An improved Serre model: Efficient simulation and comparative evaluation

Abstract

• A new Serre model with improved dispersion characteristics is considered. • A careful development of the new model allows an efficient numerical implementation. • A splitting scheme based on high order finite volume and finite difference is used. • Numerical experiments illustrate the superiority of the proposed model. The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serre α, β , presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios. A careful development of the Serre α, β model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serre α, β model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.