Numerical viscosity and the entropy condition

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 solutions. The question arises whether finite difference approximations converge to this particular solution. It is known that this is not always the case with the standard Lax‐Wendroff (L‐W) difference scheme. In this paper a simple variant of the L‐W scheme is devised which retains its desirable computational features—conservation form, three point scheme, second‐order accuracy on smooth solutions, but which has the additional property that limit solutions satisfy the entropy condition. This variant is constructed by adding a simple nonlinear artificial viscosity to the usual L‐W operator. The nature of the viscosity is deduced by first analyzing a model differential equation derived from the truncation error for the L‐W operator, keeping only terms of order (Δx)². Furthermore, this viscosity is “switched on” only when sufficiently steep discrete gradients develop in the approximate solution: The full L‐W scheme is then shown to have the desired property provided that the Courant‐Friedrichs‐Lewy restriction |λf′(u)|≤0.14 is satisfied.