High Order Efficient Algorithm for Computation of MHD Flow Ensembles (Advances in Applied Mathematics and Mechanics, 2022)
In this paper, we propose, analyze, and test a new fully discrete, efficient, decoupled, stable, and practically second-order time-stepping algorithm for computingMHD ensemble flow averages under uncertainties in the initial conditions and forcing. For each viscosity and magnetic diffusivity pair, the algorithm picks the largestpossible parameter $θ ∈ [0,1]$ to avoid the instability that arises due to the presence ofsome explicit viscous terms. At each time step, the algorithm shares the same systemmatrix with all $J$ realizations but has different right-hand-side vectors for all $J$ realizations. This saves assembling time and computer memory, allows the reuse of the samepreconditioner, and can take the advantage of block linear solvers. For the proposedalgorithm, we prove stability and convergence rigorously. To illustrate the predictedconvergence rates of our analysis, numerical experiments with manufactured solutionsare given on a unit square domain. Finally, we test the scheme on a benchmark channelflow over a step problem and it performs well.