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The simulation of long, nonlinear dispersive waves in bounded domains usuallyrequires the use of slip-wall boundary conditions. Boussinesq systems appearingin the literature are generally not well-posed when such boundary conditionsare imposed, or if they are well-posed it is very cumbersome to implement theboundary conditions in numerical approximations.  In the present paper a new Boussinesq system is proposed for the study oflong waves of small amplitude in a basin when slip-wall boundary conditions arerequired. The new system is derived using asymptotic techniques under theassumption of small bathymetric variations, and a mathematical proof ofwell-posedness for the new system is developed.  The new system is also solved numerically using a Galerkin finite-elementmethod, where the boundary conditions are imposed with the help of Nitsche'smethod. Convergence of the numerical method is analyzed, and precise errorestimates are provided. The method is then implemented, and the convergence isverified using numerical experiments. Numerical simulations for solitary wavesshoaling on a plane slope are also presented. The results are compared toexperimental data, and excellent agreement is found.

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis