Locally divergence-free discontinuous Galerkin methods for MHD equations.

*(English)*Zbl 1123.76341Summary: In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations [B. Cockburn et al., J. Comput. Phys. 194, No. 2, 588–610 (2004; Zbl 1049.78019)], to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

##### Keywords:

Discontinuous Galerkin method; divergence-free solutions; magnetohydrodynamics (MHD) equations
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\textit{F. Li} and \textit{C.-W. Shu}, J. Sci. Comput. 22--23, 413--442 (2005; Zbl 1123.76341)

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##### References:

[22] | Powell K. G. (1994). An Approximate Riemann solver for Magnetohydrodynamics (that works in more than one dimension), ICASE report No. 94–24, Langley, VA. |

[23] | Qiu, J., and Shu, C.-W. Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput., to appear. · Zbl 1077.65109 |

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