On the oscillatory solutions in hyperbolic conservation laws (Q1586542)

The authors study Riemann problems with shock initial data for two examples of \(2\times 2\) conservation laws in one-space dimension: the shallow water equations and a three-phase Buckley-Leverett flow. For shock initial data that do not possess a viscous profile, they consider a parabolic regularization of the conservation laws for which they construct approximate oscillatory solutions by the Crank-Nicholson method. Numerical examples convince them that the approximate solutions are uniformly bounded; admitting this, they prove convergence of subsequences to measure-valued solutions of the system of conservation laws with the given initial data.