A combined finite volume - finite element scheme for a dispersive shallow water system

TL;DR

This paper introduces a novel combined finite volume and finite element numerical scheme for a non-hydrostatic shallow water model, enabling accurate simulation of dispersive effects with bottom topography.

Contribution

It develops a variational framework and a prediction-correction scheme integrating finite volume and finite element methods for non-hydrostatic shallow water equations.

Findings

Numerical experiments confirm the scheme's accuracy and stability.

The method effectively captures dispersive and topographical effects.

The approach provides a reliable tool for complex shallow water simulations.

Abstract

We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation. The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces. The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition. Several numerical experiments are performed to confirm the relevance of our approach.