The stability analysis, the accuracy and the efficiency of a semi-implicit finite difference scheme for the numerical solution of a three-dimensional shallow water model are presented and discussed. The governing equations are the three-dimensional Reynolds equations in which pressure is assumed to be hydrostatic. The pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the θ-method, with θ being an implicitness parameter. It is shown that the method is stable for 1 2 ≤ θ ≤ 1, unstable for θ < 1 2 and highest accuracy and efficiency is achieved when θ = 1 2. The resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats. © 1994.
Casulli, V., & Cattani, E. (1994). Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow. Computers and Mathematics with Applications, 27(4), 99–112. https://doi.org/10.1016/0898-1221(94)90059-0