Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations.

*(English)*Zbl 1348.78028The paper under review deals with the study of the first order divergence-free method in connection with Cartesian meshes for the magnetic induction equations. This study is in strong relationship with the stability and numerical analysis of solutions of the ideal magneto-hydrodynamic equations. The authors are mainly concerned with the numerical stability, which is established through both energy and Fourier methods. This study is performed when the meshes are uniform and when the velocity field in the equations is constant. The authors also establish a priori error estimates in the \(L^2\) norm for sufficiently smooth solutions.

Reviewer: Teodora-Liliana Rădulescu (Craiova)

##### MSC:

78M25 | Numerical methods in optics (MSC2010) |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35Q60 | PDEs in connection with optics and electromagnetic theory |

35L50 | Initial-boundary value problems for first-order hyperbolic systems |

##### Keywords:

stability; error estimates; ideal magnetohydrodynamic (MHD) equations; constrained transport; divergence-free; discontinuous Galerkin; magnetic induction equations
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\textit{H. Yang} and \textit{F. Li}, ESAIM, Math. Model. Numer. Anal. 50, No. 4, 965--993 (2016; Zbl 1348.78028)

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