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High-Order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-Type Equation of State

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Abstract

All the existing entropy stable (ES) schemes for relativistic hydrodynamics (RHD) in the literature were restricted to the ideal equation of state (EOS), which however is often a poor approximation for most relativistic flows due to its inconsistency with the relativistic kinetic theory. This paper develops high-order ES finite difference schemes for RHD with general Synge-type EOS, which encompasses a range of special EOSs. We first establish an entropy pair for the RHD equations with general Synge-type EOS in any space dimensions. We rigorously prove that the found entropy function is strictly convex and derive the associated entropy variables, laying the foundation for designing entropy conservative (EC) and ES schemes. Due to relativistic effects, one cannot explicitly express primitive variables, fluxes, and entropy variables in terms of conservative variables. Consequently, this highly complicates the analysis of the entropy structure of the RHD equations, the investigation of entropy convexity, and the construction of EC numerical fluxes. By using a suitable set of parameter variables, we construct novel two-point EC fluxes in a unified form for general Synge-type EOS. We obtain high-order EC schemes through linear combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES schemes are achieved by incorporating dissipation terms into the EC schemes, based on (weighted) essentially non-oscillatory reconstructions. Additionally, we derive the general dissipation matrix for general Synge-type EOS based on the scaled eigenvectors of the RHD system. We also define a suitable average of the dissipation matrix at the cell interfaces to ensure that the resulting ES schemes can resolve stationary contact discontinuities accurately. Several numerical examples are provided to validate the accuracy and effectiveness of our schemes for RHD with four special EOSs.

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Data Availability

Programming codes and associated data for the numerical examples in Sect. 5 are available at https://github.com/PeterX3AUG1/ESRHD_gEOS_source.

References

  1. Abgrall, R.: A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. J. Comput. Phys. 372, 640–666 (2018)

    MathSciNet  Google Scholar 

  2. Abgrall, R., Öffner, P., Ranocha, H.: Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes: Application to structure preserving discretization. J. Comput. Phys. 453, 110955 (2022)

    MathSciNet  Google Scholar 

  3. Balsara, D.S., Kim, J.: A subluminal relativistic magnetohydrodynamics scheme with ADER-WENO predictor and multidimensional Riemann solver-based corrector. J. Comput. Phys. 312, 357–384 (2016)

    MathSciNet  Google Scholar 

  4. Barth, T.: Numerical methods for gasdynamic systems on unstructured meshes. In: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp. 195–285. Springer (1999)

  5. Barth, T.: On the role of involutions in the discontinuous Galerkin discretization of Maxwell and magnetohydrodynamic systems. In: Compatible Spatial Discretizations, pp. 69–88. Springer (2006)

  6. Bhoriya, D., Kumar, H.: Entropy-stable schemes for relativistic hydrodynamics equations. Z. Angew. Math. Phys. 71, 1–29 (2020)

    MathSciNet  Google Scholar 

  7. Biswas, B., Dubey, R.K.: Low dissipative entropy stable schemes using third order WENO and TVD reconstructions. Adv. Comput. Math. 44, 1153–1181 (2018)

    MathSciNet  Google Scholar 

  8. Biswas, B., Kumar, H., Bhoriya, D.: Entropy stable discontinuous Galerkin schemes for the special relativistic hydrodynamics equations. Comput. Math. Appl. 112, 55–75 (2022)

    MathSciNet  Google Scholar 

  9. Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier–Stokes equations: Discontinuous interfaces. SIAM J. Sci. Comput. 36, B835–B867 (2014)

    MathSciNet  Google Scholar 

  10. Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier–Stokes equations. Commun. Comput. Phys. 14, 1252–1286 (2013)

    MathSciNet  Google Scholar 

  11. Chandrashekar, P., Klingenberg, C.: Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian meshes. SIAM J. Numer. Anal. 54, 1313–1340 (2016)

    MathSciNet  Google Scholar 

  12. Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017)

    MathSciNet  Google Scholar 

  13. Chen, Y., Kuang, Y., Tang, H.: Second-order accurate BGK schemes for the special relativistic hydrodynamics with the Synge equation of state. J. Comput. Phys. 442, 110438 (2021)

    MathSciNet  Google Scholar 

  14. Chen, Y., Wu, K.: A physical-constraint-preserving finite volume WENO method for special relativistic hydrodynamics on unstructured meshes. J. Comput. Phys. 466, 111398 (2022)

    MathSciNet  Google Scholar 

  15. Crandall, M.G., Majda, A.: Monotone difference approximations for scalar conservation laws. Math. Comput. 34, 1–21 (1980)

    MathSciNet  Google Scholar 

  16. Del Zanna, L., Bucciantini, N.: An efficient shock-capturing central-type scheme for multidimensional relativistic flows-I. Hydrodyn. Astron. Astrophys. 390, 1177–1186 (2002)

    Google Scholar 

  17. Dolezal, A., Wong, S.: Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 120, 266–277 (1995)

    MathSciNet  Google Scholar 

  18. Duan, J., Tang, H.: High-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamics. Adv. Appl. Math. Mech. 12, 1–29 (2020)

    MathSciNet  Google Scholar 

  19. Duan, J., Tang, H.: High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics. J. Comput. Phys. 421, 109731 (2020)

    MathSciNet  Google Scholar 

  20. Duan, J., Tang, H.: Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics. J. Comput. Phys. 426, 109949 (2021)

    MathSciNet  Google Scholar 

  21. Duan, J., Tang, H.: High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics. J. Comput. Phys. 431, 110136 (2021)

    MathSciNet  Google Scholar 

  22. Duan, J., Tang, H.: High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto) hydrodynamics. J. Comput. Phys. 456, 111038 (2022)

    MathSciNet  Google Scholar 

  23. Endeve, E., Buffaloe, J., Dunham, S.J., Roberts, N., Andrew, K., Barker, B., Pochik, D., Pulsinelli, J., Mezzacappa, A.: thornado-hydro: towards discontinuous Galerkin methods for supernova hydrodynamics. In: Journal of Physics: Conference Series, vol. 1225, p. 012014. IOP Publishing (2019)

  24. Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. J. Comput. Phys. 252, 518–557 (2013)

    MathSciNet  Google Scholar 

  25. Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50, 544–573 (2012)

    MathSciNet  Google Scholar 

  26. Fjordholm, U.S., Mishra, S., Tadmor, E.: ENO reconstruction and ENO interpolation are stable. Found. Comput. Math. 13, 139–159 (2013)

    MathSciNet  Google Scholar 

  27. Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35, A1233–A1253 (2013)

    MathSciNet  Google Scholar 

  28. Gassner, G.J., Winters, A.R., Kopriva, D.A.: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291–308 (2016)

    MathSciNet  Google Scholar 

  29. Harten, A., Hyman, J.M., Lax, P.D., Keyfitz, B.: On finite-difference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29, 297–322 (1976)

    MathSciNet  Google Scholar 

  30. He, P., Tang, H.: An adaptive moving mesh method for two-dimensional relativistic hydrodynamics. Commun. Comput. Phys. 11, 114–146 (2012)

    MathSciNet  Google Scholar 

  31. Hiltebrand, A., Mishra, S.: Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws. Numer. Math. 126, 103–151 (2014)

    MathSciNet  Google Scholar 

  32. Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys. 228, 5410–5436 (2009)

    MathSciNet  Google Scholar 

  33. Ketcheson, D.I.: Relaxation Runge–Kutta methods: Conservation and stability for inner-product norms. SIAM J. Numer. Anal. 57, 2850–2870 (2019)

    MathSciNet  Google Scholar 

  34. Kidder, L.E., Field, S.E., Foucart, F., Erick, S.: SpECTRE: a task-based discontinuous Galerkin code for relativistic astrophysics. J. Comput. Phys. 335, 84–114 (2017)

    MathSciNet  Google Scholar 

  35. Lefloch, P.G., Mercier, J.-M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40, 1968–1992 (2002)

    MathSciNet  Google Scholar 

  36. Li, S., Duan, J., Tang, H.: High-order accurate entropy stable adaptive moving mesh finite difference schemes for (multi-component) compressible Euler equations with the stiffened equation of state. Comput. Methods Appl. Mech. Eng. 399, 115311 (2022)

    MathSciNet  Google Scholar 

  37. Liu, Y., Shu, C.-W., Zhang, M.: Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. J. Comput. Phys. 354, 163–178 (2018)

    MathSciNet  Google Scholar 

  38. Marquina, A., Serna, S., Ibáñez, J.M.: Capturing composite waves in non-convex special relativistic hydrodynamics. J. Sci. Comput. 81, 2132–2161 (2019)

    MathSciNet  Google Scholar 

  39. Martí, J.M., Müller, E.: Numerical hydrodynamics in special relativity. Living Rev. Relativ. 6, 7 (2003)

    MathSciNet  Google Scholar 

  40. Martí, J.M., Müller, E.: Grid-based methods in relativistic hydrodynamics and magnetohydrodynamics. Living Rev. Comput. Astrophys. 1, 3 (2015)

    Google Scholar 

  41. Mathews, W.G.: The hydromagnetic free expansion of a relativistic gas. Astrophys. J. 165, 147 (1971)

    Google Scholar 

  42. May, M.M., White, R.H.: Hydrodynamic calculations of general-relativistic collapse. Phys. Rev. 141, 1232 (1966)

    MathSciNet  Google Scholar 

  43. Mewes, V., Zlochower, Y., Campanelli, M., Baumgarte, T.W., Etienne, Z.B., Armengol, F.G.L., Cipolletta, F.: Numerical relativity in spherical coordinates: A new dynamical spacetime and general relativistic MHD evolution framework for the Einstein Toolkit. Phys. Rev. D 101, 104007 (2020)

    MathSciNet  Google Scholar 

  44. Mignone, A., Bodo, G.: An HLLC Riemann solver for relativistic flows-I. Hydrodynamics. Mont. Not. R. Astronom. Soc. 364, 126–136 (2005)

    Google Scholar 

  45. Mignone, A., Plewa, T., Bodo, G.: The piecewise parabolic method for multidimensional relativistic fluid dynamics. Astrophys. J. Suppl. Ser. 160, 199 (2005)

    Google Scholar 

  46. Osher, S.: Riemann solvers, the entropy condition, and difference. SIAM J. Numer. Anal. 21, 217–235 (1984)

    MathSciNet  Google Scholar 

  47. Osher, S., Tadmor, E.: On the convergence of difference approximations to scalar conservation laws. Math. Comput. 50, 19–51 (1988)

    MathSciNet  Google Scholar 

  48. Radice, D., Rezzolla, L.: Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes. Phys. Rev. D 84, 024010 (2011)

    Google Scholar 

  49. Radice, D., Rezzolla, L.: THC: a new high-order finite-difference high-resolution shock-capturing code for special-relativistic hydrodynamics. Astronomy Astrophys. 547, A26 (2012)

    Google Scholar 

  50. Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci. Comput. 76, 216–242 (2018)

    MathSciNet  Google Scholar 

  51. Ranocha, H., Sayyari, M., Dalcin, L., Parsani, M., Ketcheson, D.I.: Relaxation Runge–Kutta methods: fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations. SIAM J. Sci. Comput. 42, A612–A638 (2020)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  53. Roe, P.L.: Affordable, entropy consistent flux functions. In: 11th International Conference on Hyperbolic Problems: Theory, Numerics and Applications, Lyon (2006)

  54. Ryu, D., Chattopadhyay, I., Choi, E.: Equation of state in numerical relativistic hydrodynamics. Astrophys. J. Suppl. Ser. 166, 410 (2006)

    Google Scholar 

  55. Sokolov, I., Zhang, H.-M., Sakai, J.: Simple and efficient Godunov scheme for computational relativistic gas dynamics. J. Comput. Phys. 172, 209–234 (2001)

    Google Scholar 

  56. Synge, J.L.: The Relativistic Gas. North-Holland Publishing Company, Amsterdam (1957)

    Google Scholar 

  57. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49, 91–103 (1987)

    MathSciNet  Google Scholar 

  58. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)

    MathSciNet  Google Scholar 

  59. Taub, A.: Relativistic Rankine–Hugoniot equations. Phys. Rev. 74, 328 (1948)

    MathSciNet  Google Scholar 

  60. Tchekhovskoy, A., McKinney, J.C., Narayan, R.: WHAM: a WENO-based general relativistic numerical scheme-I. Hydrodynamics. Mon. Not. R. Astronom. Soc. 379, 469–497 (2007)

    Google Scholar 

  61. Teukolsky, S.A.: Formulation of discontinuous Galerkin methods for relativistic astrophysics. J. Comput. Phys. 312, 333–356 (2016)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  63. Winters, A.R., Gassner, G.J.: Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations. J. Comput. Phys. 304, 72–108 (2016)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  65. Wu, K.: Minimum principle on specific entropy and high-order accurate invariant region preserving numerical methods for relativistic hydrodynamics. SIAM J. Sci. Comput. 43, B1164–B1197 (2021)

    MathSciNet  Google Scholar 

  66. Wu, K., Shu, C.-W.: Entropy symmetrization and high-order accurate entropy stable numerical schemes for relativistic MHD equations. SIAM J. Sci. Comput. 42, A2230–A2261 (2020)

    MathSciNet  Google Scholar 

  67. Wu, K., Shu, C.-W.: Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations. Numerische Mathematik 1, 1–43 (2021)

    MathSciNet  Google Scholar 

  68. Wu, K., Shu, C.-W.: Geometric quasilinearization framework for analysis and design of bound-preserving schemes. SIAM Rev. 65, 1031–1073 (2023)

    MathSciNet  Google Scholar 

  69. 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  Google Scholar 

  70. 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  Google Scholar 

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

    Google Scholar 

  72. Zhang, W., MacFadyen, A.I.: RAM: A relativistic adaptive mesh refinement hydrodynamics code. Astrophys. J. Suppl. Ser. 164, 255 (2006)

    Google Scholar 

  73. Zhao, J., Tang, H.: Runge–Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics. J. Comput. Phys. 242, 138–168 (2013)

    MathSciNet  Google Scholar 

  74. Zhao, W.: Strictly convex entropy and entropy stable schemes for reactive Euler equations. Math. Comput. 91, 735–760 (2022)

    MathSciNet  Google Scholar 

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Funding

This work is partially supported by Shenzhen Science and Technology Program (Grant No. RCJC20 221008092757098) and National Natural Science Foundation of China (Grant No. 12171227).

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Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, People’s Republic of China

    Linfeng Xu

  2. SUSTech International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, People’s Republic of China

    Shengrong Ding

  3. Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, and National Center for Applied Mathematics Shenzhen (NCAMS), Shenzhen, 518055, People’s Republic of China

    Kailiang Wu

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  1. Linfeng Xu
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Xu, L., Ding, S. & Wu, K. High-Order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-Type Equation of State. J Sci Comput 98, 43 (2024). https://doi.org/10.1007/s10915-023-02440-x

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