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Implementation of a non-oscillatory and conservative scheme into magnetohydrodynamic equations
Earth, Planets and Space volume 61, pages895–903(2009)
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.
Brio, M. and C. C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys., 75, 400–422, 1988.
Dai, W. and P. R. Woodward, Extension of the Piecewise Parabolic Method to multidimensional ideal magnetohydrodynamics, J. Comput. Phys., 115, 485–514, 1994.
Fukuda, N. and T. Hanawa, Gravitational and parametric instabilities of the interstellar medium in which the Alfvén wave travels, Astrophys. J., 517, 226–241, 1999.
Hawley, J. F. and J. M. Stone, MOCCT: A numerical technique for astrophysical MHD, Comput. Phys. Commun., 89, 127–148, 1995.
Hirsch, C., Numerical Computation of Internal and External Flows, vol. 2: Computational Methods for Invisid and Viscous Flows, Wiley, 1990.
Kudoh, T. and K. Shibata, Magnetically driven jets from accretion disks. II. Nonsteady solutions and comparison with steady solutions, Astrophys. J., 476, 632–648, 1997.
Kudoh, T., R. Matsumoto, and K. Shibata, Magnetically driven jets from accretion disks. III. 2.5-dimensional nonsteady simulations for thick disk case, Astrophys. J., 508, 186–199, 1998.
Kudoh, T., R. Matsumoto, and K. Shibata, Numerical MHD simulation of astrophysical problems by using CIP-MOCCT method, Comput. Fluid Dyn. J., 8, 56–68, 1999.
Miyoshi, T. and K. Kusano, A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 315–344, 2005.
Ogata, Y. and T. Yabe, Multi-dimensional semi-Lagrangian characteristic approach to the shallow water equations by the CIP method, Int. J. Comput. Eng. Sci., 5, 699–730, 2004.
Ogata, Y., T. Yabe, K. Shibata, and T. Kudoh, Efficient computation of magneto-hydrodynamic phenomena in astrophysics by CCUP-MOCCT method, Int. J. Comput. Method, 1, 201–225, 2004.
Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357–372, 1981.
Ryu, D. and T. W. Jones, Numerical magetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow, Astrophys. J., 442, 228–258, 1995.
Tanaka, S., T. Ogino, and T. Umeda, Parametric decay of circularly polarized Alfven waves in the radially expanding solar wind, J. Geophys. Res., 112, A10110, doi:10.1029/2007JA012513, 2007.
Tanaka, T., Finite volume TVD scheme on an unstructured grid system for three-dimensional MHD simulation of inhomogeneous systems including strong background potential fields, J. Comput. Phys., 111, 381–390, 1994.
Umeda, T., A conservative and non-oscillatory scheme for Vlasov code simulations, Earth Planets Space, 60, 773–779, 2008.
Umeda, T., M. Ashour-Abdalla, and T. Schriver, Comparison of numerical interpolation schemes for one-dimensional electrostatic Vlasov code, J. Plasma Phys., 72, 1057–1060, 2006.
van Leer, B., Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme, J. Comput. Phys., 14, 361–370, 1974.
van Leer, B., Towards the ultimate conservative difference scheme: IV. A new approach to numerical convection, J. Comput. Phys., 23, 276–299, 1977.
Xiao, F., T. Yabe, and T. Ito, Constructing oscillation preventing scheme for advection equation by rational function, Comput. Phys. Commun., 93, 1–12, 1996.
Yabe, T., F. Xiao, and T. Utsumi, The Constrained interpolation profile method for multiphase analysis, J. Comput. Phys., 169, 556–593, 2001.
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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
- numerical scheme
- hyperbolic equation
- conservative scheme