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Second Order Unconditionally Convergent Fully Discrete Scheme for Incompressible Vector Potential MHD System

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Abstract

In this paper, we present and analyze a second order unconditionally convergent mixed finite element method for solving incompressible magnetohydrodynamic problem based on magnetic vector potential. We utilize the second-order backward difference formula within the framework of the mixed finite element method and linearize the nonlinear terms by using the extrapolated technique. The finite element approximation is proposed with the fluid equations are discretized by Taylor–Hood type elements and the magnetic vector potential equation by edge elements. As a result, the divergence-free condition on the magnetic induction preserves exactly on the discrete level. Unconditional error estimates for the velocity and magnetic vector potential are established in the sense that no any restrictions are imposed on the relationship between the time-step size and the spatial size. Finally, several three-dimensional numerical simulations are carried out to illustrate the developed scheme, including convergence tests and some benchmark problems, such as the driven cavity problems and hydromagnetic Kelvin–Helmholtz instability

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Acknowledgements

We gratefully acknowledge the anonymous referees for their pertinent and perceptive comments which have significantly improved our paper. Q. Ding was supported by the Natural Science Foundation of China (12201353), Shandong Province Natural Science Foundation (ZR2021QA054) and the Talent Fund of Beijing Jiaotong University (2023XKRC024), X. Long was supported by the Natural Science Foundation of China (12301499), S. Mao was supported by the National Natural Science Foundation of China (Nos. 12271514, 11871467, 12161141017) and National Key Research and Development Program of China (2023YFC3705701).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

    Qianqian Ding

  2. College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450045, China

    Xiaonian Long

  3. NCMIS, LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Shipeng Mao & Ruijie Xi

  4. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100190, China

    Shipeng Mao & Ruijie Xi

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  1. Qianqian Ding
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  2. Xiaonian Long
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Correspondence to Shipeng Mao.

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Ding, Q., Long, X., Mao, S. et al. Second Order Unconditionally Convergent Fully Discrete Scheme for Incompressible Vector Potential MHD System. J Sci Comput 100, 1 (2024). https://doi.org/10.1007/s10915-024-02553-x

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