This paper proposes a virtual element method (VEM) combined with a second-order implicit-explicit scheme based on the scalar auxiliary variable (SAV) method for the incompressible magnetohydrodynamics (MHD) equations. We employ the BDF2 scheme for time discretization and a conservative VEM for spatial discretization, in which the mass conservation in the velocity field is kept by taking advantage of the virtual element method's adaptability and its divergence-free characteristics. In our scheme, the nonlinear terms are handled explicitly using the SAV method, and the magnetic field is decoupled from the velocity and pressure. This decoupling only requires solving a sequence of linear systems with constant coefficient at each time step. The stability estimate of the fully discrete scheme is developed, demonstrating the scheme is unconditionally stable. Moreover, rigorous error estimates for the velocity and magnetic field are provided. Finally, numerical experiments are presented to verify the valid of theoretical analysis.

翻译：本文针对不可压缩磁流体力学方程，提出了一种结合标量辅助变量法二阶隐式-显式格式的虚拟元方法。我们采用BDF2格式进行时间离散，并采用守恒型虚拟元方法进行空间离散，其中利用虚拟元方法的适应性与无散特性保持了速度场中的质量守恒。在本方案中，非线性项通过SAV方法显式处理，磁场从速度和压力场中解耦。这种解耦仅需在每个时间步求解一系列常系数线性方程组。我们建立了全离散格式的稳定性估计，证明该格式是无条件稳定的。此外，文中给出了速度和磁场场的严格误差估计。最后，通过数值实验验证了理论分析的有效性。