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Capturing Composite Waves in Non-convex Special Relativistic Hydrodynamics

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Abstract

We deal with the numerical approximation of the complex structure in special relativistic hydrodynamics (SRHD) when the system is closed with a non-convex equation of state (EOS). We consider a recently introduced phenomenological EOS (Ibáñez et al. in MNRAS 476:1100, 2018) that mimics the loss of classical behavior when the fluid enters into a non-convex—thermodynamically—region in the relativistic regime. We introduce a flux formulation to approximate the solution of Riemann problems in SRHD such that the non-classical dynamics is detected and well resolved. We also design a strategy to recover primitive variables based on iterative procedures and present a detailed analysis providing a sufficient condition to ensure convergence. We propose a set of Riemann problems in one and two dimensions including blast waves, colliding slabs and expanding slabs, illustrating the strong complex dynamics arising in non-convex SRHD.

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References

  1. Aloy, M.A., Ibáñez, J.M., Sanchis-Gual, N., Obergaulinger, M., Font, J.A., Serna, S., Marquina, A.: Neutron star collapse and gravitational waves with a non-convex equation of state. MNRAS 484, 4980–5008 (2019)

    Google Scholar 

  2. Anile, A.M.: Relativistic Fluids and Magneto-Fluids. Cambridge Univ. Press, Cambridge (1989)

    MATH  Google Scholar 

  3. Argrow, B.M.: Computational analysis of dense gas shock tube flow. Shock Waves 6, 241–248 (1996)

    MATH  Google Scholar 

  4. Bauswein, A., Janka, H.-T., Oechslin, R.: Testing approximations of thermal effects in neutron star merger simulations. Phys. Rev. D 82, 084043 (2010)

    Google Scholar 

  5. Bethe, H.A.: The Theory of Shock Waves for an Arbitrary Equation of State. Technical Report 545 Office of Scientific Research and Development (1942)

  6. Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2), 400–422 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Cinnella P., Corre, C.: APS Division of Fluid Dynamics Meeting Abstracts (2006)

  8. Cinnella, P.: Transonic flows of dense gases over finite wings. Phys. Fluids 20, 046103 (2008)

    MATH  Google Scholar 

  9. Del Zanna, L., Bucciantini, N.: An efficient shock-capturing central-type scheme for multidimensional relativistic flows. A&A 390, 1177 (2002)

    MATH  Google Scholar 

  10. Donat, R., Marquina, A.: Capturing shock reflections: an improved flux formula. J. Comput. Phys. 125, 42–58 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Donat, R., Font, J.A., Ibáñez, J.M., Marquina, A.: A flux-split algorithm applied to relativistic flow. J. Comput. Phys. 146, 58–81 (1998)

    MATH  Google Scholar 

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

    Google Scholar 

  13. Guardone, A., Vigevano, L.: Roe Linearization for the van der Waals Gas. J. Comput. Phys. 175, 50–78 (2002)

    MATH  Google Scholar 

  14. Guardone, A., Zamfirescu, C., Colonna, P.: Maximum intensity of rarefaction shock waves for dense gases. J. Fluid Mech. 642, 127 (2010)

    MATH  Google Scholar 

  15. Haensel, P., Levenfish, K.P., Yakovlev, D.G.: Adiabatic index of dense matter and damping of neutron star pulsations. Astron. Astrophys. 394, 213 (2002)

    Google Scholar 

  16. Haensel, P., Potekhin, A.Y.: Analytical representations of unified equations of state of neutron-star matter. Astron. Astrophys. 428, 191 (2004)

    MATH  Google Scholar 

  17. Haensel, P., Potekhin, A.Y., Yakovlev, D.G. (eds.): Neutron Stars 1: Equation of State and Structure. Astrophysics and Space Science Library, vol. 326. Springer-Verlag, New York (2007)

    Google Scholar 

  18. Heuze, O., Jaouen, S., Jourdren, H.: Dissipative issue of high-order shock capturing schemes with non-convex equations of state. J. Comput. Phys. 228, 833–860 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Ibáñez, J.M., Cordero-Carrión, I., Martí, J.M., Miralles, J.A.: On the convexity of relativistic hydrodynamics. Class. Quantum Gravity 30, 057002 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Ibáñez, J.M., Marquina, A., Serna, S., Aloy, M.A.: Anomalous dynamics triggered by a non-convex equation of state in relativistic flows. MNRAS 476, 1100 (2018)

    Google Scholar 

  21. Janka, H.-T., Hanke, F., Hüdepohl, L., Marek, A., Müller, B., Obergaulinger, M.: Core-collapse supernovae: reflections and directions. Prog. Theor Exp Phys 2012(1), 01A309 (2012)

    Google Scholar 

  22. Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shocks Waves. SIAM, Philadelphia (1973)

    MATH  Google Scholar 

  23. LeVeque, R.J., Mihalas, D., Dorfi, E.A., Müller, E.: Computational Methods for Astrophysical Fluid Flow, Saas-Fee Advanced Courses, vol. 27. Springer, New York (1998)

    Google Scholar 

  24. LIGO Scientific Collaboration and Virgo Collaboration: GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017)

  25. Martí, J.M., Miralles, J.A., Diaz-Alonso, J., Ibanez, J.M.: Field theoretical model for nuclear and neutron matter. IV-Radial oscillations of warm cores in neutron stars. Astrophys. J. 329, 780 (1988)

    Google Scholar 

  26. Martí, J.M., Müller, E.: Numerical hydrodynamics in special relativity. Living Rev. Relativ. 6, 7 (2003). https://doi.org/10.12942/lrr-2003-7

    Article  MathSciNet  MATH  Google Scholar 

  27. Martí, J.M., Müller, E.: Grid-based methods in relativistic hydrodynamics and magnetohydrodynamics. Living Rev. Comput. Astrophys. 1, 3 (2015). https://doi.org/10.1007/lrca-2015-3

    Article  Google Scholar 

  28. Martí, J.M., Müller, E., Font, J.A., Ibáñez, J.M., Marquina, A.: Morphology and dynamics of relativistic jets. Astrophys. J. 479(1), 151 (1997)

    Google Scholar 

  29. Marquina, A., Martí, J.M., Ibáñez, J.M., Miralles, J.A., Donat, R.: Ultrarelativistic hydrodynamics-high-resolution shock-capturing methods. Astron. Astrophys. 258, 566 (1992)

    Google Scholar 

  30. Marquina, A.: Local piecewise hyperbolic reconstructions for nonlinear scalar conservation laws. SIAM J. Sci. Comput. 15, 892–915 (1994)

    MathSciNet  MATH  Google Scholar 

  31. Menikoff, R.: Empirical equations of state for solids. In: Horie, Y. (ed.) ShockWave Science and Technology Reference Library, pp. 143–188. Springer, Berlin (2007)

    Google Scholar 

  32. Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61(1), 75–130 (1989)

    MathSciNet  MATH  Google Scholar 

  33. Mignone, A., Bodo, G.: A HLLC Riemann solver for relativistic flows-I. Hydrodynamics. MNRAS 364, 126–136 (2005)

    Google Scholar 

  34. Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    Google Scholar 

  35. Rezzolla, L., Zanotti, O.: Relativistic Hydrodynamics. Oxford University Press, Oxford (2013)

    MATH  Google Scholar 

  36. Serna, S., Marquina, A.: Capturing shock waves in inelastic granular gases. J. Comput. Phys. 209(2), 787–795 (2005)

    MathSciNet  MATH  Google Scholar 

  37. Serna, S.: A characteristic-based nonconvex entropy-fix upwind scheme for the ideal magnetohydrodynamic equations. J. Comput. Phys 228, 4232–4247 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Serna, S., Marquina, A.: Anomalous wave structure in magnetized materials described by non-convex equations of state. Phys. Fluids 26, 016101 (2014)

    MATH  Google Scholar 

  39. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, 2. J. Comput. Phys. 83(1), 32–78 (1989)

    MathSciNet  MATH  Google Scholar 

  40. Schulz-Rinne, C.W., Collins, J.P., Glaz, H.M.: Numerical solution of the Riemann problem of two-dimensional gas dynamics. SIAM J. Sci. Comput 14, 1394–1414 (1993)

    MathSciNet  MATH  Google Scholar 

  41. Thompson, P.A.: A fundamental derivative of gas dynamics. Phys. Fluids 14, 1843–1849 (1971)

    MATH  Google Scholar 

  42. Voss A.: PhD thesis, University of Wuppertal (2005)

  43. Wu, K., Tang, H.: High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics. J. Comput. Phys. 298, 539–564 (2015)

    MathSciNet  MATH  Google Scholar 

  44. Wu, K., Tang, H.: Physical-constraint-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state. Astrophys. J. Suppl. 228, 3 (2017)

    Google Scholar 

  45. Wu, K.: Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics. Phys. Rev. D 95, 103001 (2017)

    MathSciNet  Google Scholar 

  46. Wu, K., Tang, H.: Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 27, 1871–1928 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Zeldovich, Y.: On the possibility of rarefaction shock waves. Zh. Eksp. Teor. Fiz. 4, 363 (1946)

    Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their valuable comments and recommendations which helped to improve the manuscript. Work partially supported by the Spanish Government Grants: PGC2018-101119-B-I00 and PGC2018-095984-B-I00.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universitat de València, Burjassot-València, Spain

    Antonio Marquina

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra-Barcelona, Spain

    Susana Serna

  3. Departamento de Astronomía y Astrofísica, Universitat de València, Burjassot-València, Spain

    José M. Ibáñez

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  1. Antonio Marquina
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  2. Susana Serna
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  3. José M. Ibáñez
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Correspondence to Susana Serna.

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Marquina, A., Serna, S. & Ibáñez, J.M. Capturing Composite Waves in Non-convex Special Relativistic Hydrodynamics. J Sci Comput 81, 2132–2161 (2019). https://doi.org/10.1007/s10915-019-01074-2

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