Cartesian‐MUSCL‐like face‐value reconstruction algorithm for solving the depth‐averaged 2D shallow‐water equations

Abstract

AbstractThe aim of this article is to devise a novel second‐order monotone upstream scheme for conservation law (MUSCL) scheme on unstructured quadrilateral meshes, for solving the depth‐averaged 2D nonlinear shallow‐water equations (SWEs) involving rapid wet–dry transitions on uneven bed terrains. The proposed MUSCL scheme elaborately uses the geometry property of the quadrilateral cell and the reconstruction procedure is very similar to the MUSCL scheme on a Cartesian mesh. The edge‐centred value which is to be used for calculating local variable slope in the proposed scheme, is interpolated by both its two adjacent cell‐centred values and its two edge‐node values, where the nodal values are interpolated from surrounding cell‐centred values with a pseudo‐Laplacian formula. To prevent predicting nonphysically extreme velocities near the wet–dry interfaces so as to ensure model robustness, the proposed MUSCL scheme is locally reduced to be first‐order in potentially problematic cells; these cells are detected with a proposed physically‐based criterion. The new scheme does not need to compute variable gradients at cell‐centers, and we proved that the conventional MUSCL scheme on a Cartesian mesh is just a particular case of the newly designed scheme. A 2D SWEs solver is developed based on the new MUSCL scheme. Various numerical tests demonstrate that the developed numerical model is capable of maintaining steady stationary flow at rest and robust for simulation involving large gradients, and captures well the wet–dry front motions. The developed model is also found to adapt to extremely stretched meshes. Though the proposed numerical scheme is designed originally for meshes with purely quadrilateral cells, numerical experiments show that this scheme works well on triangular meshes as well, indicating that the proposed numerical scheme is attractive for simulations on mixed meshes consisting of both quadrilateral and triangular mesh‐cells.