- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Finite-volume scheme for first-order viscoresistive relativistic magnetohydrodynamics

Ruben Lier1,2,3,*, Jay Armas1,2,3,4,†, and Oliver Porth5,‡

  • *Contact author: rlier@uva.nl
  • †Contact author: j.armas@uva.nl
  • ‡Contact author: o.j.g.porth@uva.nl
PDF

Phys. Rev. D 114, 083046 – Published 26 October, 2026

DOI: https://doi.org/10.1103/gyk4-trpr

Abstract

We present a numerical implementation of dissipative relativistic magnetohydrodynamics based on Bemfica–Disconzi–Noronha–Kovtun (BDNK) theory, in which first-order corrections render the equations causal without introducing additional dynamical variables. We show how these corrections can be incorporated in a finite-volume scheme describing the coupled dissipation of energy, momentum, and magnetic field, with the latter treated as a one-form charge. While a minimal set of BDNK terms can convert diffusive equations into telegrapher-type equations, we find that in the ultrarelativistic limit an additional correction is required for the system to behave in a stable and causal manner. With this set of equations, we develop an efficient method for primitive-variable recovery and validate the implementation through an analytical benchmark and various two-dimensional simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. A. Jaiswal and V. Roy, Relativistic hydrodynamics in heavy-ion collisions: General aspects and recent developments, Adv. High Energy Phys. 2016, 1 (2016).
  2. J. Hernandez and P. Kovtun, Relativistic magnetohydrodynamics, J. High Energy Phys. 05 (2017) 001.
  3. W. Israel and J. Stewart, Transient relativistic thermodynamics and kinetic theory, Ann. Phys. (N.Y.) 118, 341 (1979).
  4. I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Z. Phys. 198, 329 (1967).
  5. P. Kovtun, First-order relativistic hydrodynamics is stable, J. High Energy Phys. 10 (2019) 034.
  6. F. S. Bemfica, M. M. Disconzi, and J. Noronha, First-order general-relativistic viscous fluid dynamics, Phys. Rev. X 12, 021044 (2022).
  7. J. Armas and F. Camilloni, A stable and causal model of magnetohydrodynamics, J. Cosmol. Astropart. Phys. 10 (2022) 039.
  8. R. E. Hoult and P. Kovtun, Stable and causal relativistic Navier-Stokes equations, J. High Energy Phys. 06 (2020) 067.

  9. Another potential resolution to the stability problem in relativistic hydrodynamics was proposed in [72] and consists in recasting the fluid equations in an approximate Lorentzian form and in specific choice of hydrodynamic frame—the “density frame”.

  10. A. Pandya and F. Pretorius, Numerical exploration of first-order relativistic hydrodynamics, Phys. Rev. D 104, 023015 (2021).
  11. A. Pandya, E. R. Most, and F. Pretorius, Conservative finite volume scheme for first-order viscous relativistic hydrodynamics, Phys. Rev. D 105, 123001 (2022).
  12. A. Pandya, E. R. Most, and F. Pretorius, Causal, stable first-order viscous relativistic hydrodynamics with ideal gas microphysics, Phys. Rev. D 106, 123036 (2022).
  13. H. Bantilan, Y. Bea, and P. Figueras, Evolutions in first-order viscous hydrodynamics, J. High Energy Phys. 08 (2022) 298.
  14. Y. Bea and P. Figueras, Field redefinitions and evolutions in relativistic Navier-Stokes, J. High Energy Phys. 11 (2023) 110.
  15. G. S. Rocha, G. S. Denicol, and J. Noronha, Perturbative approaches in relativistic kinetic theory and the emergence of first-order hydrodynamics, Phys. Rev. D 106, 036010 (2022).
  16. G. S. Rocha, C. V. P. de Brito, and G. S. Denicol, Hydrodynamic theories for a system of weakly self-interacting classical ultrarelativistic scalar particles: Microscopic derivations and attractors, Phys. Rev. D 108, 036017 (2023).
  17. J. Bhambure, A. Mazeliauskas, J.-F. Paquet, R. Singh, M. Singh, D. Teaney, and F. Zhou, Relativistic viscous hydrodynamics in the density frame: Numerical tests and comparisons, Phys. Rev. C 111, 064910 (2025).
  18. Y. Bea, Relativistic Navier-Stokes description of the quark-gluon plasma radial flow, arXiv:2503.02931.
  19. D. Fantini and M. E. Rubio, Constraint evolution in first-order viscous relativistic fluids, Phys. Rev. D 112, 063038 (2025).
  20. N. Clarisse, E. O. Pinho, T. Patel, F. S. Bemfica, M. Hippert, and J. Noronha, Flux-conservative Bemfica-Disconzi-Noronha-Kovtun hydrodynamics and shock regularization, Phys. Rev. D 113, 024051 (2026).
  21. C. Palenzuela, L. Lehner, O. Reula, and L. Rezzolla, Beyond ideal mhd: Towards a more realistic modelling of relativistic astrophysical plasmas, Mon. Not. R. Astron. Soc. 394, 1727 (2009).
  22. A. Dash, M. Shokri, L. Rezzolla, and D. H. Rischke, Charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics, Phys. Rev. D 107, 056003 (2023).
  23. D. Schubring, Dissipative string fluids, Phys. Rev. D 91, 043518 (2015).
  24. S. Grozdanov, D. M. Hofman, and N. Iqbal, Generalized global symmetries and dissipative magnetohydrodynamics, Phys. Rev. D 95, 096003 (2017).
  25. R. Lier, A. Jain, J. Armas, and O. Porth, Resistive relativistic magnetohydrodynamics without Ampère’s law, Phys. Rev. D 112, 083059 (2025).
  26. A. J. Wright and I. Hawke, A resistive extension for ideal magnetohydrodynamics, Mon. Not. R. Astron. Soc. 491, 5510 (2019).
  27. R. E. Hoult and P. Kovtun, Causality in dissipative relativistic magnetohydrodynamics, J. High Energy Phys. 04 (2025) 009.
  28. C. Cattaneo, Sulla conduzione del calore, Atti Seminario Matematico Fisico dell’Università Modena 3, 83 (1948).
  29. T. Koide, G. S. Denicol, P. Mota, and T. Kodama, Relativistic dissipative hydrodynamics: A minimal causal theory, Phys. Rev. C 75, 034909 (2007).
  30. J. Armas and A. Jain, One-form superfluids & magnetohydrodynamics, J. High Energy Phys. 01 (2020) 041.
  31. J. Armas and A. Jain, Magnetohydrodynamics as superfluidity, Phys. Rev. Lett. 122, 141603 (2019).
  32. J. P. H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas (Cambridge University Press, Cambridge, England, 2004).
  33. W. A. Hiscock and L. Lindblom, Stability and causality in dissipative relativistic fluids, Ann. Phys. (N.Y.) 151, 466 (1983).
  34. W. A. Hiscock and L. Lindblom, Generic instabilities in first-order dissipative relativistic fluid theories, Phys. Rev. D 31, 725 (1985).
  35. L. Gavassino, Can we make sense of dissipation without causality?, Phys. Rev. X 12, 041001 (2022).
  36. M. Chandra, C. F. Gammie, F. Foucart, and E. Quataert, An extended magnetohydrodynamics model for relativistic weakly collisional plasmas, Astrophys. J. 810, 162 (2015).
  37. I. Cordeiro, E. Speranza, K. Ingles, F. S. Bemfica, and J. Noronha, Causality bounds on dissipative general-relativistic magnetohydrodynamics, Phys. Rev. Lett. 133, 091401 (2024).
  38. E. Molnár, H. Niemi, and D. H. Rischke, Numerical tests of causal relativistic dissipative fluid dynamics, Eur. Phys. J. C 65, 615 (2009).
  39. E. R. Most and J. Noronha, Dissipative magnetohydrodynamics for nonresistive relativistic plasmas: An implicit second-order flux-conservative formulation with stiff relaxation, Phys. Rev. D 104, 103028 (2021).
  40. S. Grozdanov and M. Vrbica, Algebraic structure of the gradient expansion in linearized classical hydrodynamics, Phys. Rev. D 113, 065006 (2026).
  41. L. Gavassino, M. Antonelli, and B. Haskell, Thermodynamic stability implies causality, Phys. Rev. Lett. 128, 010606 (2022).
  42. E. Krotscheck and W. Kundt, Causality criteria, Commun. Math. Phys. 60, 171 (1978).
  43. R. E. Hoult and P. Kovtun, Causality and classical dispersion relations, Phys. Rev. D 109, 046018 (2024).
  44. W. A. Hiscock and L. Lindblom, Linear plane waves in dissipative relativistic fluids, Phys. Rev. D 35, 3723 (1987).
  45. O. Porth, H. Olivares, Y. Mizuno, Z. Younsi, L. Rezzolla, M. Moscibrodzka, H. Falcke, and M. Kramer, The black hole accretion code, Comput. Astrophys. Cosmol. 4, 1 (2017).
  46. H. Olivares, O. Porth, J. Davelaar, E. R. Most, C. M. Fromm, Y. Mizuno, Z. Younsi, and L. Rezzolla, Constrained transport and adaptive mesh refinement in the black hole accretion code, Astron. Astrophys. 629, A61 (2019).
  47. G. Tóth, The ∇·b=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys. 161, 605 (2000).
  48. D. S. Balsara and D. S. Spicer, A staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys. 149, 270 (1999).
  49. P. Anninos, P. C. Fragile, and J. D. Salmonson, Cosmos++: Relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement, Astrophys. J. 635, 723 (2005).
  50. M. Brio and C. C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys. 75, 400 (1988).
  51. K. Dionysopoulou, D. Alic, C. Palenzuela, L. Rezzolla, and B. Giacomazzo, General-relativistic resistive magnetohydrodynamics in three dimensions: Formulation and tests, Phys. Rev. D 88, 044020 (2013).
  52. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Books on Physics (Dover Publications, Mineola, NY, 1981).
  53. Z. Osmanov, A. Mignone, S. Massaglia, G. Bodo, and A. Ferrari, On the linear theory of Kelvin-Helmholtz instabilities of relativistic magnetohydrodynamic planar flows, Astron. Astrophys. 490, 493 (2008).
  54. A. Ferrari, S. Massaglia, and E. Trussoni, Magnetohydrodynamic Kelvin–Helmholtz instabilities in astrophysics—III. Hydrodynamic flows with shear layers, Mon. Not. R. Astron. Soc. 198, 1065 (1982).
  55. S. A. Orszag and C. M. Tang, Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech. 90, 129 (1979).
  56. R. Keppens, O. Porth, K. Galsgaard, J. T. Frederiksen, A. L. Restante, G. Lapenta, and C. Parnell, Resistive magnetohydrodynamic reconnection: Resolving long-term, chaotic dynamics, Phys. Plasmas 20, 092109 (2013).
  57. P. Romatschke and U. Romatschke, Viscosity information from relativistic nuclear collisions: How perfect is the fluid observed at RHIC?, Phys. Rev. Lett. 99, 172301 (2007).
  58. U. Heinz and B. Schenke, Hydrodynamic description of the quark-gluon plasma (2024).
  59. M. Bleicher and E. Bratkovskaya, The quest for the quark-gluon plasma from the perspective of dynamical models of relativistic heavy-ion collisions, Astron. Nachr. 346, e70006 (2025).
  60. D. Baumann, Cosmology (Cambridge University Press, Cambridge, 2022).
  61. D. A. Uzdensky, Magnetic reconnection in extreme astrophysical environments, Space Sci. Rev. 160, 45 (2011).
  62. J. F. Mahlmann, A. A. Philippov, V. Mewes, B. Ripperda, E. R. Most, and L. Sironi, Three-dimensional dynamics of strongly twisted magnetar magnetospheres: Kinking flux tubes and global eruptions, Astrophys. J. Lett. 947, L34 (2023).
  63. K. Chatterjee, A. Philippov, A. M. Beloborodov, K. Parfrey, B. Ripperda, and E. R. Most, Relativistic magnetohydrodynamic simulations of giant magnetar bursts, Astrophys. J. Lett. 1006, L45 (2026).
  64. A. Brandenburg and K. Subramanian, Astrophysical magnetic fields and nonlinear dynamo theory, Phys. Rep. 417, 1 (2005).
  65. L. S. Keeble and F. Pretorius, First-order viscous relativistic hydrodynamics on the two-sphere, Phys. Rev. D 112, 124034 (2025).
  66. H. L. H. Shum, F. Abalos, Y. Bea, M. Bezares, P. Figueras, and C. Palenzuela, Neutron star evolution with the Bemfica-Disconzi-Noronha-Kovtun viscous hydrodynamics framework, Phys. Rev. D 113, 084029 (2026).
  67. L. S. Keeble and J. Redondo-Yuste, Radial oscillations of viscous stars, Phys. Rev. D 113, 124058 (2026).
  68. S. Bussières, J. Redondo-Yuste, J. J. Ortega Gómez, and V. Cardoso, Axial oscillations of viscous neutron stars, Phys. Rev. D 114, 044012 (2026).
  69. V. Boyanov, V. Cardoso, K. D. Kokkotas, and J. Redondo-Yuste, Dynamical response of viscous objects to gravitational waves, Phys. Rev. Lett. 135, 151402 (2025).
  70. J. Redondo-Yuste and V. Cardoso, Superradiant amplification by rotating viscous compact objects, Phys. Rev. D 112, L061501 (2025).
  71. L. Del Zanna, O. Zanotti, N. Bucciantini, and P. Londrillo, Echo: A Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics, Astron. Astrophys. 473, 11 (2007).
  72. J. Armas and A. Jain, Effective field theory for hydrodynamics without boosts, SciPost Phys. 11, 054 (2021).

Outline

Information

Phys. Rev. D 114, 083046– Published 26 October, 2026
  • Received 26 June 2026
  • Accepted 30 September 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation