Abstract
A new finite-difference method for the numerical simulation of compressible MHD flows is presented, which is applicable to a broad class of problems. The method relies on the magnetic quasi-gasdynamic equations (referred to as quasi-MHD (QMHD) equations), which are, in fact, the system of Navier–Stokes equations and Faraday’s laws averaged over a short time interval. The QMHD equations are discretized on a grid with the help of central differences. The averaging procedure makes it possible to stabilize the numerical solution and to avoid additional limiting procedures (flux limiters, etc.). The magnetic field is ensured to be free of divergence by applying Stokes’ theorem. Numerical results are presented for 3D test problems: a central blast in a magnetic field, the interaction of a shock wave with a cloud, and the three-dimensional Orszag–Tang vortex. Additionally, preliminary numerical results for a magnetic pinch in plasma are demonstrated.
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References
T. G. Elizarova, Quasi-Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer-Verlag, Berlin, 2009).
B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations (MAKS, Moscow, 2004; CIMNE, Barcelona, 2008).
Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (NITs Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2009).
T. G. Elizarova, I. S. Kalachinskaya, Yu. V. Sheretov, and I. A. Shirokov, “Numerical simulation of electrically conducting liquid flows in an external magnetic field,” J. Commun. Technol. Electron. 50 (2), 227–233 (2005).
T. G. Elizarova and S. D. Ustyugov, Preprint No. 1, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011).
T. G. Elizarova and S. D. Ustyugov, Preprint No. 30, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011).
B. Ducomet and A. Zlotnik, “On a regularization of the magnetic gas dynamics system of equations,” Kinetic Related Models 6 (3), 533–543 (2013).
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).
T. G. Elizarova, “Time averaging as an approximate technique for constructing quasi-gasdynamic and quasihydrodynamic equations,” Comput. Math. Math. Phys. 51 (11), 1973–1982 (2011).
T. A. Gardiner and J. M. Stone, “An unsplit Godunov method for ideal MHD via constrained transport in three dimensions,” J. Comput. Phys. 227, 4123–4141 (2008).
S. D. Ustyugov, M. V. Popov, A. F. Kritsuk, and M. L. Norman, “Piecewise parabolic method on a local stencil for magnetized supersonic turbulence simulation,” J. Comput. Phys. 228, 7614–7633 (2009).
W. Dai and P. Woodward, “A simple finite difference scheme for multidimensional magnetohydrodynamical equations,” J. Comput. Phys. 142, 331–369 (1998).
G. Tóth, “The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes,” J. Comput. Phys. 161, 605–652 (2000).
S. A. Orszag and C.-M. Tang, “Small-scale structure of two-dimensional magnetohydrodynamic turbulence,” J. Fluid Mech. 90, 129–143 (1979).
A. A. Vedenov, E. P. Velikhov, and R. Z. Sagdeev, “Stability of plasma” Sov. Phys. Usp. 4, 332–369 (1961).
V. E. Golant, A. P. Zhilinskii, and S. A. Sakharov, Fundamentals of Plasma Physics (Atomizdat, Moscow, 1977; Wiley, New York, 1980).
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Original Russian Text © T.G. Elizarova, M.V. Popov, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 8, pp. 1363–1379.
In blessed memory of Professor A.P. Favorskii
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Elizarova, T.G., Popov, M.V. Numerical simulation of three-dimensional quasi-neutral gas flows based on smoothed magnetohydrodynamic equations. Comput. Math. and Math. Phys. 55, 1330–1345 (2015). https://doi.org/10.1134/S0965542515080084
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DOI: https://doi.org/10.1134/S0965542515080084