The simulation of long, nonlinear dispersive waves in bounded domains usually
requires the use of slip-wall boundary conditions. Boussinesq systems appearing
in the literature are generally not well-posed when such boundary conditions
are imposed, or if they are well-posed it is very cumbersome to implement the
boundary conditions in numerical approximations.
In the present paper a new Boussinesq system is proposed for the study of
long waves of small amplitude in a basin when slip-wall boundary conditions are
required. The new system is derived using asymptotic techniques under the
assumption of small bathymetric variations, and a mathematical proof of
well-posedness for the new system is developed.
The new system is also solved numerically using a Galerkin finite-element
method, where the boundary conditions are imposed with the help of Nitsche's
method. Convergence of the numerical method is analyzed, and precise error
estimates are provided. The method is then implemented, and the convergence is
verified using numerical experiments. Numerical simulations for solitary waves
shoaling on a plane slope are also presented. The results are compared to
experimental data, and excellent agreement is found.
Israwi, S., Kalisch, H., Katsaounis, T. & Mitsotakis, D. (n.d.). A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis. http://arxiv.org/abs/2008.10754v1