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Volume 31, Issue 1
A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

Min Zhang, Weizhang Huang & Jianxian Qiu

Commun. Comput. Phys., 31 (2022), pp. 94-130.

Published online: 2021-12

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  • Abstract

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

  • AMS Subject Headings

65M50, 65M60, 76B15, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-94, author = {Min Zhang , Weizhang Huang , and Qiu , Jianxian}, title = {A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {31}, number = {1}, pages = {94--130}, abstract = {

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0127}, url = {http://global-sci.org/intro/article_detail/cicp/20019.html} }
TY - JOUR T1 - A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations AU - Min Zhang , AU - Weizhang Huang , AU - Qiu , Jianxian JO - Communications in Computational Physics VL - 1 SP - 94 EP - 130 PY - 2021 DA - 2021/12 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0127 UR - https://global-sci.org/intro/article_detail/cicp/20019.html KW - Well-balance, positivity-preserving, high-order accuracy, quasi-Lagrange moving mesh, DG method, shallow water equations. AB -

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

Min Zhang, Weizhang Huang & Jianxian Qiu. (2021). A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations. Communications in Computational Physics. 31 (1). 94-130. doi:10.4208/cicp.OA-2021-0127
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