Wave propagation and run-up in U-shaped channel bays are studied here in the framework of the quasi-1D Saint-Venant equations. Our approach is numerical, using the momentum conserving staggered-grid (MCS) scheme, as a consistent approximation of the Saint-Venant equations. We carried out simulations regarding wave focusing and run-ups in U-shaped bays. We obtained good agreement with the existing analytical results on several aspects: the moving shoreline, wave shoaling, and run-up heights. Our findings also confirm that the run-up height is significantly higher in the parabolic bay than on a plane beach. This assessment shows the merit of the MCS scheme in describing wave focusing and run-up in U-shaped bays. Moreover, the MCS scheme is also efficient because it is based on the quasi-1D Saint-Venant equations.

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Simplified models like the shallow water equations (SWE) are commonly adopted for describing a wide range of free surface flow problems, like flows in rivers,lakes, estuaries, or coastal areas. In the literature, numerical methods for the SWE are mostly mesh-based. However, this macroscopic approach is unable to accurately represent the complexity of flows near coastlines, where waves nearly break. This fact prompted the idea of coupling the mesh-based SWE model with a meshless particlemethod for solving the Euler equations. In a previous paper, a method to couple the staggered scheme SWE and the smoothed particle hydrodynamics (SPH) Euler equations was developed and discussed. In this article, this coupled model is used for simulating solitary wave run-up on a sloping beach. The results show strong agreement with the experimental data of Synolakis. Simulations of wave overtopping over aseawall were also performed.

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