An Efficient and Accurate Penalty-projection Eddy Viscosity Algorithm for Stochastic Magnetohydrodynamic Flow Problems

We propose, analyze, and test a penalty projection-based robust 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 Elsässer 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 (EEV) and grad-div stabilization terms. The unconditional stability with respect to the time-step size of the algorithm is proven rigorously. We prove 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, a regularized lid-driven cavity problem with high random Reynolds number and high random magnetic Reynolds number, and the impact of the EEV stabilization in the MHD UQ algorithm.