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

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Abstract

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.

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Acknowledgements

The authors thank Dr. Leo G. Rebholz for sharing his thoughts, and two anonymous reviewers for their comments and suggestions which greatly improved the manuscript. Alabama Supercomputer Authority (ASA) is acknowledged for generous allotment of computing time.

Funding

This work was supported by NSF grant DMS-2425308.

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, USA

    Muhammad Mohebujjaman

  2. Department of Mathematics and Physics, Texas A &M International University, Laredo, TX, 78041, USA

    Julian Miranda

  3. Department of Mathematics, Comilla University, Cumilla, 3506, Bangladesh

    Md. Abdullah Al Mahbub

  4. Department of Mathematics and Statistics, University of West Florida, Pensacosa, FL, 32514, USA

    Mengying Xiao

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Correspondence to Muhammad Mohebujjaman.

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This work is supported by NSF grant DMS-2425308. Texas A &M International University provided the logistic support.

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Muhammad Mohebujjaman was supported by the National Science Foundation (NSF) grant DMS-2425308, and Texas A &M International University. Julian Miranda was supported by the NSF grant DMS-2213274.

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Mohebujjaman, M., Miranda, J., Mahbub, M.A.A. et al. An Efficient and Accurate Penalty-projection Eddy Viscosity Algorithm for Stochastic Magnetohydrodynamic Flow Problems. J Sci Comput 101, 2 (2024). https://doi.org/10.1007/s10915-024-02633-y

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