An operator-splitting approach with a hybrid finite volume/finite difference scheme for extended Boussinesq models

Abstract

In this paper, we present a hybrid numerical scheme combining finite volume and finite difference methods (FV/FD), for solving two extended Boussinesq-type (BT) models. We propose two new reformulations specially designed to allow a separation between the hyperbolic part and dispersive part of the equations. Taking advantage of the proposed reformulations, we employ a splitting scheme specifically tailored to comprehensively capture the broad spectrum of natural events that are observed in the study of coastal oceanography. We employ a high-order FV well-balanced scheme to effectively handle the hyperbolic portion of the equations, guaranteeing non-negativity of water depth and accommodating dry regions. We solve the remaining dispersive component by employing a traditional FD approach. Several test cases in one horizontal dimension are validated showing that the proposed approach is able to accurately replicate coastal wave dynamics. In comparison to the non-linear shallow water equations (NSWE), the BT models exhibit significantly enhanced accuracy in simulating highly dispersive waves across varying water depths.