High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

doi:10.1007/s42967-021-00183-w

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (1): 485-528.doi: 10.1007/s42967-021-00183-w

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High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

Zepeng Liu¹, Yan Jiang¹, Mengping Zhang¹, Qingyuan Liu²

1 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China;
2 School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230026, Anhui, China

Received:2021-01-31 Revised:2021-12-09 Online:2023-03-20 Published:2023-03-08
Contact: Yan Jiang,E-mail:jiangy@ustc.edu.cn;Zepeng Liu,E-mail:lzp0375@mail.ustc.edu.cn;Mengping Zhang,E-mail:mpzhang@ustc.edu.cn;Qingyuan Liu,E-mail:qyliu@ahjzu.edu.cn E-mail:jiangy@ustc.edu.cn;lzp0375@mail.ustc.edu.cn;mpzhang@ustc.edu.cn;qyliu@ahjzu.edu.cn
Supported by:
This work is supported by the National Natural Science Foundation of China (11901555, 11871448, 12001009).

Abstract

Abstract: A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory (WENO) scheme. The exact C-property is investigated, and comparison with the standard finite difference WENO scheme is made. Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems. The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems, indicating smaller errors compared with the Lax-Friedrichs solver. In addition, we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

Key words: Shallow water equation, Well-balanced, High order accuracy, WENO scheme, Curvilinear meshes, Positivity-preserving limiter

CLC Number:

65M06

Zepeng Liu, Yan Jiang, Mengping Zhang, Qingyuan Liu. High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 485-528.

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High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

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