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A Characteristic-Featured Shock Wave Indicator for Conservation Laws Based on Training an Artificial Neuron

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Abstract

In this work, we use exact solutions of one-dimensional Burgers equation to train an artificial neuron as a shock wave detector. The expression of the artificial neuron detector is then modified into a practical form to reflect admissible jump of eigenvalues. We show the working mechanism of the practical form is consistent with compressing or intersecting of characteristic curves. In addition, we prove there is indeed a discontinuity inside the cell detected by the practical form, and smooth extrema and large gradient regions are never marked. As a result, we apply the practical form to numerical schemes as a shock wave indicator with its easy extension to multi-dimensional conservation laws. Numerical results are present to demonstrate the robustness of the present indicator under Runge–Kutta Discontinuous Galerkin framework, its performance is generally compared to TVB-based indicators more efficiently and accurately. To treat the initial inadmissible jumps, including linear contact discontinuities and those evolving into rarefaction waves, a preliminary strategy of combining a traditional indicator in the beginning with the present indicator is suggested. We believe the present indicator can be applied to unstructured mesh in the future.

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Acknowledgements

The authors would like to acknowledge the reviewers for the valuable comments and suggestions. This work is supported by National Numerical Wind Tunnel Project, National Nature Science Foundation of China (Nos. U1730118 and 91530325), Fundamental Research of Civil Aircraft (No. MJ-F-2012-04).

Funding

This work was funded by National Numerical Wind Tunnel Project, National Nature Science Foundation of China (Nos. U1730118 and 91530325), Fundamental Research of Civil Aircraft (No. MJ-F-2012-04).

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Correspondence to Tiegang Liu.

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Submitted to the editors on May 9 2019.

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Feng, Y., Liu, T. & Wang, K. A Characteristic-Featured Shock Wave Indicator for Conservation Laws Based on Training an Artificial Neuron. J Sci Comput 83, 21 (2020). https://doi.org/10.1007/s10915-020-01200-5

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  • DOI: https://doi.org/10.1007/s10915-020-01200-5

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