A non-local formulation for simulating the fully nonlinear Serre–Green–Naghdi equations for a solitary wave interaction with a variable slope

Abstract

In this paper, we simulate a solitary wave interaction with a variable slope with reflection on a vertical wall by integrating the fully nonlinear Serre–Green–Naghdi (SGN) equations. To this end, we first provide an iterative solution process for the SGN equations so that we can simulate a solitary wave propagating over variable bathymetry. For the purpose of the study, we examine two physical problems. The first is of a solitary wave interaction with a constant slope with reflection on a vertical wall. The simulated solutions are in good agreement with other numerical and experimental data, confirming the validity of the current work. The second is concerned with a perturbation of the first problem, where the constant slope of the first problem is varied; i.e., a variable slope is taken into account. We compare the simulated solutions of the two problems and observe the (physically realistic) effect of the variable slope on shoaling and reflection by the vertical wall.