Volume 22, Issue 4
High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

Zhen Gao & Guanghui Hu

Commun. Comput. Phys., 22 (2017), pp. 1049-1068.

Published online: 2017-10

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  • Abstract

In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

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@Article{CiCP-22-1049, author = {}, title = {High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations}, journal = {Communications in Computational Physics}, year = {2017}, volume = {22}, number = {4}, pages = {1049--1068}, abstract = {

In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0200}, url = {http://global-sci.org/intro/article_detail/cicp/9993.html} }
TY - JOUR T1 - High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations JO - Communications in Computational Physics VL - 4 SP - 1049 EP - 1068 PY - 2017 DA - 2017/10 SN - 22 DO - http://doi.org/10.4208/cicp.OA-2016-0200 UR - https://global-sci.org/intro/article_detail/cicp/9993.html KW - AB -

In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

Zhen Gao & Guanghui Hu. (2020). High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations. Communications in Computational Physics. 22 (4). 1049-1068. doi:10.4208/cicp.OA-2016-0200
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