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Well-balanced or asymptotic preserving schemes are receiving an increasing amount of interest. This paper gives a precise setting for studying both properties in the case of Euler system with friction. We derive a simple solver which, by construction, preserves discrete equilibria and reproduces at the discrete level the same asymptotic behavior as that of the solutions of the continuous system. Numerical illustrations are convincing and show that not all methods share these properties.

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

International audience; We propose a relaxation framework for general fluid models which can be understood as a natural ex- tension of the Suliciu approach in the Euler setting. In particular, the relaxation system may be totally degenerate. Several stability properties are proved. The relaxation procedure is shown to be efficient in the numerical approximation of the entropy weak solutions of the original PDEs. The numerical method is particularly simple in the case of a fully degenerate relaxation system for which the solution of the Riemann problem is explicit. Indeed, the Godunov solver for the homogeneous relaxation system results in an HLLC-type solver for the equilibirum model. Discrete entropy inequalities are established under a natural Gibbs principle.