This paper studies a time filtered finite element method for the unsteady inductionless magnetohydrodynamic (MHD) equations. The method uses the semi-implicit backward Euler scheme with a time filter in time and adopts the standard inf-sup stable fluid pairs to discretize the velocity and pressure, and the inf-sup stable face-volume elements for solving the current density and electric potential in space. Since the time filter for the velocity is added as a separate post-processing step, the scheme can be easily incorporated into the existing backward Euler code and improves the time accuracy from first order to second order. The unique solvability, unconditional energy stability, and charge conservativeness of the scheme are also proven. In terms of the energy arguments, we establish the error estimates for the velocity, current density, and electric potential. Numerical experiments are performed to verify the theoretical analysis.

本文研究了一种用于非定常无归纳磁流体动力学 （MHD） 方程的时间过滤有限元方法。该方法采用半隐式反向欧拉方案和时间过滤器，采用标准的 inf-sup 稳定流体对来离散速度和压力，并使用 inf-sup 稳定面体积单元来求解空间中的电流密度和电势。由于速度的时间滤波器是作为单独的后处理步骤添加的，因此该方案可以很容易地合并到现有的向后欧拉码中，并提高从一阶到二阶的时间精度。该方案独特的可解性、无条件能量稳定性和电荷保守性也得到了证明。根据能量参数，我们建立了速度、电流密度和电势的误差估计。进行了数值实验以验证理论分析。