Well-balanced and shock-capturing solving of 3D shallow-water equations involving rapid wetting and drying with a local 2D transition approach

Abstract

In this study, we develop a shock-capturing numerical model for solving the three-dimensional shallow-water equations with turbulence closure. Numerical discretization is performed using the Godunov-type finite-volume method in σ coordinates. An approximate Riemann solver is used for numerical flux evaluations. To efficiently and accurately resolve the wet–dry fronts, we propose a local 2D transition approach (i.e., switching from solving the three-dimensional shallow-water equations to solving the depth-averaged two-dimensional shallow-water equations) with using a rather small value of threshold water depth (1×10−6m) to distinguish dry cells from the wet ones. A fully-implicit discretization method of the bed-friction terms is developed, which was originally proposed to solve the depth-averaged two-dimensional shallow-water equations. A series of benchmark tests are used to assess the performance of the numerical models, showing that the proposed model is well-balanced and robust to resolve violent free-surface flows. The local 2D transition approach is found to significantly reduce the computational time by one to three orders in a dam-break wave propagation test, and by about 1∕2–2∕3 in a breaking solitary wave runup test, compared with a same simulation without using this approach.