynamics: Formulations, Methods and Code Tests, Phys. Rev. D 61, 044011 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  36. M. Alcubierre, B. Brügmann, T. Dramlitsch, J.A. Font, P. Papadopoulos, E. Seidel, N. Stergioulas, R. Takahashi: Towards a Stable Numerical Evolution of Strongly Gravitating Systems in General Relativity: The Conformal Treatments, Phys. Rev. D 62, 044034 (2000)

    Article  ADS  Google Scholar 

  37. N. Stergioulas, J.A. Font: Nonlinear r-modes in rapidly rotating relativistic stars, Phys. Rev. Lett. 86, 1148–1151 (2001)

    Article  ADS  Google Scholar 

  38. J.A. Font, T. Goodale, S. Iyer, M. Miller, L. Rezzolla, E. Seidel, N. Stergioulas, W.-M. Suen and M. Tobias: Three-dimensional general relativistic hydrodynamics II: long-term dynamics of single relativistic stars, Phys. Rev. D. 65, 084024 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  39. G.J. Mathews, P. Marronetti, J.R. Wilson: Relativistic Hydrodynamics in Close Binary Systems: Analysis of Neutron-Star Collapse, Phys. Rev. D 58, 043003 (1998)

    Article  ADS  Google Scholar 

  40. M. Shibata: Fully general relativistic simulation of coalescing binary neutron stars: Preparatory tests, Phys. Rev. D 60, 104052 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  41. M. Shibata, K. Uryu: Simulation of merging binary neutron stars in full general relativity: Γ = 2 case, Phys. Rev. D 61, 064001 (2000)

    Article  ADS  Google Scholar 

  42. M. Miller, W.M. Suen, M. Tobias: The Shapiro Conjecture: Prompt or Delayed Collapse in the head-on collision of neutron stars?, Phys. Rev. D 63, 121501 (2001)

    Article  ADS  Google Scholar 

  43. K. Thorne: Gravitational Wave Astronomy. Proceedings of the Yukawa Conference on Black Holes and Gravitational Waves: New Eyes in the 21st Century (1999)

    Google Scholar 

  44. R.A. Issacson, J.S. Welling, J. Winicour: Null cone computation of gravitational radiation, J. Math. Phys., 24, 1824–1834 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  45. R.J. LeVeque: Numerical Methods for Conservation Laws. Birkhäuser-Verlag (1990)

    Google Scholar 

  46. E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg (1997)

    MATH  Google Scholar 

  47. C. Hirsch: Numerical Computation of Internal and External Flows. Wiley (1988)

    Google Scholar 

  48. C.B. Laney: Computational gasdynamics, Cambridge University Press (1998)

    Google Scholar 

  49. J.M. Ibáñez, J.M. Martí: Riemann solvers in relativistic astrophysics, J. Comput. Appl. Math. 109, 173–211 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  50. J.M. Martí, E. Müller: Numerical hydrodynamics in special relativity, Liv. Rev. Relativ. 2, 1 (1999)

    ADS  Google Scholar 

  51. J. von Neumann, R.D. Richtmyer: A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21, 232 (1950)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. M.L. Norman, K-H. A. Winkler: In Astrophysical Radiation Hydrodynamics. Eds. M.L. Norman, K-H. A. Winkler, Reidel Publishing Company, 449–475 (1986)

    Google Scholar 

  53. S. Siegler, H. Riffert: Smoothed particle hydrodynamics simulations of ultrarelativistic shocks with artificial viscosity, Astrophys. J. 531, 1053 (2000)

    Article  ADS  Google Scholar 

  54. A.M. Anile: Relativistic Fluids and Magneto-fluids, Cambridge University Press, Cambridge, England (1989)

    MATH  Google Scholar 

  55. J.A. Font, J.M. Ibáñez, J.M. Martí, A. Marquina: Multidimensional relativistic hydrodynamics: characteristic fields and modern high-resolution shock-capturing schemes, Astron. Astrophys. 282, 304–314 (1994)

    ADS  Google Scholar 

  56. F. Eulderink, G. Mellema: General relativistic hydrodynamics with a Roe solver, Astron. Astrophys. Suppl. Ser. 110, 587–623 (1995)

    ADS  Google Scholar 

  57. S.A.E.G. Falle, S.S. Komissarov: An upwind numerical scheme for relativistic hydrodynamics with a general equation of state, Mon. Not. R. Astron. Soc. 278, 586–602 (1996)

    ADS  Google Scholar 

  58. F. Banyuls, J.A. Font, J.M. Ibáñez, J.M. Martí, J.A. Miralles: Numerical 3+1 General Relativistic Hydrodynamics: A Local Characteristic Approach, Astrophys. J. 476, 221–231 (1997)

    Article  ADS  Google Scholar 

  59. L. Wen, A. Panaitescu, P. Laguna: A shock-patching code for ultrarelativistic fluid flows, Astrophys. J. 486, 919–927 (1997)

    Article  ADS  Google Scholar 

  60. J.M. Ibáñez, M.A. Aloy, J.A. Font, J.M. Martí, J.A. Miralles, J.A. Pons: Riemann solvers in general relativistic hydrodynamics. In E.F. Toro ed. Godunov Methods: Theory and Applications, Kluwer Academic/Plenum Publishers, New York, 485–496 (2001)

    Google Scholar 

  61. P.D. Lax, B. Wendroff: Systems of conservation laws, Comm. Pure Appl. Math. 13, 217 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  62. P.D. Lax: Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conference Series Lectures in Applied Math. 11, SIAM, Philadelphia (1972)

    Google Scholar 

  63. R.D. Richtmyer, K.W. Morton: Difference Methods for Initial-Value Problems. Wiley-Interscience, New York (1967)

    MATH  Google Scholar 

  64. A. Harten: On a class of high resolution total-variation stable finite difference schemes, SIAM J. Numer. Anal. 21, 1–23 (1984)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  65. S.K. Godunov: A finite difference method for the numerical computation and discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47, 271 (1959) in Russian

    MathSciNet  Google Scholar 

  66. J.M. Martí, E. Müller: The analytical solution of the Riemann problem in relativistic hydrodynamics, J. Fluid Mech. 258, 317–333 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  67. J. Pons, J.M. Martí, E. Müller: The exact solution of the Riemann problem with non-zero tangential velocities in relativistic hydrodynamics, J. Fluid Mech. 422, 125 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  68. A. Harten, P.D. Lax, B. van Leer: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review 25, 35–61 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  69. B. Einfeldt: On Godunov-type methods for gas dynamics, SIAM J. Num. Anal. 25, 294–318 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  70. P.L. Roe: Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys. 43, 357–372 (1981)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  71. B. van Leer: Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, J. Comput. Phys. 32, 101–136 (1979)

    Article  ADS  Google Scholar 

  72. P. Colella, P.R. Woodward: The piecewise parabolic method (PPM) for gasdynamical simulations, J. Comput. Phys. 54, 174–201 (1984)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  73. A. Harten, B. Engquist, S. Osher, S.R. Chakrabarthy: Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys. 71, 231–303 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  74. R.J. LeVeque: Nonlinear Conservation Laws and Finite Volume Methods. In Computational Methods for Astrophysical Fluid Flow. Saas-Fee Advanced Course 27. Lecture Notes 1997. Springer-Verlag, 1 (1998)

    Google Scholar 

  75. R. Donat, J.A. Font, J.M. Ibáñez, A. Marquina: A Flux-Split Algorithm applied to Relativistic Flows, J. Comput. Phys. 146, 58–81 (1998)

    Article  MATH  ADS  Google Scholar 

  76. M.J. Rees: In Black Holes and Relativistic Stars, University of Chicago Press, Chicago (1998)

    Google Scholar 

  77. J. Kormendy: Supermassive Black Holes in Disk Galaxies, in Galaxy Disks and Disk Galaxies, ASP Conference Series, Vol. 230, Edited by J.G. Funes, S. J. and Enrico Maria Corsini. San Francisco, 247–256 (2001)

    Google Scholar 

  78. S. Chandrasekhar: The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity, Astrophys. J. 140, 417 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  79. W.F. Fowler: Massive Stars, Relativistic Polytropes, and Gravitational Radiation, Rev. Mod. Phys. 36, 545 (1964)

    Article  ADS  Google Scholar 

  80. G.M. Fuller, X. Shi: Supermassive Objects as Gamma-Ray Bursters, Astrophys. J. 502, L5 (1998)

    Article  ADS  Google Scholar 

  81. T. Zwerger, E. Müller: Dynamics and gravitational wave signature of axisymmetric rotational core collapse, Astron. Astrophys. 320, 209 (1997)

    ADS  Google Scholar 

  82. R. Gómez, P. Papadopoulos, J. Winicour: Null cone evolution of axisymmetric vacuum space-times, J. Math. Phys. 35, 4184–4204 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85740, Garching, Germany

    José A. Font

  2. Departamento de Astronomía y Astrofísica, Universidad de Valencia, Dr. Moliner 50, 46100, Burjassot (Valencia), Spain

    José A. Font

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  1. José A. Font
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Editors and Affiliations

  1. Theoretische Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Jörg Frauendiener

  2. MPI für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476, Golm, Germany

    Helmut Friedrich

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Font, J.A. (2002). Local Characteristic Algorithms for Relativistic Hydrodynamics. In: Frauendiener, J., Friedrich, H. (eds) The Conformal Structure of Space-Time. Lecture Notes in Physics, vol 604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45818-2_17

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  • DOI: https://doi.org/10.1007/3-540-45818-2_17

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