High-resolution numerical solution of 3D shallow water equations based on approximate Riemann solver

The Godunov finite volume method for solving planar 2D shallow water equations is extended to solve 3D shallow water equations, so as to establish a three-dimensional mathematical model with shock capture characteristic, the application of 3D shallow water equations will be expanded. In this paper, the 3D shallow-water equations in σ-coordinates were solved based on the Godunov-type finite-volume method. The nonlinear k-? model was employed for turbulent flow closure, the HLLC approximate Riemann solver was involved to calculate the horizontal numerical fluxes. To improve the numerical stability, the vertical diffusion-term was implicitly discretized, and the 3D model was locally switched to a horizontally depth-averaged 2D model in which the water depth was sufficiently small. The developed model was form second-order form in both space and time. The model was verified by classical tests including hydraulic jump and dam-break flood propagating in a dry river bed, and the results showed that the developed model is stable, well-balanced, capable of predicting high-resolution solution around discontinuities, and simulates wetting and drying processes well.