In this paper, a second-order backward differentiation formula (BDF) scheme for a hybrid MHD system is considered. Being different with the steady and nonstationary MHD equations, the hybrid MHD system is coupled by the time-dependent Navier-Stokes equations and the steady Maxwell equations. By using the standard extrapolation technique for the nonlinear terms, the proposed BDF scheme is a semi-implicit scheme. Furthermore, this scheme is a decoupled scheme such that the magnetic field and the velocity can be solved independently at the same time as discrete level. A rigorous error analysis is done and we prove the unconditionally optimal second-order convergence rate \(\mathcal O(h^{2}+({\Delta } t)^{2})\) in L² norm for approximations of the magnetic field and the velocity, where h and Δt are the grid mesh and the time step, respectively. Finally, the numerical results are displayed to illustrate the theoretical results.

本文考虑了混合MHD系统的二阶后向差分公式（BDF）方案。与稳态MHD方程和非稳态MHD方程不同，混合MHD系统由时间相关的Navier-Stokes方程和稳态Maxwell方程耦合。通过对非线性项使用标准外推技术，提出的BDF方案是一种半隐式方案。此外，该方案是解耦方案，使得磁场和速度可以与离散水平同时独立地求解。一个严格的错误分析完成，我们证明了无条件最佳二阶收敛速度\（\ mathcal O（H ^ {2} +（{\德尔塔}吨）^ {2}）\）在大号²磁场和速度近似的范数，其中h和Δt分别是网格和时间步长。最后，显示数值结果以说明理论结果。