In this paper, we consider the magnetohydrodynamics (MHD) equations with variable density, which are a coupled system by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. Such MHD system describes the motions of several conducting incompressible immiscible fluids without surface tension in presence of a magnetic field. A first-order Euler semi-implicit time discrete scheme is proposed to approximate the MHD system such that we only need to solve the linearized subproblems at the discrete level. Moreover, it is unconditionally stable which is a key issue for problems of multiphysical fields. A rigorous error analysis is presented and the first-order temporal convergence rate $O(\tau )$ is derived for small τ by using the discrete $L^p$-regularity technique, where τ is the time step. The numerical results are shown to confirm the unconditional stability and the convergence rate.

在本文中，我们考虑具有可变密度的磁流体动力学（MHD）方程，该方程是由不可压缩的具有可变密度的Navier-Stokes方程和Maxwell方程耦合而成的。这样的MHD系统描述了在磁场存在下没有表面张力的几种传导性不可压缩的不混溶流体的运动。提出了一阶欧拉半隐式时间离散方案，以近似MHD系统，因此我们只需要在离散级求解线性化子问题。而且，它是无条件稳定的，这是解决多物理场问题的关键问题。提出了严格的误差分析，并对一阶时间收敛速度进行了分析。$Ø（\tau ）$通过使用离散量得出小τ${\mathrm{大号}}^p$-regularity技术，其中τ是时间步长。数值结果表明了无条件稳定性和收敛速度。