# Behaviour of the Serre Equations in the Presence of Steep Gradients Revisited

## TL;DR

This study uses numerical methods to analyze the behaviour of the Serre equations with steep gradients, revealing new solution structures and assessing the accuracy of shallow water and asymptotic approximations.

## Contribution

It introduces a numerical investigation of the Serre equations with steep initial conditions, identifying new solution structures and evaluating analytical and asymptotic approximations.

## Key findings

- Four solution structures depending on steepness
- Two structures align with existing literature
- Two novel solution structures identified

## Abstract

We use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial condition problems that are a smooth approximation to the initial conditions of the dam-break problem. This class of initial condition problems allow us to observe the behaviour of the Serre equations with varying steepness of the initial conditions. The numerical solutions of the Serre equations are justified by demonstrating that as the resolution increases they converge to a solution with little error in conservation of mass, momentum and energy independent of the numerical method. We observe four different structures of the converged numerical solutions depending on the steepness of the initial conditions. Two of these structures were observed in the literature, with the other two not being commonly found in the literature. The numerical solutions are then used to assess how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem. Lastly the numerical solutions are used to evaluate the usefulness of asymptotic results in the literature to approximate the depth and location of the front of an undular bore.