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A Decoupled, Linear and Unconditionally Energy Stable Scheme with Finite Element Discretizations for Magneto-Hydrodynamic Equations

Abstract

In this paper, we consider numerical approximations for solving the nonlinear magnetohydrodynamical system, that couples the Navier–Stokes equations and Maxwell equations together. By combining the projection method and some subtle implicit–explicit treatments for nonlinear coupling terms, we develop a fully decoupled, linear and unconditionally energy stable scheme for solving this system, where a new auxiliary velocity field is specifically introduced in order to decouple the computations of the magnetic field from the velocity field. We further prove that the fully discrete scheme with finite element approximations is unconditional energy stable. By deriving the \(L^{\infty }\) bound of the numerical solution and the relation between the new auxiliary velocity field and the velocity field, and using negative norm technique, we obtain the optimal error estimates rigorously. Various numerical experiments are implemented to demonstrate the stability and the accuracy in simulating some benchmark problems, including the Kelvin–Helmholtz shear instability and the magnetic-frozen phenomenon in the lid-driven cavity.

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Funding

G. D. Zhang was supported by National Science Foundation of China under grant numbers 11601468 and 11771375 and Shandong Province Natural Science Foundation (ZR2018MA008). X. He was partially supported by the U.S. National Science Foundation under Grant NO. DMS-1818642. X. Yang was partially supported by the U.S. National Science Foundation under Grant Nos. DMS-1720212 and DMS-1818783

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Correspondence to XiaoFeng Yang.

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Zhang, GD., He, X. & Yang, X. A Decoupled, Linear and Unconditionally Energy Stable Scheme with Finite Element Discretizations for Magneto-Hydrodynamic Equations. J Sci Comput 81, 1678–1711 (2019). https://doi.org/10.1007/s10915-019-01059-1

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  • DOI: https://doi.org/10.1007/s10915-019-01059-1

Keywords

  • Magneto-hydrodynamics
  • Linear
  • Decoupled
  • Unconditional energy stability
  • First order
  • Error estimates

Mathematics Subject Classification

  • 35Q30
  • 65M12
  • 65M60