Second order ensemble simulation for MHD flow in Elsässer variable with noisy input data
This work provides an efficient numerical method for uncertainty quantification in MHD flow simulations, which is important for applications like fusion energy and astrophysics, but the approach is an incremental extension of existing ensemble methods to MHD flows.
The paper develops a second-order accurate algorithm for ensemble simulation of MHD flows with uncertain initial conditions, using Elsässer variables and a shared coefficient matrix to reduce computational cost. Numerical experiments verify predicted convergence rates and demonstrate good performance on a benchmark channel flow over a step.
We propose, analyze and test a fully discrete, efficient second-order algorithm for computing flow ensembles average of viscous, incompressible, and time-dependent magnetohydrodynamic (MHD) flows under uncertainties in initial conditions. The scheme is decoupled and based on Elsässer variable formulation. The algorithm uses the breakthrough idea of Jiang and Layton, 2014 to approximate the ensemble average of $J$ realizations. That is, at each time step, each of the $J$ realization shares the same coefficient matrix for different right-hand side matrices. Thus, storage requirements and computational time are reduced by building preconditioners once per time step and reuse them. We prove stability and optimal convergence with respect to the time step restriction. On some manufactured solutions, numerical experiments are given to verify the predicted convergence rates of our analysis. Finally, we test the scheme on a benchmark channel flow over a step and it performs well.