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Volume 16, Issue 5
A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations

Liang Li, Zhenming Wang & Jun Zhu

Adv. Appl. Math. Mech., 16 (2024), pp. 1176-1196.

Published online: 2024-07

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  • Abstract

In this paper, we propose a newly designed fifth-order finite difference well-balanced mapped unequal-sized weighted essentially non-oscillatory (WBMUS-WENO) scheme for simulating the shallow water systems on multi-dimensional structured meshes. We design new non-linear weights and a new mapping function, so that the WBMUS-WENO scheme can maintain fifth-order accuracy with a small $ε$ even nearby the extreme points in smooth regions. The truncation errors of the scheme is smaller and it has better convergence in simulating some steady-state problems. Unlike the traditional well-balanced WENO-XS scheme [29], this new WBMUS-WENO scheme uses three unequal-sized stencils, denotes the linear weights to be any positive numbers on condition that their summation is one. By incorporating a quartic polynomial on the whole big stencil into WENO reconstruction, the WBMUS-WENO scheme is simple and efficient. Extensive examples are performed to testify the exact C-property, absolute convergence property, and good representations of this new WBMUS-WENO scheme.

  • AMS Subject Headings

65N06, 65M06 ,65N22

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COPYRIGHT: © Global Science Press

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@Article{AAMM-16-1176, author = {Li , LiangWang , Zhenming and Zhu , Jun}, title = {A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2024}, volume = {16}, number = {5}, pages = {1176--1196}, abstract = {

In this paper, we propose a newly designed fifth-order finite difference well-balanced mapped unequal-sized weighted essentially non-oscillatory (WBMUS-WENO) scheme for simulating the shallow water systems on multi-dimensional structured meshes. We design new non-linear weights and a new mapping function, so that the WBMUS-WENO scheme can maintain fifth-order accuracy with a small $ε$ even nearby the extreme points in smooth regions. The truncation errors of the scheme is smaller and it has better convergence in simulating some steady-state problems. Unlike the traditional well-balanced WENO-XS scheme [29], this new WBMUS-WENO scheme uses three unequal-sized stencils, denotes the linear weights to be any positive numbers on condition that their summation is one. By incorporating a quartic polynomial on the whole big stencil into WENO reconstruction, the WBMUS-WENO scheme is simple and efficient. Extensive examples are performed to testify the exact C-property, absolute convergence property, and good representations of this new WBMUS-WENO scheme.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2022-0228}, url = {http://global-sci.org/intro/article_detail/aamm/23290.html} }
TY - JOUR T1 - A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations AU - Li , Liang AU - Wang , Zhenming AU - Zhu , Jun JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1176 EP - 1196 PY - 2024 DA - 2024/07 SN - 16 DO - http://doi.org/10.4208/aamm.OA-2022-0228 UR - https://global-sci.org/intro/article_detail/aamm/23290.html KW - Shallow water equations, exact C-property, mapping function, well-balanced unequal-sized WENO (WBMUS-WENO) scheme. AB -

In this paper, we propose a newly designed fifth-order finite difference well-balanced mapped unequal-sized weighted essentially non-oscillatory (WBMUS-WENO) scheme for simulating the shallow water systems on multi-dimensional structured meshes. We design new non-linear weights and a new mapping function, so that the WBMUS-WENO scheme can maintain fifth-order accuracy with a small $ε$ even nearby the extreme points in smooth regions. The truncation errors of the scheme is smaller and it has better convergence in simulating some steady-state problems. Unlike the traditional well-balanced WENO-XS scheme [29], this new WBMUS-WENO scheme uses three unequal-sized stencils, denotes the linear weights to be any positive numbers on condition that their summation is one. By incorporating a quartic polynomial on the whole big stencil into WENO reconstruction, the WBMUS-WENO scheme is simple and efficient. Extensive examples are performed to testify the exact C-property, absolute convergence property, and good representations of this new WBMUS-WENO scheme.

Liang Li, Zhenming Wang & Jun Zhu. (2024). A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations. Advances in Applied Mathematics and Mechanics. 16 (5). 1176-1196. doi:10.4208/aamm.OA-2022-0228
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