One-sided difference schemes and transonic flow.

Two one-sided conservation form difference approximations to a scalar one-dimensional convex conservation law are introduced. These are respectively of first- and second-order accuracy and each has the minimum possible band-width. They are nonlinearly stable, they converge only to solutions satisfying the entropy condition, and they have sharp monotone profiles. No such stable approximation of order higher than two is possible. Dimensional splitting algorithms are constructed and used to approximate the small-disturbance equation of transonic flow. These approximations are also nonlinearly stable and without nonphysical limit solutions.