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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 (ktakahashi@heap.phys.waseda.ac.jp)
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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