Volume 28, Issue 4
A High Order Central DG Method of the Two-Layer Shallow Water Equations

Yongping Cheng, Haiyun Dong, Maojun LiWeizhi Xian

Commun. Comput. Phys., 28 (2020), pp. 1437-1463.

Published online: 2020-08

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  • Abstract

In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer shallow water equations, it is not a trivial work due to the complexity of the equations. To achieve the well-balanced property of the numerical scheme easily, the two-layer shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values of the auxiliary variables suitably, we can prove that the scheme can exactly preserve the still-water solution, and thus it is a truly well-balanced scheme. To ensure the non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the numerical method will be illustrated through some numerical tests.

  • Keywords

Two-layer shallow water equations, central DG method, positivity-preserving and well-balanced, still-water solution.

  • AMS Subject Headings

65M60, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1437, author = {Cheng , Yongping and Dong , Haiyun and Li , Maojun and Xian , Weizhi}, title = {A High Order Central DG Method of the Two-Layer Shallow Water Equations}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {4}, pages = {1437--1463}, abstract = {

In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer shallow water equations, it is not a trivial work due to the complexity of the equations. To achieve the well-balanced property of the numerical scheme easily, the two-layer shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values of the auxiliary variables suitably, we can prove that the scheme can exactly preserve the still-water solution, and thus it is a truly well-balanced scheme. To ensure the non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the numerical method will be illustrated through some numerical tests.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0155}, url = {http://global-sci.org/intro/article_detail/cicp/18107.html} }
TY - JOUR T1 - A High Order Central DG Method of the Two-Layer Shallow Water Equations AU - Cheng , Yongping AU - Dong , Haiyun AU - Li , Maojun AU - Xian , Weizhi JO - Communications in Computational Physics VL - 4 SP - 1437 EP - 1463 PY - 2020 DA - 2020/08 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0155 UR - https://global-sci.org/intro/article_detail/cicp/18107.html KW - Two-layer shallow water equations, central DG method, positivity-preserving and well-balanced, still-water solution. AB -

In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer shallow water equations, it is not a trivial work due to the complexity of the equations. To achieve the well-balanced property of the numerical scheme easily, the two-layer shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values of the auxiliary variables suitably, we can prove that the scheme can exactly preserve the still-water solution, and thus it is a truly well-balanced scheme. To ensure the non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the numerical method will be illustrated through some numerical tests.

Yongping Cheng, Haiyun Dong, Maojun Li & Weizhi Xian. (2020). A High Order Central DG Method of the Two-Layer Shallow Water Equations. Communications in Computational Physics. 28 (4). 1437-1463. doi:10.4208/cicp.OA-2019-0155
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