On finite‐difference approximations and entropy conditions for shocks

Abstract

Abstract Weak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solution. The question arises whether finite‐difference approximations converge to this particular solution. It is shown in this paper that tin the case of a single conservation law, monotone schemes, when convergent, always converge to the physically relevant solution. Numerical examples show that this is not always the case with non‐monotone schemes, such as the Lax‐Wendroff scheme.