Modified Shallow Water Equations for significantly varying seabeds

TL;DR

This paper introduces a modified, hyperbolic version of the nonlinear shallow water equations that accounts for significant seabed variations, enabling more accurate modeling of surface waves such as tsunamis.

Contribution

A new non-dispersive, non-hydrostatic extension of the Saint-Venant equations derived without small parameters, suitable for complex seabed topographies.

Findings

The model is hyperbolic, allowing the use of existing numerical methods.

Finite volume discretisation is proposed for practical computations.

Test cases demonstrate the model's improved capabilities for wave simulation.

Abstract

In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.