Shock Waves

pp 1–17

Hybridized SLAU2–HLLI and hybridized AUSMPW$$+$$–HLLI Riemann solvers for accurate, robust, and efficient magnetohydrodynamics (MHD) simulations, part I: one-dimensional MHD

• K. Kitamura
• D. S. Balsara
Original Article

Abstract

SLAU2 and AUSMPW$$+$$, both categorized as AUSM-type Riemann solvers, have been extensively developed in gasdynamics. They are based on a splitting of the numerical flux into advected and pressure parts. In this paper, these two Riemann solvers have been extended to magnetohydrodynamics (MHD). The SLAU2 Riemann solver has the favorable attribute that its dissipation for low-speed flows scales as $$O(M^{2})$$, where M is the Mach number. This is the physical scaling required for low-speed flows, and the dissipation in SLAU2 for MHD is engineered to have this low Mach number scaling. The AUSMPW$$+$$, when its pressure flux is replaced with that of SLAU2, has the same low Mach number scaling. At higher Mach numbers, however, the pressure-split Riemann solvers were found not to function well for some MHD Riemann problems, despite the fact that they were engineered to have a dissipation that scales as $$O(\vert M\vert )$$ for high Mach number flows. The HLLI Riemann solver (Dumbser and Balsara in J Comput Phys 304:275–319, 2016) has a dissipation that scales as $$O(\vert M\vert )$$, which makes it unsuitable for low Mach number flows. However, it has very favorable performance for higher Mach number MHD flows. Since the two families of Riemann solvers perform very well over a range of intermediate Mach numbers, the best way to benefit from the mutually complementary strengths of both these Riemann solvers is to hybridize between them. The result is an all-speed Riemann solver for MHD. We, therefore, document hybridized SLAU2–HLLI and AUSMPW$$+$$–HLLI Riemann solvers. The hybrid Riemann solvers suppress the oscillations that appeared in single-solver solutions, and they also preserve contact discontinuities, as well as Alfvén waves, very well. Furthermore, their better resolution at low speeds has been demonstrated. We also present several stringent one-dimensional test problems.

Keywords

SLAU2 AUSMPW$$+$$ HLLI MHD Euler fluxes

Notes

Acknowledgements

This work has been conducted while the first author (K. Kitamura) was visiting at the University of Notre Dame. We would like to express gratitude to the University of Notre Dame for hosting the first author and also to Yokohama National University for financial support. The first author also thanks Shigenobu Hirose and Takashi Minoshima at JAMSTEC (Japan Agency for Marine-Earth Science and Technology), Japan (introduced through Eiji Shima, JAXA), and Nishant M. Narechania at University of Toronto, Canada, and Dan Hori at Nagoya University, Japan, for providing him with fundamental knowledge on astrophysics and nuclear fusion, respectively. Last but not least, he really appreciates Meng-Sing Liou at NASA Glenn Research Center for his continuous discussions on SLAU2, one of AUSM-family fluxes. The second author (DSB) acknowledges support via NSF Grants NSF-DMS-1361197, NSF-ACI-1533850, NSF-DMS-1622457, and NSF-ACI-1713765. Several simulations were performed on a cluster at UND that is run by the Center for Research Computing. Computer support on NSF’s XSEDE and Blue Waters computing resources is also acknowledged.

References

1. 1.
Esquivel, A., Raga, A.C., Cantó, J., Rodríguez-González, A., López-Cámara, D., Velázquez, P.F., De Colle, F.: Model of Mira’s cometary head/tail entering the local bubble. Astrophys. J. 725, 1466–1475 (2010).
2. 2.
Ohnishi, N., Kotake, K., Yamada, S.: Numerical analysis on standing accretion shock instability with neutrino heating in the supernova cores. Astrophys. J. 641(2), 1018–1028 (2006).
3. 3.
Hanawa, T., Mikami, H., Matsumoto, T.: Improving shock irregularities based on the characteristics of the MHD equations. J. Comput. Phys. 227, 7952–7976 (2008).
4. 4.
Poggie, J., Gaitonde, D.V.: Magnetic control of flow past a blunt body: numerical validation and exploration. Phys. Fluids 14(5), 1720–1731 (2002).
5. 5.
Burr, U., Barleon, L., Müller, U., Tsinober, A.: Turbulent transport of momentum and heat in magnetohydrodynamic rectangular duct flow with strong sidewall jets. J. Fluid Mech. 406, 247–279 (2000).
6. 6.
Godunov, S.K.: A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. Izd. Mosk. Mat. Obs. 47(3), 271–306 (1959)
7. 7.
van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979).
8. 8.
Colella, P.: A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput. 6(1), 104–117 (1985).
9. 9.
Colella, P., Woodward, P.R.: The Piecewise Parabolic Method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201 (1984).
10. 10.
Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981).
11. 11.
Harten, A., Hyman, J.M.: Self adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50, 235–269 (1983).
12. 12.
Harten, A., Lax, P.D., Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983).
13. 13.
Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294–318 (1988).
14. 14.
Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273–295 (1991).
15. 15.
Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)Google Scholar
16. 16.
Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL Riemann solver. Shock Waves 4, 25–34 (1994).
17. 17.
Batten, P., Clarke, N., Lambert, C., Causon, D.M.: On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18, 1553–1570 (1997).
18. 18.
Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comput. 38(158), 339–374 (1982).
19. 19.
Dumbser, M., Toro, E.F.: A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput. 48, 70–88 (2011).
20. 20.
Dumbser, M., Balsara, D.S.: A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys. 304, 275–319 (2016).
21. 21.
Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75, 400–422 (1988).
22. 22.
Ryu, D.S., Jones, T.W.: Numerical magnetohydrodynamics in astrophysics: algorithm and test for one-dimensional flow. Astrophys. J. 442, 228–258 (1995).
23. 23.
Roe, P.L., Balsara, D.S.: Notes on the eigensystem of magnetohydrodynamics. SIAM J. Appl. Math. 56, 57–67 (1996).
24. 24.
Cargo, P., Gallice, G.: Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws. J. Comput. Phys. 136, 446–466 (1997).
25. 25.
Balsara, D.S.: Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics. Astrophys. J. Suppl. 116, 119–131 (1998).
26. 26.
Miyoshi, T., Kusano, K.: A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208, 315–344 (2005).
27. 27.
Gurski, K.F.: An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics. SIAM J. Sci. Comput. 25(6), 2165–2187 (2004).
28. 28.
Li, S.: An HLLC Riemann solver for magnetohydrodynamics. J. Comput. Phys. 203, 344–357 (2005).
29. 29.
Liou, M.S., Steffen Jr., C.J.: A new flux splitting scheme. J. Comput. Phys. 107(1), 23–39 (1993).
30. 30.
Liou, M.S.: A sequel to AUSM: AUSM$$+$$. J. Comput. Phys. 129(2), 364–382 (1996).
31. 31.
Kim, K.H., Kim, C., Rho, O.H.: Methods for the accurate computations of hypersonic flows: I. AUSMPW$$+$$ scheme. J. Comput. Phys. 174, 38–80 (2001).
32. 32.
Liou, M.-S.: A sequel to AUSM, part II: AUSM$${+}$$-up for all speeds. J. Comput. Phys. 214, 137–170 (2006).
33. 33.
Shima, E., Kitamura, K.: Parameter-free simple low-dissipation AUSM-family scheme for all speeds. AIAA J. 49(8), 1693–1709 (2011).
34. 34.
Kitamura, K., Shima, E.: Towards shock-stable and accurate hypersonic heating computations: a new pressure flux for AUSM-family schemes. J. Comput. Phys. 245, 62–83 (2013).
35. 35.
Kitamura, K., Shima, E., Fujimoto, K., Wang, Z.J.: Performance of low-dissipation Euler fluxes and preconditioned LU-SGS at low speeds. Commun. Comput. Phys. 10(1), 90–119 (2011).
36. 36.
Weiss, J.M., Smith, W.A.: Preconditioning applied to variable and constant density flows. AIAA J. 33(11), 2050–2057 (1995).
37. 37.
Li, X., Gu, C.: An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behavior. J. Comput. Phys. 227, 5144–5159 (2008).
38. 38.
Kitamura, K.: Assessment of SLAU2 and other flux functions with slope limiters in hypersonic shock-interaction heating. Comput. Fluids 129, 134–145 (2016).
39. 39.
Pandolfi, M., D’Ambrosio, D.: Numerical instabilities in upwind methods: analysis and cures for the “Carbuncle” phenomenon. J. Comput. Phys. 166(2), 271–301 (2001).
40. 40.
Kitamura, K., Shima, E., Roe, P.: Carbuncle phenomena and other shock anomalies in three dimensions. AIAA J. 50(12), 2655–2669 (2012).
41. 41.
Chauvat, Y., Moschetta, J.M., Gressier, J.: Shock wave numerical structure and the carbuncle phenomenon. Int. J. Numer. Methods Fluids 47(8–9), 903–909 (2005).
42. 42.
Dumbser, M., Moschetta, J.M., Gressier, J.: A matrix stability analysis of the carbuncle phenomenon. J. Comput. Phys. 197(2), 647–670 (2004).
43. 43.
Quirk, J.J.: A contribution to the great Riemann solver debate. Int. J. Numer. Methods Fluids 18(6), 555–574 (1994).
44. 44.
Kitamura, K., Roe, P., Ismail, F.: Evaluation of Euler fluxes for hypersonic flow computations. AIAA J. 47, 44–53 (2009).
45. 45.
Barth, T.J.: Some notes on shock-resolving flux functions part 1: stationary characteristics. NASA TM-101087 (1989)Google Scholar
46. 46.
Hanayama, H., Takahashi, K., Tomisaka, K.: Generation of seed magnetic fields in primordial supernova remnants. ArXiv e-prints arXiv:0912.2686 (2009)
47. 47.
Balsara, D.S.: Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics. Astrophys. J. Suppl. Ser. 116, 133–153 (1998).
48. 48.
Stone, J.M., Gardiner, T.A., Teuben, P., Hawley, J.F., Simon, J.B.: ATHENA: a new code for astrophysical MHD. Astrophys. J. Suppl. Ser. 178(1), 137–177 (2008).
49. 49.
Matsumoto, Y., Asahina, Y., Kudoh, Y., Kawashima, T., Matsumoto, J., Takahashi, H.R., Minoshima, T., Zenitani, S., Miyoshi, T., Matsumoto, R.: Magnetohydrodynamic simulation code CANS$$+$$: assessments and applications. ArXiv e-prints arXiv:1611.01775 (2016)
50. 50.
Tzeferacos, P., Fatenejad, M., Flocke, N., Graziani, C., Gregori, G., Lamb, D.Q., Lee, D., Meinecke, J., Scopatz, A., Weide, K.: FLASH MHD simulations of experiments that study shock-generated magnetic fields. High Energy Density Phys. 17(Part A), 24–31 (2015).
51. 51.
Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O., Zanni, C., Ferrari, A.: PLUTO: a numerical code for computational astrophysics. (2007) arXiv:astro-ph/0701854
52. 52.
Shen, Y., Zha, G., Huerta, M.A.: E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO scheme. J. Comput. Phys. 231, 6233–6247 (2012).
53. 53.
Han, S.H., Lee, J.I., Kim, K.H.: Accurate and robust pressure weight advection upstream splitting method for magnetohydrodynamics equations. AIAA J. 47(4), 970–981 (2009).
54. 54.
Xisto, C.M., Páscoa, J.C., Oliveira, P.J.: A pressure-based high resolution numerical method for resistive MHD. J. Comput. Phys. 275, 323–345 (2014).
55. 55.
Kitamura, K., Hashimoto, A.: Reduced dissipation AUSM-family fluxes: HR-SLAU2 and HR-AUSM+-up for high resolution unsteady flow simulations. Comput. Fluids 126, 41–57 (2016).
56. 56.
Matsuyama, S.: Performance of all-speed AUSM-family schemes for DNS of low Mach number turbulent channel flow. Comput. Fluids 91, 130–143 (2014).
57. 57.
Batchelor, G.K.: The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere. Q. J. R. Meteorol. Soc. 79(340), 224–235 (1953).
58. 58.
Ogura, Y., Phillips, N.: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179 (1962)
59. 59.
Gough, D.O.: The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448–456 (1969).
60. 60.
Hotta, H., Rempel, M., Yokoyama, T.: High-resolution calculations of the solar global convection with the reduced speed of sound technique. I. The structure of the convection and the magnetic field without the rotation. Astrophys. J. 786, 24 (2014).
61. 61.
Brown, B.P., Vasil, G.M., Zweibel, E.G.: Energy conservation and gravity waves in sound-proof treatments of stellar interiors. Part I. Anelastic approximations. Astrophys. J. 756, 109 (2012).
62. 62.
Vasil, G.M., Lecoanet, D., Brown, B.P., Wood, T.S., Zweibel, E.G.: Energy conservation and gravity waves in sound-proof treatments of stellar interiors. II. Lagrangian constrained analysis. Astrophys. J. 773, 169 (2013).
63. 63.
Rogers, T.M., Glatzmaier, G.A.: Angular momentum transport by gravity waves in the solar interior. Astrophys. J. 653(1), 756–764 (2006).
64. 64.
Rogers, T.M., MacGregor, K.B.: On the interaction of internal gravity waves with a magnetic field—II. Convective forcing. Mon. Not. R. Astron. Soc. 410(2), 946–962 (2011).
65. 65.
MacGregor, K.B., Rogers, T.M.: Reflection and ducting of gravity waves inside the Sun. Sol. Phys. 270(2), 417–436 (2011).
66. 66.
Brun, A.S., Miesch, M.S., Toomre, J.: Modeling the dynamical coupling of solar convection with the radiative interior. Astrophys. J. 742, 79 (2011).
67. 67.
Lantz, S.R.: Dynamical behavior of magnetic fields in a stratified, convecting fluid layer. Ph.D. Thesis, Cornell University (1992)Google Scholar
68. 68.
Braginsky, S.I., Roberts, P.H.: Equations governing convection in Earth’s core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79(1), 1–97 (1995).
69. 69.
ITER Physics Basis Editors et al.: ITER physics basis. Nucl. Fusion 39, 2137 (1999).
70. 70.
Loarte, A., Liu, F., Huijsmans, G.T.A., Kukushkin, A.S., Pitts, R.A.: MHD stability of the ITER pedestal and SOL plasma and its influence on the heat flux width. J. Nucl. Mater. 463, 401–405 (2015).
71. 71.
Balsara, D.S.: Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. Ser. 151, 149–184 (2004).
72. 72.
Balsara, D.S., Spicer, D.S.: A staggered Mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamics simulation. J. Comput. Phys. 149, 270–292 (1999).
73. 73.
Dedner, A., Kemm, F., Kröner, D., Munz, C.D., Schnitzer, T., Wesenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175, 645–673 (2002).
74. 74.
Tóth, G.: The $$\nabla \cdot {\varvec {B}} = 0$$ constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161, 605–652 (2000).
75. 75.
Powell, K.G., Roe, P.L., Linde, T., Gombosi, T.I., De Zeeuw, D.L.: A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154, 284–309 (1999).
76. 76.
Balsara, D.S.: Multidimensional HLLE Riemann solver; application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 229, 1970–1993 (2010).
77. 77.
Balsara, D.S.: A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 231, 7476–7503 (2012).
78. 78.
Balsara, D.S.: Multidimensional Riemann problem with self-similar internal structure, part I: application to hyperbolic conservation laws on structured meshes. J. Comput. Phys. 277, 163–200 (2014).
79. 79.
Balsara, D.S., Nkonga, B.: Multidimensional Riemann problem with self-similar internal structure—part III—a multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems. J. Comput. Phys. 346, 25–48 (2017).
80. 80.
Balsara, D.S.: Self-adjusting, positivity preserving high order schemes for hydrodynamic and magnetohydrodynamics. J. Comput. Phys. 231, 7504–7517 (2012).
81. 81.
Shima, E., Kitamura, K.: AUSM like expression of HLLC scheme and its extension to all speed scheme. In: 6th European Conference on Computational Fluid Dynamics (ECFD VI) Barcelona, Spain (2014)Google Scholar
82. 82.
Luo, H., Baum, J.D., Löhner, R.: Extension of Harten–Lax–van Leer scheme for flows at all speeds. AIAA J. 43(6), 1160–1166 (2005).
83. 83.
Park, S.H., Lee, J.E., Kwon, J.H.: Preconditioned HLLE method for flows at all Mach numbers. AIAA J. 44(11), 2645–2653 (2006).
84. 84.
Stix, M.: The Sun: An Introduction, 2nd edn. Astronomy and Astrophysics Library. Springer, Berlin (2002). ISBN 3-540-20741-4.
85. 85.
Huang, K., Wu, H., Yu, H., Yan, D.: Cures for numerical shock instability in HLLC solver. Int. J. Numer. Methods Fluids 65(9), 1026–1038 (2011).
86. 86.
Zha, G., Shen, Y., Huerta, M.: Rotated hybrid low diffusion ECUSP-HLL scheme and its applications to hypersonic flows. In: 20th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2011-3545 (2011).
87. 87.
Ihm, S.-W., Kim, C.: Computations of homogeneous-equilibrium two-phase flows with accurate and efficient shock-stable schemes. AIAA J. 46(12), 3012–3037 (2008).
88. 88.
Dai, W., Woodward, P.R.: An approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 111, 354–372 (1994).
89. 89.
Pierce, A.D.: ACOUSTICS—An Introduction to Its Physical Principles and Applications, pp. 20–22. Acoustical Society of America, New York (1989)Google Scholar
90. 90.
Kitamura, K., Hashimoto, A., Murakami, K., Aoyama, T. and Nakamura, Y.: High resolution CFD/CAA hybrid analysis of supersonic jet interacting with walls. In: 37th AIAA Fluid Dynamics Conference and Exhibit, Miami, FL, AIAA Paper 2007-3871 (2007).
91. 91.
Happenhofer, N., Grimm-Strele, H., Kupka, F., Löw-Baselli, B., Muthsam, H.: A low Mach number solver: enhancing applicability. J. Comput. Phys. 236, 96–118 (2013).

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

• K. Kitamura
• 1
• D. S. Balsara
• 2
1. 1.Yokohama National UniversityYokohamaJapan
2. 2.University of Notre DameNotre DameUSA
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