East Asian J. Appl. Math., 15 (2025), pp. 392-414.
Published online: 2025-01
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An accurate, efficient and robust troubled-cell indicator (TCI) is always required by the discontinuous Galerkin methods for solving the nonlinear system of hyperbolic conservation laws. In this paper, we propose a new framework of TCI on general meshes using simple statistical analysis techniques which is easy to implement and efficient to run. In the framework troubled-cell indication is performed in each local stencil. We first pick the local stencils that contain the troubled cells via the range of the normalized troubled-cell indication values, and then detect the troubled cells in each picked stencil by a simple separation approach. The new TCI framework is implemented on several types of meshes with three indication variables. Numerical experiments conducted through simulating several classical shocked flows demonstrate that the new framework can capture the locations of shock waves accurately and lead to essentially nonoscillatory solutions. The results also verify the robustness of the framework.
}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2023-317.170324}, url = {http://global-sci.org/intro/article_detail/eajam/23755.html} }An accurate, efficient and robust troubled-cell indicator (TCI) is always required by the discontinuous Galerkin methods for solving the nonlinear system of hyperbolic conservation laws. In this paper, we propose a new framework of TCI on general meshes using simple statistical analysis techniques which is easy to implement and efficient to run. In the framework troubled-cell indication is performed in each local stencil. We first pick the local stencils that contain the troubled cells via the range of the normalized troubled-cell indication values, and then detect the troubled cells in each picked stencil by a simple separation approach. The new TCI framework is implemented on several types of meshes with three indication variables. Numerical experiments conducted through simulating several classical shocked flows demonstrate that the new framework can capture the locations of shock waves accurately and lead to essentially nonoscillatory solutions. The results also verify the robustness of the framework.