Using piecewise-parabolic reconstruction of physics variables to constructing a low-dissipation HLL method for numerical solution of special relativistic hydrodynamics equations

I. M. Kulikov ; D. A. Karavaev

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 1, pp. 57-75.

Résumé

A construction of the original HLL method for solving problems of relativistic hydrodynamics by using a piecewise-parabolic reconstruction of the physical variables is described. The resulting numerical method makes it possible to reproduce the numerical solutions with small dissipation at the discontinuities. The method is verified in problems of discontinuity breakdown in one-dimensional and two-dimensional formulation. The accuracy of the numerical scheme is studied in one-dimensional discontinuity breakdown problems. The method is also tested in typical astrophysical problems: interaction of relativistic jets, collision of clouds at relativistic speeds, and supernova explosion.

Bibliographie

[1] Komissarov S., Porth O., “Numerical simulations of jets”, New Astronomy Reviews, 92 (2021), 101610

[2] Siegel D.M., Metzger B.D., “Three-dimensional GRMHD simulations of neutrino-cooled accretion disks from Neutron star mergers”, The Astrophysical J., 858 (2018), 52

[3] Willingale R., Meszaros P., “Gamma-ray bursts and fast transients. Multi-wavelength observations and multi-messenger signals”, Space Science Reviews, 207 (2017), 63–86

[4] Barkov M., Lyutikov M., Klingler N., Bordas P., “Kinetic «jets» from fast-moving pulsars”, Monthly Notices of the Royal Astronomical Society, 485 (2019), 2041–2053

[5] Olmi B., Bucciantini N., “Full-3D relativistic MHD simulations of bow shock pulsar wind nebulae: dynamics”, Monthly Notices of the Royal Astronomical Society, 484 (2019), 5755–5770

[6] Huber D., Kissmann R., Reimer A., Reimer O., “Relativistic fluid modelling of gamma-ray binaries. I. The model”, Astronomy Astrophysics, 646 (2021), A91

[7] Huber D., Kissmann R., Reimer O., “Relativistic fluid modelling of gamma-ray binaries. II. Application to LS 5039”, Astronomy Astrophysics, 649 (2021), A71

[8] Busza W., Rajagopal K., van der Schee W., “Heavy ion collisions: the big picture and the big questions”, Annual Review of Nuclear and Particle Science, 68 (2018), 339–376

[9] Loffler F., Faber J., Bentivegna E. et al., “The einstein toolkit: a community computational infrastructure for relativistic astrophysics”, Classical and Quantum Gravity, 29 (2012), 115001

[10] Rivera-Paleo F.J., Guzman F.S., “CAFE-R: a code that solves the special relativistic radiation hydrodynamics equations”, The Astrophysical J. Supplement Series, 241 (2019), 28

[11] Aloy M., Ibanez J., Marti J., Muller E., “GENESIS: a high-resolution code for three-dimensional relativistic hydrodynamics”, The Astrophysical J. Supplement Series, 122 (1999), 122–151

[12] Mignone A., Bodo G., Massaglia S. et al., “PLUTO: a numerical code for computational astrophysics”, The Astrophysical J. Supplement Series, 170 (2007), 228–242

[13] Zhang W., MacFayden A., “RAM: a relativistic adaptive mesh refinement hydrodynamics code”, The Astrophysical J. Supplement Series, 164 (2006), 255–279

[14] Duffel P., MacFayden A., “TESS: a relativistic hydrodynamics code on a moving Voronoi mesh”, The Astrophysical J. Supplement Series, 197 (2011), 15

[15] Huber D., Kissmann R., “Special relativistic hydrodynamics with CRONOS”, Astronomy Astrophysics, 653 (2021), A164

[16] Popov M., Ustyugov S., “Piecewise parabolic method on local stencil for gasdynamic simulations”, Computational Mathematics and Mathematical Physics, 47:12 (2007), 1970–1989

[17] Popov M., Ustyugov S., “Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics”, Computational Mathematics and Mathematical Physics, 48:3 (2008), 477–499

[18] Ustyugov S.D., Popov M.V., Kritsuk A.G., Norman M.L., “Piecewise parabolic method on a local stencil for magnetized supersonic turbulence simulation”, J. of Computational Physics, 228 (2009), 7614–7633

[19] Kulikov I., Vorobyov E., “Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows”, J. of Computational Physics, 317 (2016), 318–346

[20] 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”, The Astrophysical J. Supplement Series, 243 (2019), 4

[21] Kulikov I. A new code for the numerical simulation of relativistic flows on supercomputers by means of a low-dissipation scheme, Computer Physics Communications, 257 (2020), 107532

[22] 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”, Numerical Analysis and Applications, 13:2 (2020), 117–126

[23] Harten A., Lax P., van Leer B., “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws”, SIAM Review, 25 (1983), 289–315

[24] Kulikov I.M., Chernykh I.G., Glinskiy B.M., Protasov V.A., “An efficient optimization of HLL method for the second generation of Intel Xeon Phi processor”, Lobachevskii J. of Mathematics, 39:4 (2018), 543–551

[25] Kulikov I.M., “Kusochno-lineinaya rekonstruktsiya peremennykh, umenshayuschaya dissipatsiyu metoda HLL pri reshenii uravnenii gazodinamiki”, Sib. zhurn. vychisl. matematiki, 25:2 (2022), 141–156

[26] Pons J., Marti J., Mueller E., “The exact solution of the Riemann problem with non-zero tangential velocities in relativistic hydrodynamics”, J. of Fluid Mechanics, 422 (2000), 125–139

[27] Marti J.. Mueller E., “Numerical hydrodynamics in special relativity”, Living Reviews in Relativity, 6 (2003), 7

[28] Godunov S.K., “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics”, Matematicheskii Sbornik, 47 (1959), 271–306

[29] Kolgan V.P., “Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous gas flows”, TsAGI Science J., 3 (1972), 68–77

[30] Collela P., Woodward P.R., “The piecewise parabolic method (PPM) gas-dynamical simulations”, J. Computational Physics, 54 (1984), 174–201

[31] Lee D., Faller H., Reyes A., “The piecewise cubic method (PCM) for computational fluid dynamics”, J. Computational Physics, 341 (2017), 230–257

[32] Deng X., Boivin P., Xiao F., “A new formulation for two-wave Riemann solver accurate at contact interfaces”, Physics of Fluids, 31 (2019), 046102

[33] Kulikov I.M., Chernykh I.G., Sapetina A.F., Lomakin S.V., Tutukov A.V., “A new Rusanov-type solver with a local linear solution reconstruction for numerical modeling of white dwarf mergers by means massive parallel supercomputers”, Lobachevskii J. of Mathematics, 41:8 (2020), 1485–1491

[34] Kriksin Yu.A., Tishkin V.F., “Chislennoe reshenie zadachi Einfeldta na osnove razryvnogo metoda Galerkina”, Preprinty IPM im. M.V. Keldysha RAN, 90 (2019)

[35] Kriksin Y.A., Tishkin V.F., “Variational entropic regularization of the discontinuous Galerkin method for gasdynamic equations”, Mathematical Models and Computer Simulations, 11 (2019), 1032–1040

[36] Mathews W., “The hydromagnetic free expansion of a relativistic gas”, The Astrophysical J., 165 (1971), 147–164

[37] Perucho M., Marti J.M., “A numerical simulation of the evolution and fate of a Fanaroff-Riley type I jet. The case of 3C 31”, Monthly Notices of the Royal Astronomical Society, 382 (2007), 526–542

[38] Perucho M., Marti J.M., Quilis V., “Long-term FRII jet evolution in dense environments”, Monthly Notices of the Royal Astronomical Society, 510 (2022), 2084–2096