Well-balanced positivity preserving adaptive moving mesh central-upwind schemes for the Saint-Venant system

¹ Department of Mathematics, SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen 518055, P.R. China
² Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA

^* Corresponding author: alexander@sustech.edu.cn

Received: 7 September 2021
Accepted: 21 April 2022

Abstract

We extend the adaptive moving mesh (AMM) central-upwind schemes recently proposed in Kurganov et al. [Commun. Appl. Math. Comput. 3 (2021) 445–479] in the context of one- (1-D) and two-dimensional (2-D) Euler equations of gas dynamics and granular hydrodynamics, to the 1-D and 2-D Saint-Venant system of shallow water equations. When the bottom topography is nonflat, these equations form hyperbolic systems of balance laws, for which a good numerical method should be capable of preserving a delicate balance between the flux and source terms as well as preserving the nonnegativity of water depth even in the presence of dry or almost dry regions. Therefore, in order to extend the AMM central-upwind schemes to the Saint-Venant systems, we develop special positivity preserving reconstruction and evolution steps of the AMM algorithms as well as special corrections of the solution projection step in (almost) dry areas. At the same time, we enforce the moving mesh to be structured even in the case of complicated 2-D computational domains. We test the designed method on a number of 1-D and 2-D examples that demonstrate robustness and high resolution of the proposed numerical approach.