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 method 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. This decoupling method greatly simplifies the computation of the whole MHD system. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Optimal error estimates are provided for the proposed numerical scheme, with second-order temporal convergence and $\mathcal{O} (h^{r+1})$ spatial convergence, where $r$ is the degree of the polynomial functions. Numerical examples are provided to illustrate the theoretical results.
翻译:在本文中,我们提出并分析磁流动力(MHD)方程式的第二顺序准确、完全离散的有限元素方法。 使用修改的crank- Nicolson 法将模型分解, 并对流体对流术语和两个组合条件适用适当的半隐含处理。 这些半隐含近似差导致一个具有可变系数的线性系统, 可以从理论上证明独特的溶解性。 此外, 我们在Stokes 解析器中使用一种脱钩方法, 该方法根据前一个时间水平的压力梯度计算中间速度字段, 并通过中间速度场的Helmholtz 分解法强制实施不易压缩限制。 这种分解方法大大简化了整个MHD系统的计算。 这个方案的能源稳定性在理论上得到证明, 需要详细分析脱钩的Stokes 溶解解码解码的Stoks 解析器。 为拟议的数字方案提供了最佳误差估计数, 其次阶的时序时间趋接近值为 美元 和 数级 数 = 数级 = 数级的理论级 。 = = yalalalalal= exl= = yalalalalalalalalalalal exollation exlate yal exlate exlationsal $=== $= expal ypal = $= = = = $= ypoll= = $=