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Submitted paper; Undercompressive shock waves arise in numerous physical applications. We propose a class of conservative finite-volume type schemes to approximate weak solutions of conservation laws that contain undercompressive shock waves. We prove the convergence of a subsequence of approximate solutions towards a generalized entropy solution if the mesh width tends to zero. The proof relies on a refined BV compactness analysis, which accounts for the effect of the kinetic relation that drives the undercompressive wave. At the same time we establish a new proof for the existence of solutions to the underlying model. Numerical experiments supplement the analytical results.

International audience; Phase transition problems in compressible media can be modelled by mixed hyperbolicelliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. In recent years several tracking-type algorithms have been suggested for numerical approximation. Typically a core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation at the location of phase boundaries. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel sub-iteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kinetic relation where exact Riemann solvers are available. For a nonlinear kinetic relation and a thermoelastic system we use the new method to gain information on the Riemann problem. Up to our knowledge an exact solution for arbitrary Riemann data is currently not available in these cases.