A free-surface correction method for simulating shallow water flows

Abstract

This paper presents a free-surface correction (FSC) method for solving the 3-D shallow water equations. Under the hydrostatic pressure assumption, the FSC method solves the governing equations for shallow water flows in two steps. First, an intermediate free surface is obtained after the horizontal momentum equations are solved with the explicit discretization of the pressure gradient terms. In the second step, the intermediate free surface is corrected by solving a five-diagonal matrix system for the free-surface change (Δ η) over the time step Δ t. The final velocity field is then corrected once the final free surface is obtained. The numerical scheme involves a semi-implicit discretization of the barotropic terms in the momentum equations and the horizontal fluxes terms in the vertically integrated continuity equation. Optinally, the FSC method is reduced to an explicit method for gravity waves, with the correction step omitted. Using the FSC method, a semi-implicit, 3-D finite difference model for free-surface flows has been developed. The model is mass conservative both locally and globally and is unconditionally stable with respect to gravity waves, wind and bottom stresses, and vertical eddy viscosity terms. Because both steps are straightforward and can be easily carried out, the FSC method presented here is an efficient method for simulating shallow water flows.