Использование кусочно-параболической реконструкции физических переменных в схеме Русанова. II. Уравнения специальной релятивистской магнитной гидродинамики

Использование кусочно-параболической реконструкции физических переменных в схеме Русанова. II. Уравнения специальной релятивистской магнитной гидродинамики

Куликов И. М.
Сибирский журнал индустриальной математики, 27, 1(97), 29–42 (2024)
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УДК 519.4 
DOI: 10.33048/SIBJIM.2024.27.103

Аннотация:

Как известно, схема Русанова для решения гидродинамических уравнений является одной из самых робастных в классе численных схем решения задачи Римана. Ранее было показано, что схема Русанова с использованием кусочно-параболической реконструкции физических переменных даёт малодиссипативную схему, качественно и количественно соответствующую схемам типа Рое и семейству схем Хартена—Лакса—Ван Леера при использовании аналогичной реконструкции. При этом в отличие от этих схем численное решение свободно от артефактов. В случае уравнений специальной релятивистской магнитной гидродинамики спектральное разложение для разрешения задачи Римана является достаточно сложным и не имеет аналитического решения. В статье предлагается развитие схемы Русанова с использованием кусочно-параболического представления решения на уравнения специальной релятивистской магнитной гидродинамики. Проведена верификация разработанной схемы на восьми классических задачах о распаде произвольного разрыва, описывающих основные особенности релятивистских замагниченных течений.

Литература:
  1. Ferrari A. Modeling extragalactic jets // Annu. Rev. Astron. Astrophys. 1998. V. 36. P. 539–598.
     
  2. Gabuzda D. C., Pushkarev A. B., Garnich N. N. Unusual radio properties of the BL Lac object 0820+225 // Mon. Notices Royal Astron. Soc. 2001. V. 327, N 1. P. 1–9.
     
  3. Gabuzda D. C., Gomez J. L. VSOP polarization observations of the BL Lacertae object OJ 287 // Mon. Notices Royal Astron. Soc. 2001. V. 320, N 4. P. L49–L54.
     
  4. Attridge J., Roberts D., Wardle J. Radio Jet-Ambient Medium Interactions on Parsec Scales in the Blazar 1055+018 // Astrophys. J. 1999. V. 518, N 2. P. L87–L90.
     
  5. Krause M., Lohr A. The magnetic field along the jets of NGC 4258 // Astron. Astrophys. 2004. V. 420, N 1. P. 115–123.
     
  6. Kigure H., Uchida Y., Nakamura M., Hirose S., Cameron R. Distribution of Faraday Rotation Measure in Jets from Active Galactic Nuclei. II. Prediction from Our Sweeping Magnetic Twist Model for the Wiggled Parts of Active Galactic Nucleus Jets and Tails // Astrophys. J. 2004. V. 608, N 1. P. 119–135.
     
  7. Anton L., Miralles J., Marti J., Ibanez J., Aloy M., Mimica P. Relativistic magnetohydrodynamics: renormalized eigenvectors and full wave decomposition Riemann solver // Astrophys. J. Suppl. Ser. 2010. V. 188, N 1. P. 1–31.
     
  8. Leismann T., Anton L., Aloy M., Mueller E., Marti J., Miralles J., Ibanez J. Relativistic MHD simulations of extragalactic jets // Astron. Astrophys. 2005. V. 436, N 2. P. 503–526.
     
  9. Nunez-de la Rosa J., Munz C.-D. XTROEM-FV: a new code for computational astrophysics based on very high order finite-volume methods – II. Relativistic hydro- and magnetohydrodynamics // Mon. Notices Royal Astron. Soc. 2016. V. 460, N 1. P. 535–559.
     
  10. Lora-Clavijo F., Cruz-Osorio A., Guzman F. CAFE: a new relativistic MHD code // Astrophys. J. Suppl. Ser. 2015. V. 218, N 2. Article 24.
     
  11. Kulikov I. M. Using Piecewise Parabolic Reconstruction of Physical Variables in the Rusanov Solver. I. The Special Relativistic Hydrodynamics Equations // J. Appl. Ind. Math. 2023. V. 17, N 4. P. 737–749.
     
  12. Komissarov S. S. A Godunov-type scheme for relativistic magnetohydrodynamics // Mon. Notices Royal Astron. Soc. 1999. V. 303, N 2. P. 343–366.
     
  13. Balsara D. Total Variation Diminishing Scheme for Relativistic Magnetohydrodynamics // Astrophys. J. Suppl. Ser. 2001. V. 132, N 1. P. 83–101.
     
  14. Giacomazzo B., Rezzolla L. The exact solution of the Riemann problem in relativistic magnetohydrodynamics // J. Fluid Mech. 2006. V. 562. P. 223–259.
     
  15. Brio M., Wu C. An upwind differencing scheme for the equations of ideal magnetohydrodynamics // J. Comput. Phys. 1988. V. 75, N 2. P. 400–422.
     
  16. Kriksin Y. A., Tishkin V. F. Variational Entropic Regularization of the Discontinuous Galerkin Method for Gasdynamic Equations // Math. Models Comput. Simul. 2019. V. 11, N 6. P. 1032–1040.
     
  17. Godunov S. K. Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field // Comput. Math. Math. Phys. 2012. V. 52, N 5. P. 787–799.
     
  18. Godunov S. K. About inclusion of Maxwell’s equations in systems relativistic of the invariant equations // Comput. Math. Math. Phys. 2013. V. 53, N 8. P. 1179–1182.
     
  19. Godunov S. K., Kulikov I. M. Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee // Comput. Math. Math. Phys. 2014. V. 54, N 6. P. 1012–1024.
     
  20. Freistuehler H., Trakhinin Yu. Symmetrizations of RMHD equations and stability of relativistic currentvortex sheets // Class. Quantum Gravity. 2013. V. 30, N 8. Article 085012.
     
  21. Trakhinin Yu. Local Existence of Contact Discontinuities in Relativistic Magnetohydrodynamics // Siberian Adv. Math. 2020. V. 30, N 2. P. 55–76.
     
  22. Lee D., Faller H., Reyes A. The Piecewise Cubic Method (PCM) for computational fluid dynamics // J. Comput. Phys. 2017. V. 341, N 1. P. 230–257.

Исследование выполнено при финансовой поддержке Российского научного фонда (проект № 23-11-00014; https://rscf.ru/project/23-11-00014/). Других источников финансирования проведения или руководства данным конкретным исследованием не было.

И. М. Куликов
  1. Институт вычислительной математики и математической геофизики СО РАН, 
    просп. Акад. Лаврентьева, 6, г. Новосибирск 630090, Россия

E-mail: kulikov@ssd.sscc.ru

Статья поступила 10.06.2023 г.
После доработки — 14.07.2023 г.
Принята к публикации 07.02.2024 г.

Abstract:

Rusanov’s scheme for solving hydrodynamic equations is one of the most robust in the class of Riemann solvers. It was previously shown that Rusanov’s scheme based on piecewise parabolic reconstruction of primitive variables gives a low-dissipative scheme relevant to Roe and Harten—Lax—Van Leer solvers when using a similar reconstruction. Moreover, unlike these solvers, the numerical solution is free from artifacts. In the case of equations of special relativistic magnetohydrodynamics, the spectral decomposition for solving the Riemann problem is quite complex and does not have an analytical solution. The present paper proposes the development of Rusanov’s scheme using a piecewise parabolic reconstruction of primitive variables to use in the equations of special relativistic magnetohydrodynamics. The developed scheme was verified using eight classical problems on the decay of an arbitrary discontinuity that describe the main features of relativistic magnetized flows.

References:
  1. A. Ferrari, “Modeling extragalactic jets,” Annu. Rev. Astron. Astrophys. 36, 539–598 (1998).
     
  2. D. C. Gabuzda, A. B. Pushkarev, and N. N. Garnich, “Unusual radio properties of the BL Lac object 0820+225,” Mon. Not. R. Astron. Soc. 327 (1), 1–9 (2001).
     
  3. D. C. Gabuzda and J. L. Gomez, “VSOP polarization observations of the BL Lacertae object OJ 287,” Mon. Not. R. Astron. Soc. 320 (4), L49–L54 (2001).
     
  4. J. Attridge, D. Roberts, and J. Wardle, “Radio jet-ambient medium interactions on parsec scales in the Blazar 1055+018,” Astrophys. J. 518 (2), L87–L90 (1999).
     
  5. M. Krause and A. Lohr, “The magnetic field along the jets of NGC 4258,” Astron. Astrophys. 420 (1), 115–123 (2004).
     
  6. H. Kigure, Y. Uchida, M. Nakamura, S. Hirose, and R. Cameron, “Distribution of Faraday rotation measure in jets from active galactic nuclei. II. Prediction from our sweeping magnetic twist model for the wiggled parts of active galactic nucleus jets and tails,” Astrophys. J. 608 (1), 119–135 (2004).
     
  7. L. Anton, J. Miralles, J. Marti, J. Ibanez, M. Aloy, and P. Mimica, “Relativistic magnetohydrodynamics: renormalized eigenvectors and full wave decomposition Riemann solver,” Astrophys. J. Suppl. Ser. 188 (1), 1–31 (2010).
     
  8. T. Leismann, L. Anton, M. Aloy, E. Mueller, J. Marti, J. Miralles, and J. Ibanez, “Relativistic MHD simulations of extragalactic jets,” Astron. Astrophys. 436 (2), 503–526 (2005).
     
  9. J. Nunez-de la Rosa and C.-D. Munz, “XTROEM-FV: A new code for computational astrophysics based on very high order finite-volume methods — II. Relativistic hydro- and magnetohydrodynamics,” Mon. Not. R. Astron. Soc. 460 (1), 535–559 (2016).
     
  10. F. Lora-Clavijo, A. Cruz-Osorio, and F. Guzman, “CAFE: A new relativistic MHD code,” Astrophys. J. Suppl. Ser. 218 (2), 24 (2015).
     
  11. I. M. Kulikov, “Using piecewise parabolic reconstruction of physical variables in the Rusanov solver. I. The special relativistic hydrodynamics equations,” J. Appl. Ind. Math. 17 (4), 737–749 (2023).
     
  12. S. S. Komissarov, “A Godunov-type scheme for relativistic magnetohydrodynamics,” Mon. Not. R. Astron. Soc. 303 (2), 343–366 (1999).
     
  13. D. Balsara, “Total variation diminishing scheme for relativistic magnetohydrodynamics,” Astrophys. J. Suppl. Ser. 132 (1), 83–101 (2001).
     
  14. B. Giacomazzo and L. Rezzolla, “The exact solution of the Riemann problem in relativistic magnetohydrodynamics,” J. Fluid Mech. 562, 223–259 (2006).
     
  15. M. Brio and C. Wu, “An upwind differencing scheme for the equations of ideal magnetohydrodynamics,” J. Comput. Phys. 75 (2), 400–422 (1988).
     
  16. Y. A. Kriksin and V. F. Tishkin, “Variational entropic regularization of the discontinuous Galerkin method for gasdynamic equations,” Math. Models Comput. Simul. 11 (6), 1032–1040 (2019).
     
  17. S. K. Godunov, “Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field,” Comput. Math. Math. Phys. 52 (5), 787–799 (2012).
     
  18. S. K. Godunov, “About inclusion of Maxwell’s equations in systems relativistic of the invariant equations,” Comput. Math. Math. Phys. 53 (8), 1179–1182 (2013).
     
  19. S. K. Godunov and I. M. Kulikov, “Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee,” Comput. Math. Math. Phys. 54 (6), 1012–1024 (2014).
     
  20. H. Freistuehler and Yu. Trakhinin, “Symmetrizations of RMHD equations and stability of relativistic current-vortex sheets,” Class. Quantum Gravity 30 (8), 085012 (2013).
     
  21. Yu. Trakhinin, “Local existence of contact discontinuities in relativistic magnetohydrodynamics,” Sib. Adv. Math. 30 (2), 55–76 (2020).
     
  22. D. Lee, H. Faller, and A. Reyes, “The Piecewise Cubic Method (PCM) for computational fluid dynamics,” J. Comput. Phys. 341 (1), 230–257 (2017).