This article explores the unsteady magnetohydrodynamic (MHD) model within the framework of porous media flow. This model consists of the Brinkman–Forchheimer equations and Maxwell equations in the porous media domain, which are coupled by the Lorentz force. We propose and analyze a numerical discretization method for MHD porous model. The second-order backward difference formula is utilized for temporal derivative terms, and the mixed finite element method is employed for spatial discretization. Rigorous proofs of stability and uniqueness are provided for the numerical solutions. We establish optimal L2-error estimates for the velocity and magnetic induction without imposing constraints on the relationship between the time step and mesh size. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and validate the theoretical findings. To our knowledge, this is the first error analysis and simulation to address unsteady MHD flow through porous media.

中文翻译：

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一种用于多孔介质中非定常磁流体动力学流动的全离散有限元方法


本文探讨了多孔介质流动框架内的非定常磁流体动力学 （MHD） 模型。该模型由多孔介质域中的 Brinkman-Forchheimer 方程和 Maxwell 方程组成，它们由洛伦兹力耦合。我们提出并分析了一种用于 MHD 多孔模型的数值离散化方法。二阶向后差分公式用于时间导数项，混合有限元方法用于空间离散化。为数值解提供了稳定性和唯一性的严格证明。我们为速度和磁感应建立了最佳的 L2 误差估计，而没有对时间步长和网格尺寸之间的关系施加约束。最后，进行了几次三维数值实验，以说明所提数值方法的特点并验证了理论结果。据我们所知，这是第一个解决通过多孔介质的 MHD 流的误差分析和仿真。