Optimal dispersion with minimized Poisson equations for non-hydrostatic free surface flows

A non-hydrostatic shallow-water model is proposed to simulate the wave propagation in situations where the ratio of the wave length to the water depth is small. It exploits the reduced-size stencil in the Poisson pressure solver to make the model less expensive in terms of memory and CPU time. We refer to this new technique as the minimized Poisson equations formulation. In the simplest case when the method applied to a two-layer model, the new model requires the same computational effort as depth-integrated non-hydrostatic models, but can provide a much better description of dispersive waves. To allow an easy implementation of the new method in depth-integrated models, the governing equations are transformed into a depth-integrated system, in which the velocity difference serves as an extra variable. The non-hydrostatic shallow-water model with minimized Poisson equations formulation produces good results in a series of numerical experiments, including a standing wave in a basin, a non-linear wave test, solitary wave propagation in a channel and a wave propagation over a submerged bar.