A well-balanced shock-capturing hybrid finite volume–finite difference numerical scheme for extended 1D Boussinesq models

A formally fourth-order well-balanced hybrid finite volume/difference (FV/FD) numerical scheme for approximating the conservative form of two 1D extended Boussinesq systems is presented. The FV scheme is of the Godunov type and utilizes Roeʼs approximate Riemann solver for the advective fluxes along with well-balanced topography source term upwinding, while FD discretizations are applied to the dispersive terms in the systems. Special attention is given to the accurate numerical treatment of moving wet/dry fronts. To access the performance and applicability, by exposing the merits and differences of the two formulations, the numerical models have been applied to idealized and challenging experimental test cases. Special attention is paid in comparing both Boussinesq models to the nonlinear shallow water equations (NSWE) in the simulation of the experimental results. The outcomes from this work confirm that, although the NSWE can be sufficient in some cases to predict the general characteristics of propagating waves, the two Boussinesq models provided considerable more accurate results for highly dispersive waves over increasing water depths.