A class of boundary conditions for time-discrete Green-Naghdi equations with bathymetry

TL;DR

This paper develops a boundary condition framework for time-discrete Green-Naghdi equations with bathymetry, ensuring well-posedness and entropy stability, and demonstrates its practical effectiveness through numerical experiments.

Contribution

It introduces a novel boundary condition structure for semi-discrete Green-Naghdi equations that guarantees stability and robustness, extending applicability beyond existing theories.

Findings

Boundary conditions ensuring well-posedness are characterized.

The proposed scheme is entropy stable by design.

Numerical tests confirm practical effectiveness.

Abstract

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