In this paper, we investigate the fractional magnetohydrodynamic (MHD) flow model of a generalized second-grade fluid through a porous medium with Hall current. The fully discrete numerical scheme for solving the model is developed using the formula and Legendre spectral method in time and space, respectively. On the basis of sum-of-exponentials (SOE) technique, a fast formula for the Caputo fractional derivative is employed to reduce the computational and storage cost. The stability and convergence in -norm of the fast spectral scheme are proved relatively simply, with the accuracy order , where is the time step, is the polynomial degree and is the SOE tolerance error. Besides, we also consider the nonuniform meshes in time to effectively deal with the weak singularity of the solution at the initial time. Finally, theoretical analysis and the effectiveness of the fast algorithm are verified by numerical examples, the effect of various physical parameters on the fluid flow velocity is analyzed.

中文翻译：

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广义二级流体MHD流动模型的快速求解方法及收敛性分析


在本文中，我们研究了广义二级流体通过霍尔电流多孔介质的分数磁流体动力学（MHD）流动模型。分别使用时间和空间上的公式和勒让德谱方法开发了求解模型的全离散数值方案。在指数和（SOE）技术的基础上，采用Caputo分数阶导数的快速公式来降低计算和存储成本。快速谱格式的稳定性和范数收敛性证明相对简单，精度阶为 ，其中 为时间步长， 为多项式次数， 为 SOE 容差误差。此外，我们还及时考虑了非均匀网格，以有效处理初始时解的弱奇异性。最后通过数值算例验证了理论分析和快速算法的有效性，分析了各种物理参数对流体流速的影响。