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A simple a posteriori indicator for discontinuous Galerkin method on unstructured grids

非结构网格间断Galerkin方法的一种简单有效后验激波指示器

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Abstract

A simple and efficient troubled-cell indicator based on a posteriori limiting paradigm is proposed for the discontinuous Galerkin (DG) method on the triangular grids. The developed methodology utilizes discrete solution from different time levels in the von Neumann neighborhood to maintain the compactness of the DG schemes. Effective technique is suggested to provide further information about the troubled cells. Different limitation can then be applied to the resulting troubled cells. Favorable numerical characteristic including positivity-preserving and oscillation-suppressing can be achieved. The present indicator has been implemented with both the simple limiter such as the total variation bounded (TVB) limiter and more sophisticated limiter such as the Hermite weighted essentially non-oscillatory (HWENO) limiter. The resulting limiting strategy, compared with the minmod based TVB indicator, has been examined for DG schemes of up to fourth order of accuracy in solving the two dimensional Euler equations on the unstructured grids. Numerical results demonstrate the effectiveness and robustness of the current a posteriori indication method.

摘要

针对间断Galerkin (DG)方法的激波捕捉问题, 在三角形网格上提出了一种基于后验方法的简单有效的问题单元指示器. 方法利用了von Neumann单元中不同时刻的离散解来保持DG格式的紧凑性. 本文采用有效的技术来提供有关问题单元的进一步信息, 并且可以对不同的问题单元应用不同的限制方法, 因此可以获得良好的数值特性, 包括保正性质和振荡抑制性质. 本文采用了TVB限制器和Hermite WENO限制器, 对比了提出的指示器与TVB指示器的性能. 数值结果表明了当前后验激波指示器的有效性和鲁棒性.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11702015 and 11721202). The authors acknowledge the support by the high performance computing (HPC) resources at Beihang University. The first author also acknowledges the support by the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. College of Aeronautics Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Zhen-Hua Jiang  (姜振华), Chao Yan  (闫超) & Jian Yu  (于剑)

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  1. Zhen-Hua Jiang  (姜振华)
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  2. Chao Yan  (闫超)
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  3. Jian Yu  (于剑)
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Contributions

Zhen-Hua Jiang and Chao Yan designed the research. Zhen-Hua Jiang wrote the first draft of the manuscript. Zhen-Hua Jiang and Jian Yu set up the experiment. Zhen-Hua Jiang processed the experiment data. Jian Yu helped organize the manuscript. Chao Yan and Zhen-Hua Jiang revised and edited the final version.

Corresponding author

Correspondence to Chao Yan  (闫超).

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Jiang, ZH., Yan, C. & Yu, J. A simple a posteriori indicator for discontinuous Galerkin method on unstructured grids. Acta Mech. Sin. 39, 322296 (2023). https://doi.org/10.1007/s10409-022-22296-x

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