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 Zhu , and Wenxiu Han , 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 - Hongqiang Zhu , AU - Wenxiu Han , AU - Haijin Wang , 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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