An Eulerian-Lagrangian method for tidal current computation

In this paper, a semi-implicit finite difference method for the 2-D shallow water equations is derived and applied. A characteristic analysis of the governing equations indicates those terms to be discretised implicitly so that the stability of the method will not depend on the celerity. Such terms are the gradient of the water surface elevation in the momentum equations, and the velocity divergence in the continuity equation. The convective terms are discretised explicitly by using either an upwind or an Eulerian-Lagrangian formula. The resulting algorithm, at each time step, required the solution of a linear give-diagonals system which is diagonally dominant with positive elements on the main diagonal and negative ones elsewhere. Thus, existence and uniqueness of the numerical solution is assured at each time step. An application of this method is finally considered for computing the tidal circulation in the MARE PICCOLO, an embayment in the Gulf of Taranto.