# Constructive Approach of the Solution of Riemann Problem for Shallow Water Equations with Topography and Vegetation

## TL;DR

This paper develops a constructive method for solving the Riemann problem in shallow water equations considering topography and vegetation, extending solutions beyond small initial jumps and addressing degeneracy issues.

## Contribution

It introduces a novel constructive approach to solve the Riemann problem for complex shallow water models with terrain and porosity, handling large initial data and degeneracy.

## Key findings

- Solution construction for large initial jumps
- Identification of conditions for existence and non-existence of solutions
- Addressing degeneracy in hyperbolic systems

## Abstract

We investigate the Riemann Problem for a shallow water model with porosity and terrain data. Based on recent results on the local existence, we build the solution in the large settings (the magnitude of the jump in the initial data is not supposed to be ``small enough''). One difficulty for the extended solution arises from the double degeneracy of the hyperbolic system describing the model. Another difficulty is given by the fact that the construction of the solution assumes solving an equation which has no global solution. Finally, we present some cases to illustrate the existence and non-existence of the solution.