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Article Dans Une Revue Journal of Evolution Equations Année : 2007

Mathematical and numerical study of a system of conservation laws

Elodie Tillier
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Résumé

The system of equations $(f(u))_t - \left( a(u) v + b(u) \right)_x = 0$ and $u_t - \left( c(u) v + d(u) \right)_x = 0$, where the unknowns $u$ and $v$ are functions depending on $(x,t)\in\R\times\R_+$, arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution $(u,v)$, inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on $x/t$. This property results from the proof of the existence, by passing to the limit on some approximations, of a function $g$ such that $u$ is the classical entropy solution of $u_t - ((c g + d)(u))_x = 0$ and simultaneously $w=f(u)$ is the entropy solution of $w_t - ((a g + b)(f^{(-1)}(w)))_x = 0$. We then take $v=g(u)$, and the proof that $(u,v)$ is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by $u$ and $f(u)$. %We then give an example with explicit values, showing that $v$ cannot be %replaced by $\psi(u)$ for some continuous function $\psi$. We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.
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Dates et versions

hal-00108526 , version 1 (22-10-2006)

Identifiants

Citer

Robert Eymard, Elodie Tillier. Mathematical and numerical study of a system of conservation laws. Journal of Evolution Equations, 2007, http://www.springerlink.com/content/wrj46l34231r8k66/?MUD=MP. ⟨10.1007/s00028-006-0271-x⟩. ⟨hal-00108526⟩
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