A well-balanced and positivity-preserving adaptive unstaggered central scheme for two-dimensional shallow water equations with nonflat bottom topography on irregular quadrangles is presented. The irregular quadrilateral mesh adds to the difficulty of designing unstaggered central schemes. In particular, the integral of the source term needs to subtly be dealt with. A new method of discretizing the source term for the well-balanced property is proposed, which is one of the main contributions of this work. The spacial second-order accuracy is obtained by constructing piecewise bilinear functions. Another novelty is that we introduce a strong-stability-preserving Unstaggered-Runge–Kutta method to improve the accuracy in time integration. Adaptive moving mesh strategies are introduced to couple with the current unstaggered central scheme. The well-balanced property is still valid. The positivity-preserving property can be proved when the cells shrink. We prove that the current adaptive unstaggered central scheme can obtain the stationary solution (“lake at rest” steady solutions) and guarantee the water depth to be nonnegative. Several classical problems of shallow water equations are shown to demonstrate the properties of the current numerical scheme.
Keywords: Two-dimensional shallow water equations, unstaggered central schemes, irregular quadrangles, physical-constraints-preserving, adaptive moving mesh methods
@article{M2AN_2022__56_6_2297_0,
author = {Dong, Jian and Qian, Xu and Song, Songhe},
title = {Adaptive physical-constraints-preserving unstaggered central schemes for shallow water equations on quadrilateral meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2297--2338},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {6},
doi = {10.1051/m2an/2022076},
mrnumber = {4519225},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022076/}
}
TY - JOUR AU - Dong, Jian AU - Qian, Xu AU - Song, Songhe TI - Adaptive physical-constraints-preserving unstaggered central schemes for shallow water equations on quadrilateral meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 2297 EP - 2338 VL - 56 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022076/ DO - 10.1051/m2an/2022076 LA - en ID - M2AN_2022__56_6_2297_0 ER -
%0 Journal Article %A Dong, Jian %A Qian, Xu %A Song, Songhe %T Adaptive physical-constraints-preserving unstaggered central schemes for shallow water equations on quadrilateral meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 2297-2338 %V 56 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022076/ %R 10.1051/m2an/2022076 %G en %F M2AN_2022__56_6_2297_0
Dong, Jian; Qian, Xu; Song, Songhe. Adaptive physical-constraints-preserving unstaggered central schemes for shallow water equations on quadrilateral meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2297-2338. doi: 10.1051/m2an/2022076
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