Undercompressive Shocks and Riemann Problems for Scalar Conservation Laws with Nonconvex Fluxes

The Riemann initial value problem is studied for scalar conservation laws whose uxes have a single innection point. For a regularization consisting of balanced diiusive and dispersive terms, the traveling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley-Leverett ux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution has several diierent forms, depending on the initial data; the diierent forms are illustrated by numerical computations. Qualitatively similar behavior is observed in Lax-Wendroo approximations of solutions of the Buckley-Leverett equation with no dissipation or dispersion.