High order efficient algorithm for computation of MHD flow ensembles

TL;DR

This paper introduces a new efficient, stable, and second-order time-stepping algorithm for computing magnetohydrodynamic (MHD) flow ensembles under uncertainty, optimizing computational resources and ensuring rigorous stability and convergence.

Contribution

The paper presents a novel decoupled, fully discrete algorithm for MHD ensemble flow computation that is both efficient and stable, with proven convergence and practical testing.

Findings

Algorithm achieves second-order accuracy in time.

Shared system matrix reduces computational cost.

Performs well on benchmark MHD flow problems.

Abstract

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.