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A New S-M Limiter Entropy Stable Scheme Based on Moving Mesh Method for Ideal MHD and SWMHD Equations

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Abstract

For magnetohydrodynamics (MHD) equations, the existing entropy stable (ES) flux effectively eliminates the spurious oscillation, but it still has potential for improvement such as the resolution of discontinuous solutions. We propose a high-resolution scheme based on a moving mesh strategy for approximating solutions of ideal MHD and shallow water MHD (SWMHD) in this paper. Firstly, a new S-M flux limiter is constructed to weigh the ES flux and the anti-diffusive flux to obtain a new flux with high resolution and entropy stability properties. By adjusting the amplitude of anti-diffusive flux adaptively, the new flux can reduce its dissipation in smooth regions and increase dissipation at discontinuities, which is more consistent with physical laws. Secondly, we introduce the moving mesh strategy to optimize mesh generation and eliminate the defects of structured mesh. A new monitor function is defined to identify structural features of solutions at each particular time level and to assign appropriate weights to all regions with large numerical solution gradients, to construct a mesh evolution equation and increase the mesh density of these regions. Finally, we combine the moving mesh strategy with the high-resolution ES scheme and evaluate more accurate solutions according to the order of “mesh redistribution-update of solution-equation solving on new mesh”. Numerical results show that the new algorithm can achieve strong robustness and high resolution, and can track various waves effectively (especially the shock and rarefaction waves).

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Data Availability

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

All authors are grateful to Professor Yulong Xing of Ohio State University for his useful comments and helpful suggestions.

Funding

This work is supported by the National Natural Science Foundation of China (Grant Number [11971075]).

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Authors and Affiliations

  1. College of Science, Chang’an University, Xi’an, 710064, Shaanxi, China

    Mengqing Zhai, Supei Zheng, Chengzhi Zhang & Mangmang Jian

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Contributions

Mengqing Zhai derived the scheme, did the numerical experiments, wrote the main manuscript text, and prepared all the figures. Supei Zheng provided the original data and programs, and reviewed and edited the manuscript. Chengzhi Zhang did several two-dimensional experiments and provided numerical results. All authors reviewed the manuscript.

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Correspondence to Supei Zheng.

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Zhai, M., Zheng, S., Zhang, C. et al. A New S-M Limiter Entropy Stable Scheme Based on Moving Mesh Method for Ideal MHD and SWMHD Equations. J Sci Comput 98, 68 (2024). https://doi.org/10.1007/s10915-024-02458-9

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  • DOI: https://doi.org/10.1007/s10915-024-02458-9

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