Volume 11, Issue 4
A New Robust High-Order Weighted Essentially Non-Oscillatory Scheme for Solving Well-Balanced Shallow Water Equations

Adv. Appl. Math. Mech., 11 (2019), pp. 911-941.

Published online: 2019-06

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• Abstract

A new simple and robust type of finite difference well-balanced weighted essentially non-oscillatory (WENO) schemes is designed for solving the one- and two-dimensional shallow water equations with or without source terms on structured meshes in this paper. Compared with the classical WENO schemes [5] in this field, the set of linear weights of these new WENO schemes could be chosen arbitrarily with one constraint that their summation equals one, maintain the optimal order of accuracy in smooth regions and keep essentially non-oscillatory property in non-smooth regions. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure for balancing the flux gradients and the source terms and then these new WENO schemes with any set of linear weights will satisfy the exact C-property for still stationary solutions and maintain the other advantages of other high-order WENO schemes at the same time. Some benchmark numerical examples are performed to obtain high-order accuracy in smooth regions, keep exact C-property, sustain good convergence property for some steady-state problems and show sharp shock transitions by such new type of finite difference WENO schemes.

• Keywords

Shallow water equation, high-order WENO scheme, well-balanced procedure, exact $C$-property, convergence property.

65M60, 35L65

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@Article{AAMM-11-911, author = {Wang , ZhenmingZhu , Jun and Zhao , Ning}, title = {A New Robust High-Order Weighted Essentially Non-Oscillatory Scheme for Solving Well-Balanced Shallow Water Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {4}, pages = {911--941}, abstract = {

A new simple and robust type of finite difference well-balanced weighted essentially non-oscillatory (WENO) schemes is designed for solving the one- and two-dimensional shallow water equations with or without source terms on structured meshes in this paper. Compared with the classical WENO schemes [5] in this field, the set of linear weights of these new WENO schemes could be chosen arbitrarily with one constraint that their summation equals one, maintain the optimal order of accuracy in smooth regions and keep essentially non-oscillatory property in non-smooth regions. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure for balancing the flux gradients and the source terms and then these new WENO schemes with any set of linear weights will satisfy the exact C-property for still stationary solutions and maintain the other advantages of other high-order WENO schemes at the same time. Some benchmark numerical examples are performed to obtain high-order accuracy in smooth regions, keep exact C-property, sustain good convergence property for some steady-state problems and show sharp shock transitions by such new type of finite difference WENO schemes.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0184}, url = {http://global-sci.org/intro/article_detail/aamm/13194.html} }
TY - JOUR T1 - A New Robust High-Order Weighted Essentially Non-Oscillatory Scheme for Solving Well-Balanced Shallow Water Equations AU - Wang , Zhenming AU - Zhu , Jun AU - Zhao , Ning JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 911 EP - 941 PY - 2019 DA - 2019/06 SN - 11 DO - http://doi.org/10.4208/aamm.OA-2018-0184 UR - https://global-sci.org/intro/article_detail/aamm/13194.html KW - Shallow water equation, high-order WENO scheme, well-balanced procedure, exact $C$-property, convergence property. AB -

A new simple and robust type of finite difference well-balanced weighted essentially non-oscillatory (WENO) schemes is designed for solving the one- and two-dimensional shallow water equations with or without source terms on structured meshes in this paper. Compared with the classical WENO schemes [5] in this field, the set of linear weights of these new WENO schemes could be chosen arbitrarily with one constraint that their summation equals one, maintain the optimal order of accuracy in smooth regions and keep essentially non-oscillatory property in non-smooth regions. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure for balancing the flux gradients and the source terms and then these new WENO schemes with any set of linear weights will satisfy the exact C-property for still stationary solutions and maintain the other advantages of other high-order WENO schemes at the same time. Some benchmark numerical examples are performed to obtain high-order accuracy in smooth regions, keep exact C-property, sustain good convergence property for some steady-state problems and show sharp shock transitions by such new type of finite difference WENO schemes.

Zhenming Wang, Jun Zhu & Ning Zhao. (2019). A New Robust High-Order Weighted Essentially Non-Oscillatory Scheme for Solving Well-Balanced Shallow Water Equations. Advances in Applied Mathematics and Mechanics. 11 (4). 911-941. doi:10.4208/aamm.OA-2018-0184
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