This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

中文翻译：

---

非定常不可压缩磁流体动力学方程的克兰克-尼科尔森蛙跳式

本文提出了非定常不可压缩磁流体动力学（MHD）方程的克兰克-尼科尔森蛙跳（CNLF）方案。空间离散化采用伽辽金有限元法(FEM)，时间离散化线性项采用CNLF法，非线性项采用半隐式法。第一步采用Stokes风格的方案，第二步采用Crank-Nicolson外推方案，其他采用CNLF方案。我们确定，当时间步长小于或等于正常数时，完全离散方案是稳定且收敛的。首先，我们通过数学归纳法证明了该方案的稳定性。接下来重点分析CNLF方法的误差估计，其中速度和磁场的收敛阶数达到二阶精度，压力为一阶收敛精度。最后，数值例子证明了所提出算法的最佳误差估计。