Об одной численной схеме типа Годунова для описания газовой и пылевой компонент в задачах звездообразования

Об одной численной схеме типа Годунова для описания газовой и пылевой компонент в задачах звездообразования

Куликов И. М., Черных И. Г., Сапетина А. Ф., Воробьёв Э. И., Элбакян В. Г.
Сибирский журнал индустриальной математики, 26, 1(93), 85-97 (2023)
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УДК 519.63 
DOI: 10.33048/SIBJIM.2023.26.108

Аннотация:

Изложена одна конструкция метода типа Годунова на основе схемы разделения операторов, описывающих работу сил давления и адвективного переноса. Отдельный учёт адвективного переноса позволяет в рамках единой численной схемы описать движение как газовой, так и пылевой компонент. В случае описания динамики газа учёт работы сил давления производится на отдельном этапе независимо от переноса, что позволяет использовать численную схему при решении задач звездообразования, где приходится совместно решать уравнения гидродинамики и уравнения для движения пыли. Для уменьшения диссипации численного метода испольузется кусочно-параболическое представление физических переменных по всем направлениям. Численный метод верифицирован на задачах о распаде гидродинамического и пылевого разрывов, задаче Седова о точечном взрыве и задаче о коллапсе облака пыли, которые имеют аналитическое решение.

Литература:
  1. Vorobyov E. I. Ejection of gaseous clumps from gravitationally unstable protostellar disks // Astron. Astrophys. 2016. V. 590. Article 115.
     
  2. Vorobyov E. I., Elbakyan V. G., Plunkett A. L., Dunham M. M., Audard M., Guedel M., Dionatos O. Knotty protostellar jets as a signature of episodic protostellar accretion? // Astron. Astrophys. 2018. V. 613. Article 18. 
     
  3. Vorobyov E. I., Akimkin V., Stoyanovskaya O. P., Pavlyuchenkov Y., Liu H. B. Early evolution of viscous and self-gravitating circumstellar disks with a dust component // Astron. Astrophys. 2018. V. 614. Article 98.
     
  4. Vorobyov E. I., Elbakyan V. G. Gravitational fragmentation and formation of giant protoplanets on orbits of tens of au // Astron. Astrophys. 2018. V. 618. Article 7.
     
  5. Vorobyov E. I., Elbakyan V. G., Omukai K., Hosokawa T., Matsukoba R., Guedel M. Accretion bursts in low-metallicity protostellar disks // Astron. Astrophys. 2020. V. 641. Article 72.
     
  6. Bate M. Collapse of a molecular cloud core to stellar densities: the formation and evolution of pre-stellar discs // Monthly Notices Royal Astron. Soc. 2011. V. 417. P. 2036–2056.
     
  7. Kulikov I., Vorobyov E. Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows // J. Comput. Phys. 2016. V. 317. P. 318–346.
     
  8. Годунов С. К. Разностный метод численного расчета разрывных решений уравнений гидродинамики // Мат. сб. 1959. Т. 47(89), № 3. С. 271–306.
     
  9. Einfeldt D., Munz C., Roe P., Sjoegreen B. On Godunov-type methods near low densities // J. Comput. Phys. 1991. V. 92. P. 273–295.
     
  10. Godunov S. K., Manuzina Y. D., Nazar’eva M. A. Experimental analysis of convergence of the numerical solution to a generalized solution in fluid dynamics // Comput. Math. Math. Phys. 2011. V. 51. P. 88–95.
     
  11. 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.
     
  12. Godunov S. K., Denisenko V. V., Klyuchinskii D. V., Fortova S. V., Shepelev V. V. Study of entropy properties of a linearized version of Godunov’s method // Comput. Math. Math. Phys. 2020. V. 60. P. 628–640.
     
  13. Gallice G., Chan A., Loubere R., Maire P. Entropy stable and positivity preserving Godunov-type schemes for multidimensional hyperbolic systems on unstructured grid // J. Comput. Phys. 2022. V. 468. Article 111493.
     
  14. Dellacherie S. Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number // J. Comput. Phys. 2010. V. 229. P. 978–1016.
     
  15. Godunov S. K., Klyuchinskii D. V., Fortova S. V., Shepelev V. V. Experimental studies of difference gas dynamics models with shock waves // Comput. Math. Math. Phys. 2018. V. 58. P. 1201–1216.
     
  16. Chen S., Li J., Li Z., Yuan W., Gao Z. Anti-dissipation pressure correction under low Mach numbers for Godunov-type schemes // J. Comput. Phys. 2022. V. 456. Article 111027.
     
  17. Sekora M., Stone J. A hybrid Godunov method for radiation hydrodynamics // J. Comput. Phys. 2010. V. 229. P. 6819–6852.
     
  18. Gardiner T., Stone J. An unsplit Godunov method for ideal MHD via constrained transport // J. Comput. Phys. 2005. V. 205. P. 509–539.
     
  19. Gardiner T., Stone J. An unsplit Godunov method for ideal MHD via constrained transport in three dimensions // J. Comput. Phys. 2008. V. 227. P. 4123–4141.
     
  20. Mignone A., Tzeferacos P. A second-order unsplit Godunov scheme for cell-centered MHD: The CTUGLM scheme // J. Comput. Phys. 2010. V. 229. P. 2117–2138.
     
  21. Kulikov I. M. A low-dissipation numerical scheme based on a piecewise parabolic method on a local stencil for mathematical modeling of relativistic hydrodynamic flows // Numer. Anal. Appl. 2020. V. 13. P. 117–126.
     
  22. Kulikov I. M. On a modification of the Rusanov solver for the equations of special relativistic magnetic hydrodynamics // J. Appl. Indust. Math. 2020. V. 14. P. 524–531.
     
  23. Godunov S. K., Kiselev S. P., Kulikov I. M., Mali V. I. Numerical and experimental simulation of wave formation during explosion welding // Proc. Steklov Inst. Math. 2013. V. 281. P. 12–26.
     
  24. Aganin A. A., Khismatullina N. A. UNO modifications of the Godunov method for calculating the dynamics of an elastic-plastic body // Lobachevskii J. Math. 2019. V. 40. P. 256–262.
     
  25. Khismatullina N. A. Calculation of waves in an elastic-plastic body based on ENO modifications of the Godunov method // Lobachevskii J. Math. 2020. V. 41. P. 1228–1234.
     
  26. Barbas A., Velarde P. Development of a Godunov method for Maxwell’s equations with adaptive mesh refinement // J. Comput. Phys. 2015. V. 300. P. 186–201.
     
  27. Moreno J., Oliva E., Velarde P. EMcLAW: An unsplit Godunov method for Maxwell’s equations including polarization, metals, divergence control and AMR // Comput. Phys. Comm. 2021. V. 260. Article 107268.
     
  28. Lei X., Li J. A staggered-projection Godunov-type method for the Baer—Nunziato two-phase model // J. Comput. Phys. 2021. V. 437. Article 110312.
     
  29. Ghitti B., Berthon C., Hoang Le M., Toro E. A fully well-balanced scheme for the 1D blood flow equations with friction source term // J. Comput. Phys. 2020. V. 421. Article 109750.
     
  30. Kulikov I. Molecular cloud collapse to stellar densities: Models on moving geodesic vs. Unstructured tetrahedron vs. nested meshes // J. Phys. Conf. Ser. 2021. V. 2028. Article 012001.
     
  31. Chernykh I., Vorobyov E., Elbakyan V., Kulikov I. The impact of compiler level optimization on the performance of iterative Poisson solver for numerical modeling of protostellar disks // Comm. Comput. Inform. Sci. 2021. V. 1510. P. 415–426.

Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (проект 19-51-14002 АНФ_а) и Австрийского научного фонда (проект I4311-N27).

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

E-mail: kulikov@ssd.sscc.ru

И. Г. Черных
  1. Институт вычислительной математики и математической геофизики СО РАН, 
    просп. Лаврентьева, 6, г. Новосибирск 630090, Россия

E-mail: chernykh@parbz.sscc.ru

А. Ф. Сапетина
  1. Институт вычислительной математики и математической геофизики СО РАН, 
    просп. Лаврентьева, 6, г. Новосибирск 630090, Россия

E-mail: afsapetina@gmail.com

Э. И. Воробьёв
  1. Институт астрономии, Университет города Вена, 
    ул. Тюркеншанцштрассе, 17, г. Вена 1180, Австрия

E-mail: eduard.vorobiev@univie.ac.at

В. Г. Элбакян
  1. НИИ физики Южного федерального университета, 
    просп. Стачки, 194, г. Ростов-на-Дону 344090, Россия

E-mail: vgelbakyan@sfedu.ru

Статья поступила 16.08.2022 г.
После доработки — 20.09.2022 г.
Принята к публикации 29.09.2022 г.

Abstract:

The paper presents one construction of the Godunov-type method based on the separation of operators describing the work of pressure forces and advective transfer. Separate consideration of advective transfer makes it possible to describe the motion of both gas and dust components within the framework of a single numerical scheme. In the case of describing gas dynamics, the work of pressure forces is taken into account at a separate stage, regardless of transfer. This makes it possible to use the numerical scheme in solving star formation problems, where it is necessary to jointly solve the equations of hydrodynamics and equations for dust motion. A piecewise parabolic representation of physical variables in all directions is used to reduce the dissipation of the numerical method. The numerical method has been verified on the Riemann problems for a hydrodynamic and dust discontinuity, the Sedov problem of point explosion, and the problem of dust cloud collapse, which have an analytical solution.

References:
  1. Vorobyov E. I. Ejection of gaseous clumps from gravitationally unstable protostellar disks. Astron. Astrophys., 2016, Vol. 590, article 115.
     
  2. Vorobyov E. I., Elbakyan V. G., Plunkett A. L., Dunham M. M., Audard M., Guedel M., Dionatos O. Knotty protostellar jets as a signature of episodic protostellar accretion? Astron. Astrophys., 2018, Vol. 613, article 18. 
     
  3. Vorobyov E. I., Akimkin V., Stoyanovskaya O. P., Pavlyuchenkov Y., Liu H. B. Early evolution of viscous and self-gravitating circumstellar disks with a dust component. Astron. Astrophys., 2018, Vol. 614, article 98.
     
  4.  Vorobyov E. I., Elbakyan V. G. Gravitational fragmentation and formation of giant protoplanets on orbits of tens of au. Astron. Astrophys., 2018. V. 618, article 7.
     
  5. Vorobyov E. I., Elbakyan V. G., Omukai K., Hosokawa T., Matsukoba R., Guedel M. Accretion bursts in low-metallicity protostellar disks. Astron. Astrophys., 2020. V. 641, article 72.
     
  6. Bate M. Collapse of a molecular cloud core to stellar densities: the formation and evolution of pre-stellar discs. Monthly Notices Royal Astron. Soc., 2011, Vol. 417, pp. 2036–2056.
     
  7. Kulikov I., Vorobyov E. Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows. J. Comput. Phys., 2016, Vol. 317, pp. 318–346.
     
  8. Годунов С. К. Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki [A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics], Mat. Sb., 1959, Vol. 47(89), No. 3, pp. 271–306 (in Russian).
     
  9. Einfeldt D., Munz C., Roe P., Sjoegreen B. On Godunov-type methods near low densities. J. Comput. Phys., 1991, Vol. 92, pp. 273–295.
     
  10. Godunov S. K., Manuzina Y. D., Nazar’eva M. A. Experimental analysis of convergence of the numerical solution to a generalized solution in fluid dynamics. Comput. Math. Math. Phys., 2011, Vol. 51, pp. 88–95.
     
  11. Godunov S. K., Kulikov I. M. Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee. Comput. Math. Math. Phys., 2014, Vol. 54, No. 6, pp. 1012–1024.
     
  12. Godunov S. K., Denisenko V. V., Klyuchinskii D. V., Fortova S. V., Shepelev V. V. Study of entropy properties of a linearized version of Godunov’s method. Comput. Math. Math. Phys., 2020, Vol. 60, pp. 628– 640.
     
  13. Gallice G., Chan A., Loubere R., Maire P. Entropy stable and positivity preserving Godunov-type schemes for multidimensional hyperbolic systems on unstructured grid. J. Comput. Phys., 2022, Vol. 468, article 111493.
     
  14. Dellacherie S. Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys., 2010, Vol. 229, pp. 978–1016.
     
  15. Godunov S. K., Klyuchinskii D. V., Fortova S. V., Shepelev V. V. Experimental studies of difference gas dynamics models with shock waves. Comput. Math. Math. Phys., 2018, Vol. 58, pp. 1201–1216.
     
  16. Chen S., Li J., Li Z., Yuan W., Gao Z. Anti-dissipation pressure correction under low Mach numbers for Godunov-type schemes. J. Comput. Phys., 2022, Vol. 456, article 111027.
     
  17. Sekora M., Stone J. A hybrid Godunov method for radiation hydrodynamics. J. Comput. Phys., 2010, Vol. 229, pp. 6819–6852.
     
  18. Gardiner T., Stone J. An unsplit Godunov method for ideal MHD via constrained transport. J. Comput. Phys., 2005, Vol. 205, pp. 509–539.
     
  19. Gardiner T., Stone J. An unsplit Godunov method for ideal MHD via constrained transport in three dimensions. J. Comput. Phys., 2008, Vol. 227, pp. 4123–4141.
     
  20. Mignone A., Tzeferacos P. A second-order unsplit Godunov scheme for cell-centered MHD: The CTUGLM scheme. J. Comput. Phys., 2010, Vol. 229, pp. 2117–2138.
     
  21. Kulikov I. M. A low-dissipation numerical scheme based on a piecewise parabolic method on a local stencil for mathematical modeling of relativistic hydrodynamic flows. Numer. Anal. Appl., 2020, Vol. 13, pp. 117–126.
     
  22. Kulikov I. M. On a modification of the Rusanov solver for the equations of special relativistic magnetic hydrodynamics. J. Appl. Indust. Math., 2020, Vol. 14, pp. 524–531.
     
  23. Godunov S. K., Kiselev S. P., Kulikov I. M., Mali V. I. Numerical and experimental simulation of wave formation during explosion welding. Proc. Steklov Inst. Math., 2013, Vol. 281, pp. 12–26.
     
  24. Aganin A. A., Khismatullina N. A. UNO modifications of the Godunov method for calculating the dynamics of an elastic-plastic body. Lobachevskii J. Math., 2019, Vol. 40, pp. 256–262.
     
  25. Khismatullina N. A. Calculation of waves in an elastic-plastic body based on ENO modifications of the Godunov method. Lobachevskii J. Math., 2020, Vol. 41, pp. 1228–1234.
     
  26. Barbas A., Velarde P. Development of a Godunov method for Maxwell’s equations with adaptive mesh refinement. J. Comput. Phys., 2015, Vol. 300, pp. 186–201.
     
  27. Moreno J., Oliva E., Velarde P. EMcLAW: An unsplit Godunov method for Maxwell’s equations including polarization, metals, divergence control and AMR. Comput. Phys. Comm., 2021, Vol. 260, article 107268.
     
  28. Lei X., Li J. A staggered-projection Godunov-type method for the Baer—Nunziato two-phase model. J. Comput. Phys., 2021, Vol. 437, article 110312.
     
  29. Ghitti B., Berthon C., Hoang Le M., Toro E. A fully well-balanced scheme for the 1D blood flow equations with friction source term. J. Comput. Phys., 2020, Vol. 421, article 109750.
     
  30. Kulikov I. Molecular cloud collapse to stellar densities: Models on moving geodesic vs. Unstructured tetrahedron vs. nested meshes. J. Phys. Conf. Ser., 2021, Vol. 2028, article 012001.
     
  31. Chernykh I., Vorobyov E., Elbakyan V., Kulikov I. The impact of compiler level optimization on the performance of iterative Poisson solver for numerical modeling of protostellar disks. Comm. Comput. Inform. Sci., 2021, Vol. 1510, pp. 415–426.