Mathematical and numerical study of a system of conservation laws |
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Authors: | R Eymard E Tillier |
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Institution: | (1) Université de Marne-la-Vallée, 5, boulevard Descartes Champs-sur-Marne, F-77454 Marne-la-Vallée CEDEX 2, France;(2) 1 & 4, avenue de Bois-Préau, F-92852 Rueil-Malmaison CEDEX, France |
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Abstract: | The system of equations (f (u))t − (a(u)v + b(u))x = 0 and ut − (c(u)v + d(u))x = 0, where the unknowns u and v are functions depending on
, 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 ut − ((cg + d)(u))x = 0 and simultaneously w = f (u) is the entropy solution of wt − ((ag + 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 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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Keywords: | Mathematics Subject Classifications (2000):" target="_blank">Mathematics Subject Classifications (2000): 35A35 35L45 65M12 |
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