Shock Waves

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

  • K. KitamuraEmail author
  • D. S. Balsara
Original Article


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.


SLAU2 AUSMPW\(+\) HLLI MHD Euler fluxes 



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.


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

Authors and Affiliations

  1. 1.Yokohama National UniversityYokohamaJapan
  2. 2.University of Notre DameNotre DameUSA

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