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High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

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Abstract

In this paper, we develop a high-order semi-implicit (SI) structure-preserving finite difference weighted essentially nonoscillatory (WENO) scheme for magnetohydrodynamic (MHD) equations with a gravitational source. The proposed scheme is well-balanced for magnetic steady states, divergence-free for the magnetic field, conservative in the high Mach regime, and exhibits asymptotic preserving (AP) and asymptotically accurate (AA) properties in the incompressible low sonic Mach regime. The constrained transport method is applied to maintain a discrete divergence-free magnetic field. The sonic Mach number \(\varepsilon \) ranging from 0 to \(\mathcal {O}(1)\) is taken into account for all Mach flows. One of the crucial and novel ingredients is the addition of an evolution equation for the perturbation of potential temperature as an auxiliary equation to the conservative MHD system. This addition ensures a correct asymptotic low sonic Mach limit and helps to effectively capture shocks in the compressible high Mach regime. A well-balanced finite difference WENO scheme is designed for conservative variables of the resulting system. With stiffly accurate SI implicit-explicit Runge–Kutta time discretizations, the AP and AA properties are formally proven. Numerical experiments are provided to validate the effectiveness and structure-preserving properties of the proposed scheme.

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Code availability

The codes generated in this study are available from the corresponding author on reasonable request.

Data availability

No data sets were generated or analyzed during the current study.

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Funding

The first and the third authors are partially supported by National Key R & D Program of China No. 2022YFA1004500, NSFC grant No. 11971025, NSF of Fujian Province No. 2023J02003, and the Strategic Priority Research Program of Chinese Academy of Sciences Grant No. XDA25010401. The work of Kailiang Wu is partially supported by National Natural Science Foundation of China (Grant No. 12171227) and Shenzhen Science and Technology Program (Grant No. RCJC20221008092757098).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Wei Chen

  2. Department of Mathematics & SUSTech International Center for Mathematics, Southern University of Science and Technology, National Center for Applied Mathematics Shenzhen (NCAMS), Shenzhen, 518055, Guangdong, China

    Kailiang Wu

  3. School of Mathematical Sciences, Xiamen University, Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen, 361005, Fujian, People’s Republic of China

    Tao Xiong

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  1. Wei Chen
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Wei Chen: Conceptualization, Formal analysis, Investigation, Methodology, Writing. Kailiang Wu: Conceptualization, Formal analysis, Methodology, Writing. Tao Xiong: Conceptualization, Formal analysis, Methodology, Writing, Funding acquisition.

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Chen, W., Wu, K. & Xiong, T. High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers. J Sci Comput 99, 36 (2024). https://doi.org/10.1007/s10915-024-02492-7

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