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Volume 12, Issue 5
A Generalization of a Troubled-Cell Indicator to $h$-Adaptive Meshes for Discontinuous Galerkin Methods

Hongqiang Zhu, Wenxiu Han & Haijin Wang

Adv. Appl. Math. Mech., 12 (2020), pp. 1224-1246.

Published online: 2020-07

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  • Abstract

We generalize the troubled-cell indicator on unstructured triangular meshes recently introduced by Fu and Shu (J. Comput. Phys., 347 (2017), pp. 305--327) to $h$-adaptive rectangular meshes where hanging nodes exist. The generalized troubled-cell indicator keeps the good properties of simplicity, compactness and insensitivity to particular test cases. Numerical tests on the two-dimensional scalar Burgers' equation and hyperbolic systems of Euler equations demonstrate the good performance of the generalized indicator. The results on both uniform and $h$-adaptive meshes indicate that the generalized indicator is able to capture shocks effectively without any PDE-sensitive parameter to tune.

  • Keywords

Troubled-cell indicator, discontinuous Galerkin method, adaptive mesh, conservation law.

  • AMS Subject Headings

65M60, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-12-1224, author = {Hongqiang and Zhu and and 8334 and and Hongqiang Zhu and Wenxiu and Han and and 8335 and and Wenxiu Han and Haijin and Wang and and 8336 and and Haijin Wang}, title = {A Generalization of a Troubled-Cell Indicator to $h$-Adaptive Meshes for Discontinuous Galerkin Methods}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {5}, pages = {1224--1246}, abstract = {

We generalize the troubled-cell indicator on unstructured triangular meshes recently introduced by Fu and Shu (J. Comput. Phys., 347 (2017), pp. 305--327) to $h$-adaptive rectangular meshes where hanging nodes exist. The generalized troubled-cell indicator keeps the good properties of simplicity, compactness and insensitivity to particular test cases. Numerical tests on the two-dimensional scalar Burgers' equation and hyperbolic systems of Euler equations demonstrate the good performance of the generalized indicator. The results on both uniform and $h$-adaptive meshes indicate that the generalized indicator is able to capture shocks effectively without any PDE-sensitive parameter to tune.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0149}, url = {http://global-sci.org/intro/article_detail/aamm/17746.html} }
TY - JOUR T1 - A Generalization of a Troubled-Cell Indicator to $h$-Adaptive Meshes for Discontinuous Galerkin Methods AU - Zhu , Hongqiang AU - Han , Wenxiu AU - Wang , Haijin JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1224 EP - 1246 PY - 2020 DA - 2020/07 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0149 UR - https://global-sci.org/intro/article_detail/aamm/17746.html KW - Troubled-cell indicator, discontinuous Galerkin method, adaptive mesh, conservation law. AB -

We generalize the troubled-cell indicator on unstructured triangular meshes recently introduced by Fu and Shu (J. Comput. Phys., 347 (2017), pp. 305--327) to $h$-adaptive rectangular meshes where hanging nodes exist. The generalized troubled-cell indicator keeps the good properties of simplicity, compactness and insensitivity to particular test cases. Numerical tests on the two-dimensional scalar Burgers' equation and hyperbolic systems of Euler equations demonstrate the good performance of the generalized indicator. The results on both uniform and $h$-adaptive meshes indicate that the generalized indicator is able to capture shocks effectively without any PDE-sensitive parameter to tune.

Hongqiang Zhu, Wenxiu Han & Haijin Wang. (2020). A Generalization of a Troubled-Cell Indicator to $h$-Adaptive Meshes for Discontinuous Galerkin Methods. Advances in Applied Mathematics and Mechanics. 12 (5). 1224-1246. doi:10.4208/aamm.OA-2019-0149
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