A Conservative and Convergent Scheme for Undercompressive Shock Waves

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.