Volume 18, Issue 2
Numerical Simulations for Shallow Water Flows over Erodible Beds by Central DG Methods

Weizhi Xian, Aimin Chen, Yongping Cheng & Haiyun Dong

Int. J. Numer. Anal. Mod., 18 (2021), pp. 143-164.

Published online: 2021-03

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  • Abstract

In this paper, we investigate the shallow water flows over erodible beds by using a fully coupled mathematical model in two-dimensional space. This model combines the nonlinear shallow water equations, the sediment transport equation and the bed evolution equation. The purpose of this paper is to design a well-balanced and positivity-preserving scheme for this model. In order to achieve the well-balanced property, the coupled system is first reformulated as a new form by introducing an auxiliary variable. The central discontinuous Galerkin method is applied to discretize the model. By choosing the value of the auxiliary variable suitably, the scheme can exactly balance the flux gradients and source terms in the "still-water" case, and thus the well-balanced property of the proposed scheme can be proved. Moreover, the non-negativity of the volumetric sediment concentration in the sediment transport equation is maintained by choosing a suitable time step and using a positivity-preserving limiter. Numerical tests are presented to illustrate the validity of the proposed scheme.

  • Keywords

Shallow water equation, sediment transport equation, bed evolution, central discontinuous Galerkin method, well-balanced and positivity-preserving scheme.

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@Article{IJNAM-18-143, author = {Xian , Weizhi and Chen , Aimin and Cheng , Yongping and Dong , Haiyun}, title = {Numerical Simulations for Shallow Water Flows over Erodible Beds by Central DG Methods}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {2}, pages = {143--164}, abstract = {

In this paper, we investigate the shallow water flows over erodible beds by using a fully coupled mathematical model in two-dimensional space. This model combines the nonlinear shallow water equations, the sediment transport equation and the bed evolution equation. The purpose of this paper is to design a well-balanced and positivity-preserving scheme for this model. In order to achieve the well-balanced property, the coupled system is first reformulated as a new form by introducing an auxiliary variable. The central discontinuous Galerkin method is applied to discretize the model. By choosing the value of the auxiliary variable suitably, the scheme can exactly balance the flux gradients and source terms in the "still-water" case, and thus the well-balanced property of the proposed scheme can be proved. Moreover, the non-negativity of the volumetric sediment concentration in the sediment transport equation is maintained by choosing a suitable time step and using a positivity-preserving limiter. Numerical tests are presented to illustrate the validity of the proposed scheme.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/18712.html} }
TY - JOUR T1 - Numerical Simulations for Shallow Water Flows over Erodible Beds by Central DG Methods AU - Xian , Weizhi AU - Chen , Aimin AU - Cheng , Yongping AU - Dong , Haiyun JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 143 EP - 164 PY - 2021 DA - 2021/03 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/18712.html KW - Shallow water equation, sediment transport equation, bed evolution, central discontinuous Galerkin method, well-balanced and positivity-preserving scheme. AB -

In this paper, we investigate the shallow water flows over erodible beds by using a fully coupled mathematical model in two-dimensional space. This model combines the nonlinear shallow water equations, the sediment transport equation and the bed evolution equation. The purpose of this paper is to design a well-balanced and positivity-preserving scheme for this model. In order to achieve the well-balanced property, the coupled system is first reformulated as a new form by introducing an auxiliary variable. The central discontinuous Galerkin method is applied to discretize the model. By choosing the value of the auxiliary variable suitably, the scheme can exactly balance the flux gradients and source terms in the "still-water" case, and thus the well-balanced property of the proposed scheme can be proved. Moreover, the non-negativity of the volumetric sediment concentration in the sediment transport equation is maintained by choosing a suitable time step and using a positivity-preserving limiter. Numerical tests are presented to illustrate the validity of the proposed scheme.

Weizhi Xian, Aimin Chen, Yongping Cheng & Haiyun Dong. (2021). Numerical Simulations for Shallow Water Flows over Erodible Beds by Central DG Methods. International Journal of Numerical Analysis and Modeling. 18 (2). 143-164. doi:
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