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An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations. (English) Zbl 1369.76061
Summary: Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must either confront the challenge of controlling errors in the discrete divergence of the magnetic field, or else be faced with nonlinear numerical instabilities. One approach for controlling the discrete divergence is through a so-called constrained transport method, which is based on first predicting a magnetic field through a standard finite volume solver, and then correcting this field through the appropriate use of a magnetic vector potential. In this work we develop a constrained transport method for the 3D ideal MHD equations that is based on a high-resolution wave propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme developed by J.A. Rossmanith [SIAM J. Sci. Comput. 28, No. 5, 1766–1797 (2006; Zbl 1344.76092)], and is based on the high-resolution wave propagation method of J. O. Langseth and R. J. LeVeque [J. Comput. Phys. 165, No.1, 126–166 (2000; Zbl 0967.65095)]. In particular, in our extension we take great care to maintain the three most important properties of the 2D scheme: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as cell-centered; (2) we develop a high-resolution wave propagation scheme for evolving the magnetic potential; and (3) we develop a wave limiting approach that is applied during the vector potential evolution, which controls unphysical oscillations in the magnetic field. One of the key numerical difficulties that is novel to 3D is that the transport equation that must be solved for the magnetic vector potential is only weakly hyperbolic. In presenting our numerical algorithm we describe how to numerically handle this problem of weak hyperbolicity, as well as how to choose an appropriate gauge condition. The resulting scheme is applied to several numerical test cases.

76W05 Magnetohydrodynamics and electrohydrodynamics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M25 Other numerical methods (fluid mechanics) (MSC2010)
82D10 Statistical mechanics of plasmas
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