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A stabilized Gauge-Uzawa discontinuous Galerkin method for the magneto-hydrodynamic equations

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Abstract

In this paper, we develop a stabilized Gauge-Uzawa discontinuous Galerkin (DG) method with second-order time-accurate for the incompressible magnetohydrodynamic equations. The proposed scheme here possess the properties of the linearity, fully decoupling and unconditional energy stability, which is employed by combining a second-order projection-type Gauge-Uzawa method for the Navier–Stokes equations, a stabilization strategy for Maxwell’s equations, the implicit-explicit frameworks for the nonlinear coupling terms and the interior penalty DG method for the spatial approximation. We rigorously prove the unique solvability, unconditional energy stability and optimal error estimates of the proposed scheme. Finally, several numerical examples are provided to demonstrate the accuracy, stability, and efficiency of the proposed scheme.

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Acknowledgements

The authors are grateful to the reviewers for the constructive comments and valuable suggestions which have improved the paper. Guang-an Zou is supported by the Foundation for University Youth Key Teacher of Henan Province of China (2023GGJS017) and the Key Scientific Research Projects of Colleges and Universities in Henan Province, China (23A110006). Xiaofeng Yang is partially supported by the U.S. National Science Foundation under the grant number DMS-2012490 and DMS-2309731.

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Authors and Affiliations

  1. Henan Key Laboratory of Earth System Observation and Modeling, Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Guang-an Zou

  2. Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou, 450046, China

    Guang-an Zou

  3. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Yuanhong Wei

  4. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Yuanhong Wei

  5. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

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Zou, Ga., Wei, Y. & Yang, X. A stabilized Gauge-Uzawa discontinuous Galerkin method for the magneto-hydrodynamic equations. Bit Numer Math 64, 40 (2024). https://doi.org/10.1007/s10543-024-01044-7

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