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

Abstract

We discuss solutions of the one dimensional scalar conservation law with the flux function $y⟼G_{c,\ddot{I}}\left(y\right)=((1-\ddot{I})c-y)1_{\{y>c\}}-\ddot{I}y{1}_{\{y⩽c\}}$ 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.