A third-order finite volume semi-implicit method for the Shallow Water-Exner model

TL;DR

This paper introduces a third-order semi-implicit finite volume method on staggered meshes for the shallow water and Saint-Venant-Exner systems, emphasizing stability, accuracy, and efficiency in simulating subcritical flows.

Contribution

It presents a novel third-order semi-implicit scheme with pressure gradient approximation and a new semi-analytical solution for the Saint-Venant-Exner system, enhancing simulation accuracy and stability.

Findings

Achieves third-order accuracy in tests

Demonstrates stability in subcritical flow regimes

Effectively simulates slow bedload processes

Abstract

In this work, third-order semi-implicit schemes on staggered meshes for the shallow water and Saint-Venant-Exner systems are presented. They are based on a third-order extension of the technique introduced in Cassulli \& Cheng [1]. The stability conditions for these schemes depend on the velocity and not on the celerity, allowing us to reduce computational efforts, especially in subcritical flow simulations, which is the regime we are mainly interested in. The main novelty consists in the third-order approximation of the pressure gradient term in the momentum equation through appropriate polynomial reconstructions. Concretely, CWENO conservative reconstruction is considered for the water thickness $h$ and a centered fourth-degree polynomial is adopted interpolating the cell averages of the free surface $\eta$. For time discretization, a third-order IMEX scheme is applied. In addition, a novel time-dependent semi-analytical solution for Saint-Venant-Exner system is introduced and compared with the numerical ones. Several tests are performed, including accuracy tests showing third-order accuracy, well-balance tests, and simulations of slow bedload processes for large time.