In this paper, we propose a novel structure-preserving finite element scheme for the three-dimensional incompressible magnetohydrodynamic (MHD) equations. The contribution of our research is three-fold. Firstly, the proposed scheme exactly preserves critical physical properties, including the incompressibility condition, the solenoidal condition of magnetic field, current density conservation, energy conservation and magnetic/fluid helicity conservation in their respective physical limits. To the best of our knowledge, this is the first numerical method that preserves all these properties simultaneously. Secondly, it introduces the first linear scheme that upholds the helicity-preserving property for MHD, thus eliminating the necessity for fixed-point iterations as seen in existing literature. Last but not least, for the resulting large linear systems, we develop efficient block preconditioners that remain robust at high fluid and magnetic Reynolds numbers by incorporating techniques such as the augmented Lagrangian method and mass augmentation. Finally, a series of numerical experiments demonstrate that our method is accurate, stable, robust under extreme physical parameters and capable of preserving all the stated physical properties, including benchmark problems of Orszag Tang vortex and driven magnetic reconnection with fluid and magnetic Reynolds numbers up to 10^5–10^6.
Publication:
JOURNAL OF COMPUTATIONAL PHYSICS
http://dx.doi.org/10.1016/j.jcp.2025.114130
Author:
Shipeng Mao
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
maosp@lsec.cc.ac.cn
Ruijie Xi
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
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