  We consider solutions of hyperbolic conservation laws regularized with
vanishing diffusion and dispersion terms. Following a pioneering work by
Schonbek, we establish the convergence of the regularized solutions toward
discontinuous solutions of the hyperbolic conservation law. The proof relies on
the method of compensated compactness in the $L^2$ setting. Our result improves
upon Schonbek's earlier results and provides an optimal condition on the
balance between the relative sizes of the diffusion and the dispersion
parameters. A convergence result is also established for multi-dimensional
conservation laws by relying on DiPerna's uniqueness theorem for entropy
measure-valued solutions.
