In this paper, we propose and analyze a temporally second-order accurate, fully discrete finite element method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Optimal-order convergence of $\mathcal{O} (\tau^2+h^{r+1})$ in the discrete $L^\infty(0,T;L^2)$ norm is proved for the proposed decoupled projection finite element scheme, where $\tau$ and $h$ are the time stepsize and spatial mesh size, respectively, and $r$ is the degree of the finite elements. Existing error estimates of second-order projection methods of the Van Kan type \cite{vankan1986} were only established in the discrete $L^2(0,T;L^2)$ norm for the Navier--Stokes equations. Numerical examples are provided to illustrate the theoretical results.

翻译：在本文中, 我们提议并分析磁力流体动力( MHD) 方程式的暂时第二顺序准确、 完全离散的有限元素方法 。 使用修改的 Crank- Nicolson 方法将模型分解, 并对流体对流体对流术语和两个组合条件应用适当的半隐含处理 。 这些半隐含的近似结果产生了一个具有可变系数的线性系统, 可以从理论上证明独有的溶解性。 此外, 我们在Stokes 解析器中使用Van Kan 类型\ cite{vankan1986} 的解析预测方法。 根据前一时间级压力的梯度计算中间速度字段, 通过 Helmholtz 对中间速度字段进行分解。 这个办法的能量稳定性在理论上得到证明, 其中解析的Stokeys Stokeys Stalalalal $( mathcal) $( $); (tau2+_ +_r) rodeal rodual roup $; rodudeal dealal $L2 rodude rodudeal rodude rodude rodude rodudeal $; rodududeal $; rol_ rol= roildal_l_ rol_ rol_ rol_ rol_ rodudal_ rol_ rodal_ rodal_ $= 美元; rol_ rodaldal_ 美元 rodal_ rodal_l_ rod 美元 美元 美元= 美元; rodal_ 美元; rodal_l_ rodal_l_l_l_l_l_ rodal_ rodal_ rodal rodal rodal_ routdal rodal rodal rod rodal rodal rodal rodal_ rodal rodal_ rodal_ rod_ 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元 美元