Communications in Mathematical Sciences

Volume 22 (2024)

Number 6

An efficient interface hydrostatic reconstruction for the two-layer shallow flows with arbitrary wet-dry fronts

Pages: 1657 – 1684

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n6.a9

Authors

Jian Dong (Department of Mathematics, College of Science, National University of Defense Technology, Changsha, China)

Dingfang Li (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Abstract

This paper aims to propose a well-balanced positivity-preserving numerical scheme for the two-layer shallow water systems with arbitrary wet-dry fronts based on interface hydrostatic reconstructions (IHR). One key difficulty in solving the two-layer shallow water systems is the nonconservative product term which cannot be evaluated on the cell boundaries. Another difficulty is that the well-balanced property for the still water maybe missed when the computational domain has wet-dry fronts, especially, the wet-dry front is located at the discontinuous bottom topography. For the nonlinear stability of the numerical scheme, the positivity of the water height is vital. To this end, we discretize the nonconservative product term based on the IHR method, which is a particular choice of path-conservative methods. The intermediate bottom level used in the discretization of the bed source term of two layers is different. The nonconservative product term due to the momentum exchange between two layers is discretized using the intermediate interface water height.

The resulting numerical scheme can preserve the positivity of two-layered heights and maintain the still water even when the computational domain has wet-dry fronts. The numerical scheme performs well in solving the complex problems, such as the Kelvin-Helmholtz instable problems. We demonstrate these properties of the current scheme through several classical problems of the two-layer shallow water systems with arbitrary wet-dry fronts.

Keywords

well-balanced, positivity preserving, arbitrary wet-dry fronts, two-layer shallow water systems, interface hydrostatic reconstruction

2010 Mathematics Subject Classification

35L65, 76M12

Received 4 May 2021

Received revised 12 December 2023

Accepted 25 January 2024

Published 18 July 2024