Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911562996908032 |
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| author | Franke, Brice Lagnaoui, Majid Rainer, Catherine |
| author_facet | Franke, Brice Lagnaoui, Majid Rainer, Catherine |
| contents | We discuss solutions of the one dimensional scalar conservation law with the flux function $y\longmapsto G_{c,ρ}\left(y\right)=((1-ρ)c-y)\mathbb{1}_{\{y>c\}}-ρy\mathbb{1}_{\{y\leqslant c\}}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context Franke, Brice Lagnaoui, Majid Rainer, Catherine Analysis of PDEs We discuss solutions of the one dimensional scalar conservation law with the flux function $y\longmapsto G_{c,ρ}\left(y\right)=((1-ρ)c-y)\mathbb{1}_{\{y>c\}}-ρy\mathbb{1}_{\{y\leqslant c\}}$ 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. |
| title | Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.01976 |