# Hyperbolic Model for Free Surface Shallow Water Flows with Effects of Dispersion, Vorticity and Topography

## TL;DR

This paper introduces a hyperbolic system to model dispersive shallow water flows, simplifying numerical computations and accurately capturing wave phenomena like undular bores and solitary waves.

## Contribution

A novel hyperbolic model for dispersive shallow water flows that avoids elliptic problems and reduces computation time, validated against the original dispersive model.

## Key findings

- Model closely matches dispersive solutions over time
- Effectively simulates undular bores and wave breaking
- Reduces numerical complexity and computation time

## Abstract

We derive a hyperbolic system of equations approximating the two-layer dispersive shallow water model for shear flows recently proposed by Gavrilyuk, Liapidevskii \& Chesnokov (J. Fluid Mech., vol. 808, 2016, pp. 441--468). The use of this system for modelling the evolution of surface waves makes it possible to avoid the major numerical challenges in solving dispersive shallow water equations, which are connected with the resolution of an elliptic problem at each time instant and realization of non-reflecting conditions at the boundary of the calculation domain. It also allows one to reduce the computation time. The velocities of the characteristics of the obtained model are determined and the linear analysis is performed. Stationary solutions of the model are constructed and studied. Numerical solutions of the hyperbolic system are compared with solutions of the original dispersive model. It is shown that they almost coincide for large time intervals. The system obtained is applied to study non-stationary undular bores produced after interaction of a uniform flow with an immobile wall, non-hydrostatic shear flows over a local obstacle and the evolution of breaking solitary wave on a sloping beach.