SUMMARYA numerical technique for the modelling of shallow water ow in one and two dimensions is presented in this work along with the results obtained in di erent applications involving unsteady ows in complex geometries. A cell-centred ÿnite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured cells is presented. The discretization of the bed slope source terms is done following an upwind approach. In some applications a problem arises when the ow propagates over adverse dry bed slopes, so a special procedure has been introduced to model the advancing front. It is shown that this modiÿcation reproduces exactly steady state of still water in conÿgurations with strong variations in bed slope and contour. The applications presented are mainly related with unsteady ow problems. The scheme is capable of handling complex ow domains as will be shown in the simulations corresponding to the test cases that are going to be presented. Comparisons of experimental and numerical results are shown for some of the tests.

SUMMARYA wetting-drying condition (WDC) for unsteady shallow water ow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the e ectiveness of the WDC in ood propagation and dam break ows over real geometries. The WDC has been incorporated into a cell centred ÿnite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting-drying condition based on steady-state conditions lead to numerical errors in unsteady cases over conÿgurations with strong variations on bed slope. A modiÿcation of the wetting-drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and e cient 2D river ow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications.

SUMMARYThe two-dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two-dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second-order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case.The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant-Friedrichs-Lewy (CFL) condition.

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