In this paper, we propose, analyze, and test a new fully discrete, efficient, decoupled, stable, and practically second-order time-stepping algorithm for computing MHD ensemble flow averages under uncertainties in the initial conditions and forcing. For each viscosity and magnetic diffusivity pair, the algorithm picks the largest possible parameter $\theta\in[0,1]$ to avoid the instability that arises due to the presence of some explicit viscous terms. At each time step, the algorithm shares the same system matrix with all $J$ realizations but with different right-hand-side vectors. That saves assembling time and computer memory, allows the reuse of the same preconditioner, and can take the advantage of block linear solvers. For the proposed algorithm, we prove stability and convergence rigorously. To illustrate the predicted convergence rates of our analysis, numerical experiments with manufactured solutions are given on a unit square domain. Finally, we test the scheme on a benchmark channel flow over a step problem and it performs well.
翻译:在本文中,我们提议、分析并测试一种新的完全独立的、高效的、脱钩的、稳定的、实际的二级时间步算法,用于在初始条件和强迫的不确定性下计算MHD混合流平均值。对于每种粘度和磁异性对配方,算法选择了最大可能的参数$\theta\in[0,1]美元,以避免由于存在某些明显的粘度条件而出现不稳定。在每一个步骤中,算法都与所有J美元的实现额共享相同的系统矩阵,但与不同的右侧矢量共享。这节省了时间和计算机记忆的集合,允许重新使用相同的先决条件,并能够利用块线性解算法的优势。对于拟议的算法,我们证明稳定性和趋同性。为了说明我们分析的预测趋同率,用制造的解决方案进行的数字实验是在一个单位正方域上进行。最后,我们用一个基准通道对一个步骤问题进行测试,并且运行良好。