Volume 56, Number 6, November-December 2022
|Page(s)||2297 - 2338|
|Published online||08 December 2022|
Adaptive physical-constraints-preserving unstaggered central schemes for shallow water equations on quadrilateral meshes
Department of Mathematics, College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, Hunan, P.R. China
2 State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China
* Corresponding author: email@example.com
Accepted: 7 September 2022
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.
Mathematics Subject Classification: 76M12 / 35L65 / 65L05 / 65M08
Key words: Two-dimensional shallow water equations / unstaggered central schemes / irregular quadrangles / physical-constraints-preserving / adaptive moving mesh methods
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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