The numerical viscosity of entropy stable schemes for systems of conservation laws. I

TLDR

Discrete approximations to hyperbolic conservation laws are investigated. The study quantifies numerical viscosity in these schemes and relates it to entropy stability. The authors construct entropy‑conservative, second‑order accurate schemes that can be viewed as piecewise linear finite element methods on various meshes. Conservative schemes are entropy stable only when they contain more viscosity than the entropy‑conservative schemes, at least for three‑point schemes.

Abstract

Discrete approximations to hyperbolic systems of conservation laws are studied. We quantify the amount of numerical viscosity present in such schemes, and relate it to their entropy stability by means of <italic>comparison</italic>. To this end, conservative schemes which are also entropy conservative are constructed. These <italic>entropy conservative</italic> schemes enjoy second-order accuracy; moreover, they can be interpreted as piecewise linear finite element methods, and hence can be formulated on various mesh configurations. We then show that conservative schemes are entropy stable, if and—for three-point schemes—only if they contain <italic>more</italic> viscosity than that present in the above-mentioned entropy conservative ones.