Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes
The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010)”, the authors constructed high order discontinuous Galerkin methods for the shallow water...
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2014
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| Online Access: | http://hdl.handle.net/1721.1/85659 |
| _version_ | 1826206820484513792 |
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| author | Xing, Yulong Zhang, Xiangxiong |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Xing, Yulong Zhang, Xiangxiong |
| author_sort | Xing, Yulong |
| collection | MIT |
| description | The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010)”, the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions. |
| first_indexed | 2024-09-23T13:38:45Z |
| format | Article |
| id | mit-1721.1/85659 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:38:45Z |
| publishDate | 2014 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/856592022-10-01T16:14:03Z Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes Xing, Yulong Zhang, Xiangxiong Massachusetts Institute of Technology. Department of Mathematics Zhang, Xiangxiong Zhang, Xiangxiong The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010)”, the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions. 2014-03-14T20:11:50Z 2014-03-14T20:11:50Z 2013-03 2013-01 Article http://purl.org/eprint/type/JournalArticle 0885-7474 1573-7691 http://hdl.handle.net/1721.1/85659 Xing, Yulong, and Xiangxiong Zhang. “Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes.” Journal of Scientific Computing 57, no. 1 (October 2013): 19–41. en_US http://dx.doi.org/10.1007/s10915-013-9695-y Journal of Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Springer-Verlag Zhang |
| spellingShingle | Xing, Yulong Zhang, Xiangxiong Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes |
| title | Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes |
| title_full | Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes |
| title_fullStr | Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes |
| title_full_unstemmed | Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes |
| title_short | Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes |
| title_sort | positivity preserving well balanced discontinuous galerkin methods for the shallow water equations on unstructured triangular meshes |
| url | http://hdl.handle.net/1721.1/85659 |
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