Asymptotic shallow water models with non smooth topographies

We present new models to describe shallow water flows over non smooth topographies. The water waves problem is formulated as a system of two equations on surface quantities in which the topography is involved in a Dirichlet-Neumann operator. Starting from this formulation and using the joint analyticity of this operator with respect to the surface and the bottom parametrizations, we derive a nonlocal shallow water model which only includes smoothing contributions of the bottom. Under additional small amplitude assumptions, Boussinesq-type systems are also derived. Using these alternative shallow water models as references, we finally present numerical tests to assess the precision of the classical shallow water approximations over rough bottoms. In the case of a polygonal bottom, we show numerically that our new model is consistent with the approach developed by Nachbin.