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Second order unconditionally convergent and energy stable linearized scheme for MHD equations

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Abstract

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L 2 and H 1 fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.

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Acknowledgments

We gratefully acknowledge the anonymous referees for their pertinent and perceptive comments which have significantly improved our paper.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Guo-Dong Zhang & Chunjia Bi

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Jinjin Yang

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  1. Guo-Dong Zhang
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Correspondence to Guo-Dong Zhang.

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Communicated by: Long Chen

Guodong Zhang is supported by National Science Foundation of China (11601468); Chunjia Bi is supported by National Science Foundation of China (11571297) and Shandong Province Natural Science Foundation (ZR2014AM003).

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Zhang, GD., Yang, J. & Bi, C. Second order unconditionally convergent and energy stable linearized scheme for MHD equations. Adv Comput Math 44, 505–540 (2018). https://doi.org/10.1007/s10444-017-9552-x

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