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Regular and non-regular solutions of the Riemann problem in ideal magnetohydrodynamics

Published online by Cambridge University Press:  28 November 2012

K. TAKAHASHI  and
S. YAMADA
K. TAKAHASHI
Affiliation:
Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, 169-8555, Japan ([email protected])
S. YAMADA
Affiliation:
Department of Science & Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, 169-8555, Japan Advanced Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, 169-8555, Japan
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Abstract

We have built a code to numerically solve the Riemann problem in ideal magnetohydrodynamics (MHD) for an arbitrary initial condition to investigate a variety of solutions more thoroughly. The code can handle not only regular solutions, in which no intermediate shocks are involved, but also all types of non-regular solutions if any. As a first application, we explored the neighborhood of the initial condition that was first picked up by Brio and Wu (1988) (Brio, M. and Wu, C. C. 1988 An upwind differencing scheme for the equation of ideal magnetohydrodynamics. J. Comput. Phys. 75, 400–422) and has been frequently employed in the literature as a standard problem to validate numerical codes. Contrary to the conventional wisdom that there will always be a regular solution, we found an initial condition for which there is no regular solution but a non-regular one. The latter solution has only regular solutions in its neighborhood and actually sits on the boundary of regular solutions. This implies that the regular solutions are not sufficient to solve the ideal MHD Riemann problem and suggests that at least some types of non-regular solutions are physical. We also demonstrate that the non-regular solutions are not unique. In fact, we found for the Brio and Wu initial condition that there are uncountably many non-regular solutions. This poses an intriguing question: Why a particular non-regular solution is always obtained in numerical simulations? This has important ramifications to the discussion of which intermediate shocks are really admissible.

Type
Papers
Information
Journal of Plasma Physics , Volume 79 , Issue 3 , June 2013 , pp. 335 - 356
Copyright
Copyright © Cambridge University Press 2012 

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References

Andreev, R., Torrilhon, M. and Jorosch, T. 2008 Exact Riemann Solver for Ideal MHD. https://web.mathcces.rwth-aachen.de/mhdsolver/. Accessed July 17, 2012.Google Scholar
Barmin, A. A., Kulikovskiy, A. G. and Pogorelov, N. V. 1996 Shock-caputuring approach and nonevolutionary solutions in magnetohydrodynamics. J. Comput. Phys. 126 (0121), 77–90.CrossRefGoogle Scholar
Brio, M. and Wu, C. C. 1988 An upwind differencing scheme for the equation of ideal magnetohydrodynamics. J. Comput. Phys. 75, 400–422.CrossRefGoogle Scholar
Chao, J. K. 1995 Intermediate shocks: observations. Adv. Space Res. 15 (8/9), 521–530.CrossRefGoogle Scholar
Falle, S. A. E. G. and Komissarov, S. S. 1997 On the existence of intermediate shocks. In: 12th ‘Kingston Meeting’: Computational Astrophysics (ed. Clark, David A. and West, Michael J.), ASP Conference Series, vol. 123. San Francisco, CA: ASP, pp. 66–71.Google Scholar
Falle, S. A. E. G. and Komissarov, S. S. 2001 On the inadmissibility of non-evolutionary shocks. J. Plasma Phys. 65 (1), 29–58.CrossRefGoogle Scholar
Feng, H. and Wang, J. M. 2008 Observations of a 2 → 3 type interplanetary intermediate shock. Solar Phys. 247, 195–201.CrossRefGoogle Scholar
Feng, H. Q., Wang, J. M. and Chao, J. K. 2009 Observations of a subcritical switch-on shock. A&A 503, 203–206.Google Scholar
Hada, T. 1994 Evolutionary conditions in the dissipative MHD system: stability of intermediate MHD shock waves. Geophys. Res. Lett. 21 (21), 2275–2278.CrossRefGoogle Scholar
Inoue, T. and Inutsuka, S. 2007 Evolutionary conditions in dissipative MHD systems revisited. Prog. Theor. Phys. 118 (1), 47–58.CrossRefGoogle Scholar
Jeffrey, A. and Taniuti, T. 1964 Non-linear Wave Propagation. New York: Academic Press.Google Scholar
Landau, L. D., Lifshitz, E. M. and Pitaevskii, L. P. 1984 Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media, Landau and Lifshitz, 2nd edn.Oxford, UK: Elsevier Butterworth-Heinemann.Google Scholar
Lax, P. D. 1957 Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. X, 537–566.CrossRefGoogle Scholar
Markovskii, S. A. 1998 Nonevolutionary discontinuous magnetohydrodynamic flows in a dissipative medium. Phys. Plasmas 5 (7), 2596–2604.CrossRefGoogle Scholar
Polovin, R. V. and Demutskii, V. P. 1990 Fundamentals of Magnetohydrodynamics. New York: Consultants Bureau.Google Scholar
Takahashi, K. and Yamada, S. 2012 in preparation.Google Scholar
Torrilhon, M. 2002 Exact Solver and Uniqueness Conditions for Riemann Problems of Ideal Magnetohydrodynamics. Research report 2002-06, Seminar for Applied Mathematics, ETH, Zurich.Google Scholar
Torrilhon, M. 2003 Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics. J. Plasma Phys. 69 (3), 253–276.CrossRefGoogle Scholar
Wu, C. C. 1987 On MHD intermediate shocks. Geophy. Res. Lett. 14 (6), 668–671.CrossRefGoogle Scholar
Wu, C. C. 1988a Effects of dissipation on rotational discontinuities. J. Geophy. Res. 93 (A5), 3969–3982.CrossRefGoogle Scholar
Wu, C. C. 1988b The MHD intermediate shock interaction with an intermediate wave: are intermediate shocks physical? J. Geophys. Res. 93 (A2), 987–990.CrossRefGoogle Scholar
Wu, C. C. 1990 Formation, structure, and stability of MHD intermediate shocks. J. Geophys. Res. 95 (A6), 8149–8175.CrossRefGoogle Scholar
Wu, C. C. and Kennel, C. F. 1992 Structural relations for time-dependent intermediate shocks. Geophys. Res. Lett. 19 (20), 2087–2090.CrossRefGoogle Scholar