A Class of Boundary Conditions for Time-Discrete Green--Naghdi Equations with Bathymetry

This work is devoted to the structure of time-discrete Green--Naghdi equations including bathymetry. We use projection characterize homogeneous and inhomogeneous boundary conditions for which semidiscrete are well posed. allows us propose efficient robust numerical treatment that ensures entropy stability scheme by construction. Numerical evidence provided illustrate our approach suitable situations practical interest not covered existing theory.