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This paper deals with the analysis of some numerical difficulties related to the appearance of wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids. The fluid is supposed to be governed by the Shallow Water equations and the discretization of the equations is performed, when wet/dry fronts do not appear, by means of the Q-scheme of Roe upwinding the source terms introduced in Ref. 40. This scheme is well-balanced in the sense that it solves exactly stationary solutions corresponding to water at rest. Wet/dry fronts cannot be correctly treated with this scheme: it can produce negative values of the thickness of the fluid layer and stationary solutions corresponding to water at rest including wet/dry transitions are not exactly solved. In Refs. 3–5 some variants of this numerical scheme have been proposed that partially solve these difficulties. Here we propose a new variant: at intercells where wet/dry transitions occur, a Nonlinear Riemann Problem is considered instead of a Linear one. The exact solutions of these nonlinear problems, which are easy to calculate, are used in order to define the numerical fluxes. We investigate the properties of the resulting scheme and present some comparisons with the numerical results obtained with some other modified numerical schemes proposed previously.

In this work, a characterization of the hyperbolicity region for the two layer shallow-water system is proposed and checked. Next, some path-conservative finite volume schemes (see [11]) that can be used even if the system is not hyperbolic are presented, but they are not in general L 2 linearly stable in that case. Then, we introduce a simple but efficient strategy to enforce the hyperbolicity of the two-layer shallow-water system consisting in adding to the system an extra amount of friction at every cell in which complex eigenvalues are detected at a given time step. The implementation is performed by a predictor/corrector strategy: first a numerical scheme is applied to the unmodified two-layer system, regardless of the hyperbolic character of the system. Next, we check if the predicted cell averages are in the hyperbolic region or not. If not, the mass-fluxes are corrected by adding a quadratic friction law between layers whose coefficient is computed so that the corrected cell average is as near as possible of the boundary of the hyperbolicity region. Finally, some numerical test have been performed to assess the efficiency of the proposed strategy.Short title : Numerical treatment of the loss of hyperbolicity of the two-layer SWS.

Abstract-The Tsunami-HySEA model is used to perform some of the numerical benchmark problems proposed and documented in the ''Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop''. The final aim is to obtain the approval for Tsunami-HySEA to be used in projects funded by the National Tsunami Hazard Mitigation Program (NTHMP). Therefore, this work contains the numerical results and comparisons for the five benchmark problems (1, 4, 6, 7, and 9) required for such aim. This set of benchmarks considers analytical, laboratory, and field data test cases. In particular, the analytical solution of a solitary wave runup on a simple beach, and its laboratory counterpart, two more laboratory tests: the runup of a solitary wave on a conically shaped island and the runup onto a complex 3D beach (Monai Valley) and, finally, a field data benchmark based on data from the 1993 Hokkaido Nansei-Oki tsunami.