Using piecewise-linear reconstruction to constructing a low-dissipation HLL method for numerical solution of hydrodynamics equation

I. M. Kulikov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 2, pp. 141-156.

Résumé

In this paper, one construction of the original «Harten-Lax-van Leer» method using a piecewise-linear reconstruction of physical variables is described. The obtained numerical method makes possible to reproduce a low-dissipation solution at discontinuities. To verify the method, we used the classical problems with an analytical solution based on various configurations of shock waves, contact discontinuities, and rarefaction waves. On the Sod-like problem, the order of accuracy of the developed numerical method was studied, it was shown that the main suppression of the order of accuracy occurs when the rarefaction wave is reproduced. The numerical method was verified by means of a three-dimensional Sedov test of a point explosion, and on the problem of a supernova Ia type explosion with two symmetric ignition points, leading to the formation of a G1.9+0.3 like remnant.

Bibliographie

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

[2] Balsara D., “Higher-order accurate space-time schemes for computational astrophysics Part I: finite volume methods”, Living Reviews in Computational Astrophysics, 3 (2017), 2 | DOI | MR

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

[4] Simon S., Mandal J. C., “A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control”, Computer Fluids, 174 (2018), 144–166 | DOI | MR | Zbl

[5] Rusanov V. V., “The calculation of the interaction of non-stationary shock waves with barriers”, Computational Mathematics and Mathematical Physics, 1 (1961), 267–279 | MR

[6] Einfeldt B., “On Godunov-type methods for gas dynamics”, SIAM J. Numerical Analysis, 25 (1988), 294–318 | DOI | MR | Zbl

[7] Einfeldt B., Munz C.-D., Roe P., Sjogreen B., “On Godunov-type methods near low densities”, J. Computational Physics, 92 (1991), 273–295 | DOI | MR | Zbl

[8] Toro E., Spruce M., Speares W., “Restoration of the contact surface in the Harten-Lax-van Leer Riemann solver”, Shock Waves, 4 (1994), 25–34 | DOI | Zbl

[9] Mandal J. C., Panwar V., “Robust HLL-type Riemann solver capable of resolving contact discontinuity”, Computer Fluids, 63 (2012), 148–164 | DOI | MR | Zbl

[10] Xie W., Li W., Li H., “On numerical instabilities of Godunov-type schemes for strong shocks”, J. Computational Physics, 350 (2017), 607–637 | DOI | MR | Zbl

[11] Dumbser M., Balsara D., “A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems”, J. Computational Physics, 304 (2016), 275–319 | DOI | MR | Zbl

[12] Miyoshi T., Kusano K., “A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics”, J. Computational Physics, 208 (2005), 315–344 | DOI | MR | Zbl

[13] Pandolfi M., D'Ambrosio D., “Numerical instabilities in upwind methods: analysis and cures for the «carbuncle» phenomenon”, J. Computational Physics, 166 (2000), 271–301 | DOI | MR

[14] Chauvat Y., Moschetta J.-M., Gressier J., “Shock wave numerical structure and the carbuncle phenomenon”, Intern. J. Numerical Methods in Fluids, 47 (2005), 903–909 | DOI | Zbl

[15] Roe P., “Approximate Riemann solver, parameter vectors and difference schemes”, J. Computational Physics, 43 (1981), 357–372 | DOI | MR | Zbl

[16] Davis S. F., “A rotationally biased upwind difference scheme for the Euler equations”, J. Computational Physics, 56 (1984), 65–92 | DOI | Zbl

[17] Levy D. W., Powell K. G., van Leer B., “Use of a rotated Riemann solver for the two-dimensional Euler equations”, J. Computational Physics, 106 (1993), 201–214 | DOI | Zbl

[18] Nishikawa H., Kitamura K., “Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers”, J. Computational Physics, 227 (2008), 2560–2581 | MR | Zbl

[19] 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. Mathematics, 39:4 (2018), 543–551 | MR | Zbl

[20] Kulikov I. M., Chernykh I. G., Tutukov A. V., “A new parallel Intel Xeon Phi hydrodynamics code for massively parallel supercomputers”, Lobachevskii J. Mathematics, 39:9 (2018), 1207–1216 | DOI | MR | Zbl

[21] 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 | DOI

[22] Chernykh I., Kulikov I., Tutukov A., “Hydrogen-helium chemical and nuclear galaxy collision: hydrodynamic simulations on AVX-512 supercomputers”, J. Computational and Applied Mathematics, 391 (2021), 113395 | DOI | MR | Zbl

[23] Kim S. D., Lee B. J., Lee H. J., Jeung I. S., “Robust HLLC Riemann solver with weighted average flux scheme for strong shock”, J. Computational Physics, 228 (2009), 7634–7642 | DOI | MR | Zbl

[24] Simon S., Mandal J. C., “A simple cure for numerical shock instability in the HLLC Riemann solver”, J. Computational Physics, 378 (2019), 477–496 | DOI | MR | Zbl

[25] Rodionov A. V., “Artificial viscosity in Godunov-type schemes to cure the carbuncle phenomenon”, J. Computational Physics, 345 (2017), 308–329 | DOI | MR | Zbl

[26] Rodionov A. V., “Artificial viscosity to cure the shock instability in high-order Godunov-type schemes”, Computer Fluids, 190 (2019), 77–97 | DOI | MR | Zbl

[27] 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. Mathematics, 41:8 (2020), 1485–1491 | DOI | MR | Zbl

[28] Guy C., “A HLL-Rankine–Hugoniot Riemann solver for complex non-linear hyperbolic problems”, J. Computational Physics, 251 (2013), 156–193 | MR | Zbl

[29] Capdeville G., “A high-order multi-dimensional HLL-Riemann solver for non-linear Euler equations”, J. Computational Physics, 230 (2011), 2915–2951 | DOI | MR | Zbl

[30] Toro E., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer-Verlag, 2009 | MR | Zbl

[31] Kriksin Yu. A., Tishkin V. F., “Chislennoe reshenie zadachi Einfeldta na osnove razryvnogo metoda Galerkina”, Preprinty IPM im. M. V. Keldysha, 2019, 090, 22 pp.

[32] 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 | MR

[33] Reinecke M., Hillebrandt W., Niemeyer J. C., “Three-dimensional simulations of type Ia supernovae”, Astronomy Astrophysics, 391 (2002), 1167–1172 | DOI