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Implementation of a non-oscillatory and conservative scheme into magnetohydrodynamic equations

Abstract

We present a magnetohydrodynamic (MHD) simulation technique with a new non-oscillatory and conservative interpolation scheme. Several high-resolution and stable numerical schemes have recently been proposed for solving the MHD equations. To apply the CIP scheme to the hydrodynamic equations, we need to add a certain diffusion term to suppress numerical oscillations at discontinuities. Although the TVD schemes can automatically avoid numerical oscillations, they are not appropriate for profiles with a local maximum or minimum, such as waves. To deal with the above problems, we implement a new non-oscillatory and conservative interpolation scheme in MHD simulations. Several numerical tests are carried out in order to compare our scheme with other recent high-resolution schemes. The numerical tests suggest that the present scheme can follow long-term evolution of both Alfvén waves and compressive shocks. The present scheme has been used for a numerical modeling of Alfvén waves in the solar wind, in which sinusoidal Alfvén waves decay into compressive sound waves that steepen into shocks.

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Correspondence to Shin Tanaka.

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Tanaka, S., Umeda, T., Matsumoto, Y. et al. Implementation of a non-oscillatory and conservative scheme into magnetohydrodynamic equations. Earth Planet Sp 61, 895–903 (2009). https://doi.org/10.1186/BF03353200

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Key words

  • MHD
  • numerical scheme
  • hyperbolic equation
  • conservative scheme