Volume 14, Issue 5
High Order Efficient Algorithm for Computation of MHD Flow Ensembles

Adv. Appl. Math. Mech., 14 (2022), pp. 1111-1137.

Published online: 2022-06

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• 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 $θ ∈ [0,1]$ 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 has different right-hand-side vectors for all $J$ realizations. This 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.

• Keywords

Magnetohydrodynamics, uncertainty quantification, fast ensemble calculation, finite element method, Elsässer variables, second rder scheme.

65M12, 65M22, 65M60, 76W05

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@Article{AAMM-14-1111, author = {Muhammad and Mohebujjaman and and 23521 and and Muhammad Mohebujjaman}, title = {High Order Efficient Algorithm for Computation of MHD Flow Ensembles}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {5}, pages = {1111--1137}, 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 $θ ∈ [0,1]$ 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 has different right-hand-side vectors for all $J$ realizations. This 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0154}, url = {http://global-sci.org/intro/article_detail/aamm/20554.html} }
TY - JOUR T1 - High Order Efficient Algorithm for Computation of MHD Flow Ensembles AU - Mohebujjaman , Muhammad JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1111 EP - 1137 PY - 2022 DA - 2022/06 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0154 UR - https://global-sci.org/intro/article_detail/aamm/20554.html KW - Magnetohydrodynamics, uncertainty quantification, fast ensemble calculation, finite element method, Elsässer variables, second rder scheme. AB -

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 $θ ∈ [0,1]$ 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 has different right-hand-side vectors for all $J$ realizations. This 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.

Muhammad Mohebujjaman. (2022). High Order Efficient Algorithm for Computation of MHD Flow Ensembles. Advances in Applied Mathematics and Mechanics. 14 (5). 1111-1137. doi:10.4208/aamm.OA-2021-0154
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