Discontinuous Galerkin methods for solving Boussinesq–Green–Naghdi equations in resolving non-linear and dispersive surface water waves

A local discontinuous Galerkin method for Boussinesq–Green–Naghdi equations is presented and validated against experimental results for wave transformation over a submerged shoal. Currently Green–Naghdi equations have many variants. In this paper a numerical method in one dimension is presented for the Green–Naghdi equations based on rotational characteristics in the velocity field. Stability criterion is also established for the linearized Green–Naghdi equations for both the analytical problem and the numerical method. Verification is done against a linearized standing wave problem in flat bathymetry and h, p (denoted by K in this paper) error rates are plotted. Validation plots show good agreement of the numerical results with the experimental ones.