The Crank–Nicolson (CN) finite element method is examined with a linear time relaxation term in this study. The linear differential filter term is added to simplified magnetohydrodynamics (SMHD) equations for numerical regularization and it introduced SMHD linear time relaxation model (SMHDLTRM). The SMHDLTRM model is discretized by CN method in time and the finite element method in space. The stability and convergency of the method are also conducted. The method is unconditionally stable and convergent under the small time step condition. Additionally, this study summarizes the effectiveness of four methods for SMHD and SMHDLTRM. In previous works SMHD is solved with CN and BE methods and SMHDLTRM is solved with BE method. In this study, the CN solutions of the SMHDLTRM are obtained and compared with the other solutions. All computations are conducted by using FreeFem++.

中文翻译：

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具有线性时间弛豫的简化磁流体动力学的 Crank-Nicolson 方法的数值分析

在本研究中，使用线性时间松弛项检查了 Crank-Nicolson (CN) 有限元方法。线性差分滤波器项被添加到简化的磁流体动力学 (SMHD) 方程以进行数值正则化，并引入了 SMHD 线性时间弛豫模型 (SMHDLTRM)。SMHDLTRM模型在时间上用CN方法离散化，在空间上用有限元方法离散化。还进行了方法的稳定性和收敛性。该方法在小时间步长条件下无条件稳定收敛。此外，本研究总结了 SMHD 和 SMHDLTRM 的四种方法的有效性。在以前的工作中，SMHD 用 CN 和 BE 方法求解，SMHDLTRM 用 BE 方法求解。在这项研究中，获得了 SHDLTRM 的 CN 解决方案，并与其他解决方案进行了比较。所有计算均使用 FreeFem++ 进行。