Использование кусочно-параболической реконструкции физических переменных в схеме Русанова. I.

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

Куликов И. М.
Сибирский журнал индустриальной математики, 26, 4(96), 49-64 (2023)
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УДК 519.6 
DOI: 10.33048/SIBJIM.2023.26.404

Аннотация:

Схема Русанова для решения гидродинамических уравнений является одной из самых робастных в классе схем численного решения задачи Римана. В случае уравнений специальной релятивистской гидродинамики условие робастности схемы является наиболее ключевым свойством, особенно при достаточно высоких значениях фактора Лоренца. В то же время известно, что схема Русанова достаточно диссипативна. В статье предлагается использование кусочно-параболического представления физических переменных для уменьшения диссипации схемы Русанова. Использование такого подхода позволило получить схему с такими же диссипативными свойствами как схемы типа Рое и семейство схем Хартена—Лакса—Ван Леера. На задаче о распаде релятивистского гидродинамического разрыва показано преимущество авторского варианта схемы Русанова при воспроизведении контактного разрыва. Проведена верификация схемы на классических задачах о распаде разрыва и на задаче о взаимодействии двух релятивистских джетов в трёхмерной постановке.

Литература:
  1. Chechetkin V. M., D’yachenko V. F., Ginzburg S. L., Paleichik V. V., Fimin N. N., Sudarikov A. L. On the generation mechanism of hard cosmic gamma-ray emission from AGN jets // Astron. Rep. 2009. V. 53. P. 501–509.
     
  2. Barkov M. V., Bisnovatyi-Kogan G. S. Interaction of a cosmological gamma-ray burst with a dense molecular cloud and the formation of jets // Astron. Rep. 2005. V. 49. P. 24–35.
     
  3. Sotomayor P., Romero G. Nonthermal radiation from the central region of super-accreting active galactic nuclei // Astron. Astrophys. 2022. V. 664. Article A178.
     
  4. Sokolov V. V., Bisnovatyi-Kogan G. S., Kurt V. G., Gnedin Yu. N., Baryshev Yu. V. Observational constraints on the angular and spectral distributions of photons in gamma-ray burst sources // Astron. Rep. 2006. V. 50. P. 612–625.
     
  5. Komissarov S. S. Simulations of the axisymmetric magnetospheres of neutron stars // Mon. Not. R. Astron. Soc. 2006. V. 367. P. 19–31.
     
  6. Tutukov A. V., Bogomazov A. I. Radio pulsars in close binaries with neutron stars // Astron. Rep. 2008. V. 52. P. 390–402.
     
  7. Istomin Ya. N., Komberg B. V. Gamma-ray bursts as a result of the interaction of a shock from a supernova and a neutron-star companion // Astron. Rep. 2002. V. 46. P. 908–917.
     
  8. Fateev V. F., Davlatov R. A. Space-Based Gravitational-Wave Detectors: Development of Ground-Breaking Technologies for Future Space-Based Gravitational Gradiometers // Astron. Rep. 2019. V. 63. P. 699–709.
     
  9. Belyaev V. S., Bisnovatyi-Kogan G. S., Gromov A. I., Zagreev B. V., Lobanov A. V., Matafonov A. P., Moiseenko S. G., Toropina O. D. Numerical Simulations of Magnetized Astrophysical Jets and Comparison with Laboratory Laser Experiments // Astron. Rep. 2018. V. 62. P. 162–182.
     
  10. Krauz V. I., Mitrofanov K. N., Kharrasov A. M., Il’ichev I. V., Myalton V. V., Anan’ev S. S., Beskin V. S. Laboratory Modeling of the Rotation of Jets Ejected from Young Stellar Objects at Studies the Azimuthal Structure of an Axial Jet at the PF-3 Facility // Astron. Rep. 2021. V. 65. P. 26–44.
     
  11. Kulikov I. A new code for the numerical simulation of relativistic flows on supercomputers by means of a low-dissipation scheme // Comput. Phys. Commun. 2020. V. 257. Article 107532.
     
  12. Rusanov V. V. The calculation of the interaction of non-stationary shock waves with barriers // Comput. Math. Math. Phys. 1961. V. 1. P. 267–279.
     
  13. Mohamed K., Benkhaldoun F. A modified Rusanov scheme for shallow water equations with topography and two phase flows // Eur. Phys. J. Plus. 2016. V. 131. Article 207.
     
  14. Wu C., Walker R., Dawson J. A three dimensional MHD model of the Earth’s magnetosphere // Geophys. Res. Lett. 1981. V. 8, N 5. P. 523–526.
     
  15. Mohamed K., Abdelrahman M. The modified Rusanov scheme for solving the ultra-relativistic Euler equations // Eur. J. Mech. B Fluids. 2021. V. 90. P. 89–98.
     
  16. Mohammadian S., Moghaddam A., Sahaf A. On the performance of HLL, HLLC, and Rusanov solvers for hyperbolic traffic models // Comput. Fluids. 2021. V. 231. Article 105161.
     
  17. Nishikawa H., Kitamura K. Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers // J. Comput. Phys. 2008. V. 227. P. 2560–2581.
     
  18. Townsend J., Koenoezsy L., Jenkins K. On the development of a rotated-hybrid HLL/HLLC approximate Riemann solver for relativistic hydrodynamics // Mon. Not. R. Astron. Soc. 2020. V. 496, N 2. P. 2493– 2505.
     
  19. Jaisankar S., Raghurama Rao S. Diffusion regulation for Euler solvers // J. Comput. Phys. 2007. V. 221, N 2. P. 577–599.
     
  20. Mazaheri A., Nishikawa H. High-order shock-capturing hyperbolic residual-distribution schemes on irregular triangular grids // Comput. Fluids. 2016. V. 131. P. 29–44.
     
  21. Rossi G., Dumbser M., Armanini A. A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows // Comput. Fluids. 2017. V. 154. P. 102–122.
     
  22. Balsara D. Three dimensional HLL Riemann solver for conservation laws on structured meshes; Application to Euler and magnetohydrodynamic flows // J. Comput. Phys. 2015. V. 295. P. 1–23.
     
  23. Popov M., Ustyugov S. Piecewise parabolic method on local stencil for gasdynamic simulations // Comput. Math. Math. Phys. 2007. V. 47, N 12. P. 1970–1989.
     
  24. Popov M., Ustyugov S. Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics // Comput. Math. Math. Phys. 2008. V. 48, N 3. P. 477–499.
     
  25. 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.
     
  26. Kulikov I., Chernykh I., Tutukov A. A New Hydrodynamic Code with Explicit Vectorization Instructions Optimizations that Is Dedicated to the Numerical Simulation of Astrophysical Gas Flow. I. Numerical Method, Tests, and Model Problems // Astrophys. J. Suppl. Ser. 2019. V. 243. Article 4.
     
  27. Kriksin Y. A., Tishkin V. F. Variational Entropic Regularization of the Discontinuous Galerkin Method for Gasdynamic Equations // Math. Models Comput. Simul. 2019. V. 11. P. 1032–1040.
     
  28. Kulikov I., Chernykh I., Karavaev D., Prigarin V., Sapetina A., Ulyanichev I., Zavyalov O. A New Parallel Code Based on a Simple Piecewise Parabolic Method for Numerical Modeling of Colliding Flows in Relativistic Hydrodynamics // Mathematics. 2022. V. 10, N 11. Article 1865.
     
  29. Kulikov I. M., Karavaev D. A. A Piecewise-Parabolic Reconstruction of the Physical Variables in a Low-Dissipation HLL Method for the Numerical Solution of the Equations of Special Relativistic Hydrodynamics // Numer. Anal. Appl. 2023. V. 16. P. 45–60.
     
  30. 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. 
     
  31. Lee D., Faller H., Reyes A. The Piecewise Cubic Method (PCM) for computational fluid dynamics // J. Comput. Phys. 2017. V. 341. P. 230–257.
     
  32. Wang P., Abel T., Zhang W. Relativistic Hydrodynamic Flows Using Spatial and Temporal Adaptive Structured Mesh Refinement // Astrophys. J. Suppl. Ser. 2008. V. 176. P. 467–483.
     
  33. Bhoriya D., Kumar H. Entropy-stable schemes for relativistic hydrodynamics equations // Z. Angew. Math. Phys. 2020. V. 71. Article 29.
     
  34. Guercilena F., Radice D., Rezzolla L. Entropy-limited hydrodynamics: a novel approach to relativistic hydrodynamics // Comput. Astrophys. Cosmol. 2017. V. 4. Article 3.
     
  35. Zlotnik A. A. Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation // Comput. Math. Math. Phys. 2012. V. 52. P. 1060–1071.
     
  36. Gavrilin V. A., Zlotnik A. A. On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance // Comput. Math. Math. Phys. 2015. V. 55. P. 264–281.
     
  37. Zlotnik A. A. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations // Comput. Math. Math. Phys. 2017. V. 57. P. 706–725.

Работа выполнена при финансовой поддержке Российского научного фонда (проект 23-11-00014; https://rscf.ru/project/23-11-00014/).

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 04.06.2023 г.
После доработки — 07.07.2023 г.
Принята к публикации 09.08.2023 г.

Abstract:

The Rusanov solver for solving hydrodynamic equations is one of the most robust schemes in the class of Riemann solvers. For special relativistic hydrodynamics, the robustness condition of the scheme is the most important property, especially for sufficiently high values of the Lorentz factor. At the same time, the Rusanov solver is known to be very dissipative. It is proposed to use a piecewise parabolic representation of physical variables to reduce the dissipation of the Rusanov scheme. Using this approach has made it possible to obtain a scheme with the same dissipative properties as Roe-type schemes and the family of Harten—Lax—van Leer schemes. Using the problem of the decay of a relativistic hydrodynamic discontinuity, it is shown that the present author’s version of the Rusanov scheme is advantageous in terms of reproducing a contact discontinuity. The scheme is verified on classical problems of discontinuity decay and on the problem of the interaction of two relativistic jets in the three-dimensional formulation.

References:
  1. V. M. Chechetkin, V. F. D’yachenko, S. L. Ginzburg, V. V. Paleichik, N. N. Fimin, and A. L. Sudarikov, “On the generation mechanism of hard cosmic gamma-ray emission from AGN jets,” Astron. Rep. 53, 501–509 (2009). 
     
  2. M. V. Barkov and G. S. Bisnovatyi-Kogan, “Interaction of a cosmological gamma-ray burst with a dense molecular cloud and the formation of jets,” Astron. Rep. 49, 24–35 (2005).
     
  3. P. Sotomayor and G. Romero, “Nonthermal radiation from the central region of super-accreting active galactic nuclei,” Astron. Astrophys. 664, A178 (2022).
     
  4. V. V. Sokolov, G. S. Bisnovatyi-Kogan, V. G. Kurt, Yu. N. Gnedin, and Yu. V. Baryshev, “Observational constraints on the angular and spectral distributions of photons in gamma-ray burst sources,” Astron. Rep. 50, 612–625 (2006).
     
  5. S. S. Komissarov, “Simulations of the axisymmetric magnetospheres of neutron stars,” Mon. Not. R. Astron. Soc. 367, 19–31 (2006).
     
  6. A. V. Tutukov and A. I. Bogomazov, “Radio pulsars in close binaries with neutron stars,” Astron. Rep. 52, 390–402 (2008).
     
  7. Ya. N. Istomin and B. V. Komberg, “Gamma-ray bursts as a result of the interaction of a shock from a supernova and a neutron-star companion,” Astron. Rep. 46, 908–917 (2002).
     
  8. V. F. Fateev and R. A. Davlatov, “Space-based gravitational-wave detectors: Development of groundbreaking technologies for future space-based gravitational gradiometers,” Astron. Rep. 63, 699– 709 (2019).
     
  9. V. S. Belyaev, G. S. Bisnovatyi-Kogan, A. I. Gromov, B. V. Zagreev, A. V. Lobanov, A. P. Matafonov, S. G. Moiseenko, and O. D. Toropina, “Numerical simulations of magnetized astrophysical jets and comparison with laboratory laser experiments,” Astron. Rep. 62, 162–182 (2018).
     
  10. V. I. Krauz, K. N. Mitrofanov, A. M. Kharrasov, I. V. Il’ichev, V. V. Myalton, S. S. Anan’ev, and V. S. Beskin, “Laboratory modeling of the rotation of jets ejected from young stellar objects at studies the azimuthal structure of an axial jet at the PF-3 facility,” Astron. Rep. 65, 26–44 (2021).
     
  11. I. Kulikov, “A new code for the numerical simulation of relativistic flows on supercomputers by means of a low-dissipation scheme,” Comput. Phys. Commun. 257, 107532 (2020).
     
  12. V. V. Rusanov, “The calculation of the interaction of non-stationary shock waves with barriers,” Comput. Math. Math. Phys. 1, 267–279 (1961).
     
  13. K. Mohamed and F. Benkhaldoun, “A modified Rusanov scheme for shallow water equations with topography and two phase flows,” Eur. Phys. J. Plus 131, 207 (2016).
     
  14. C. Wu, R. Walker, and J. Dawson, “A three dimensional MHD model of the Earth’s magnetosphere,” Geophys. Res. Lett. 8 (5), 523–526 (1981).
     
  15. K. Mohamed and M. Abdelrahman, “The modified Rusanov scheme for solving the ultra-relativistic Euler equations,” Eur. J. Mech. — B/Fluids 90, 89–98 (2021).
     
  16. S. Mohammadian, A. Moghaddam, and A. Sahaf, “On the performance of HLL, HLLC, and Rusanov solvers for hyperbolic traffic models,” Comput. Fluids 231, 105161 (2021).
     
  17. H. Nishikawa and K. Kitamura, “Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers,” J. Comput. Phys. 227, 2560–2581 (2008).
     
  18. J. Townsend, L. Koenoezsy, and K. Jenkins, “On the development of a rotated-hybrid HLL/HLLC approximate Riemann solver for relativistic hydrodynamics,” Mon. Not. R. Astron. Soc. 496 (2), 2493– 2505 (2020).
     
  19. S. Jaisankar and S. Raghurama Rao, “Diffusion regulation for Euler solvers,” J. Comput. Phys. 221 (2), 577–599 (2007).
     
  20. A. Mazaheri and H. Nishikawa, “High-order shock-capturing hyperbolic residual-distribution schemes on irregular triangular grids,” Comput. Fluids 131, 29–44 (2016).
     
  21. G. Rossi, M. Dumbser, and A. Armanini, “A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows,” Comput. Fluids 154, 102–122 (2017).
     
  22. D. Balsara, “Three dimensional HLL Riemann solver for conservation laws on structured meshes: Application to Euler and magnetohydrodynamic flows,” J. Comput. Phys. 295, 1–23 (2015).
     
  23. M. Popov and S. Ustyugov, “Piecewise parabolic method on local stencil for gasdynamic simulations,” Comput. Math. Math. Phys. 47 (12), 1970–1989 (2007).
     
  24. M. Popov and S. Ustyugov, “Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics,” Comput. Math. Math. Phys. 48 (3), 477–499 (2008).
     
  25. I. Kulikov and E. Vorobyov, “Using the PPML approach for constructing a low-dissipation, operatorsplitting scheme for numerical simulations of hydrodynamic flows,” J. Comput. Phys. 317, 318–346 (2016).
     
  26. I. Kulikov, I. Chernykh, and A. Tutukov, “A new hydrodynamic code with explicit vectorization instructions optimizations that is dedicated to the numerical simulation of astrophysical gas flow. I. Numerical method, tests, and model problems,” Astrophys. J. Suppl. Ser. 243, 4 (2019).
     
  27. Y. A. Kriksin and V. F. Tishkin, “Variational entropic regularization of the discontinuous Galerkin method for gasdynamic equations,” Math. Models Comput. Simul. 11, 1032–1040 (2019).
     
  28. I. Kulikov, I. Chernykh, D. Karavaev, V. Prigarin, A. Sapetina, I. Ulyanichev, and O. Zavyalov, “A new parallel code based on a simple piecewise parabolic method for numerical modeling of colliding flows in relativistic hydrodynamics,” Mathematics 10 (11), 1865 (2022).
     
  29. I. M. Kulikov and D. A. Karavaev, “A piecewise-parabolic reconstruction of the physical variables in a low-dissipation HLL method for the numerical solution of the equations of special relativistic hydrodynamics,” Numer. Anal. Appl. 16, 45–60 (2023).
     
  30. I. M. Kulikov, “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. 13, 117–126 (2020).
     
  31. D. Lee, H. Faller, and A. Reyes, “The piecewise cubic method (PCM) for computational fluid dynamics,” J. Comput. Phys. 341, 230–257 (2017).
     
  32. P. Wang, T. Abel, and W. Zhang, “Relativistic hydrodynamic flows using spatial and temporal adaptive structured mesh refinement,” Astrophys. J. Suppl. Ser. 176, 467–483 (2008).
     
  33. D. Bhoriya and H. Kumar, “Entropy-stable schemes for relativistic hydrodynamics equations,” Z. Angew. Math. Phys. 71, 29 (2020).
     
  34. F. Guercilena, D. Radice, and L. Rezzolla, “Entropy-limited hydrodynamics: A novel approach to relativistic hydrodynamics,” Comput. Astrophys. Cosmol. 4, 3 (2017).
     
  35. A. A. Zlotnik, “Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation,” Comput. Math. Math. Phys. 52, 1060–1071 (2012).
     
  36. V. A. Gavrilin and A. A. Zlotnik, “On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance,” Comput. Math. Math. Phys. 55, 264–281 (2015).
     
  37. A. A. Zlotnik, “Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations,” Comput. Math. Math. Phys. 57, 706–725 (2017).