We propose, analyze, and test a penalty projection-based efficient and accurate algorithm for the Uncertainty Quantification (UQ) of the time-dependent Magnetohydrodynamic (MHD) flow problems in convection-dominated regimes. The algorithm uses the Els\"asser variables formulation and discrete Hodge decomposition to decouple the stochastic MHD system into four sub-problems (at each time-step for each realization) which are much easier to solve than solving the coupled saddle point problems. Each of the sub-problems is designed in a sophisticated way so that at each time-step the system matrix remains the same for all the realizations but with different right-hand-side vectors which allows saving a huge amount of computer memory and computational time. Moreover, the scheme is equipped with ensemble eddy-viscosity and grad-div stabilization terms. The stability of the algorithm is proven rigorously. We prove that the proposed scheme converges to an equivalent non-projection-based coupled MHD scheme for large grad-div stabilization parameter values. We examine how Stochastic Collocation Methods (SCMs) can be combined with the proposed penalty projection UQ algorithm. Finally, a series of numerical experiments are given which verify the predicted convergence rates, show the algorithm's performance on benchmark channel flow over a rectangular step, and a regularized lid-driven cavity problem with high random Reynolds number and magnetic Reynolds number.

翻译：我们提出、分析并测试了一种基于罚投影的高效精确算法，用于对流主导区域中时间依赖的磁流体动力学（MHD）流动问题的不确定性量化（UQ）。该算法采用埃尔塞瑟变量公式和离散霍奇分解，将随机MHD系统解耦为四个子问题（在每一时间步的每次实现中），这些子问题比求解耦合的鞍点问题更易处理。每个子问题经过精心设计，使得在每一时间步中，系统矩阵对所有实现保持相同，仅右端向量不同，从而大幅节省计算机内存和计算时间。此外，该方案配备了系综涡粘度和梯度-散度稳定项。算法的稳定性得到了严格证明。我们证明了当梯度-散度稳定化参数较大时，所提方案收敛于等价的非投影耦合MHD方案。我们还研究了随机配置法（SCMs）如何与所提罚投影UQ算法相结合。最后，通过一系列数值实验验证了预期的收敛速率，展示了算法在矩形台阶上的基准通道流以及高随机雷诺数和磁雷诺数下的正则化顶盖驱动空腔问题中的性能。