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

Brice Franke, Majid Lagnaoui, Catherine Rainer

Abstract

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.

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 for two specific initial conditions . This equation arises as the limit of a specific conceptual hydrological model. For initial data strictly below (resp. above) the threshold level , the equation reduces to a constant-speed transport equation with velocity (resp. ). Our goal is to understand precisely what happens when the initial condition crosses the threshold , which corresponds to a generalisation of the Riemann problem, and to provide, in such cases, quasi-closed-form expressions for the corresponding solutions.

Paper Structure

This paper contains 16 sections, 8 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context and Problem Statement
  3. Problem Statement
  4. Some definitions of the general theory
  5. The Constant-Speed Advection Equation
  6. First Type of Initial Condition : Down crossing
  7. Second Type of Initial Condition : Up crossing
  8. Parametrization of the Initial Condition
  9. Notations and auxilliary functions
  10. Definition of the functions y, x and t.
  11. On the regularity and integrability assumptions for $u_0$
  12. Weak entropy solution for the Cauchy problem with Down-crossing initial data
  13. Examples
  14. First example
  15. Second example
  16. ...and 1 more sections

Key Result

Proposition 1

The function $u$ is an entropy solution in the sense of the definition (definitionnotionsolution). if and only if, for all $k\in\mathbb{R}$ the equation (entropyequation) holds with $\eta(s)=|s-k|$ and $\Phi(s)=\operatorname{sgn}(s-k)(f(s)-f(k))$, i.e: The function $\eta$ is called Krǔzhkov's entropy. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Initial profile and its evolution under the scalar conservation law (Down-Crossing Case). The curve $u(.,t)$ shows the phenomen of rarefaction on the interval $]x_0-t,x_0-\rho t[$.
  • Figure 2: Illustration of the undderlying idea (Up-Crossing Case).

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1: Krǔzhkov's entropy condition
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 1
  • Remark 1
  • Lemma 2
  • ...and 4 more