Mathematical and numerical study of a system of conservation laws

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 (x, t) ∈ R × 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 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.Keywords: nonlinear hyperbolic equations, existence of a solution, Riemann problem, admissible shocks, entropy solutions, convergence of a finite volume scheme.