Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context

TL;DR

This paper analyzes solutions to a scalar conservation law with a piecewise flux function, modeling a hydrological process, and provides explicit solutions for initial conditions crossing a threshold.

Contribution

It introduces a semi-explicit entropic solution approach for a generalized Riemann problem in a hydrological context, extending classical solutions.

Findings

Derived quasi-closed-form solutions for crossing threshold initial data.

Identified the equation reduces to simple transport equations away from the threshold.

Clarified the behavior of solutions at the threshold crossing.

Abstract

We discuss solutions of the one dimensional scalar conservation law with the flux function $y\longmapsto G_{c,\rho}\left(y\right)=((1-\rho)c-y)\mathbb{1}_{\{y>c\}}-\rho y\mathbb{1}_{\{y\leqslant c\}}$ for two specific initial conditions $u(\cdot,0)=u_0$. This equation arises as the limit of a specific conceptual hydrological model. For initial data strictly below (resp. above) the threshold level $c$, the equation reduces to a constant-speed transport equation with velocity $p$ (resp. $1$). Our goal is to understand precisely what happens when the initial condition crosses the threshold $c$, which corresponds to a generalisation of the Riemann problem, and to provide, in such cases, quasi-closed-form expressions for the corresponding solutions.