On Entropy-Stable Discretizations and the Entropy Adjoint

Abstract

For conservation laws that admit a convex entropy, Fidkowski and Roe (SIAM J Sci Comput 32(3):1261–1287, 2010) showed that the entropy variables satisfy an adjoint equation corresponding to an entropy-balance functional. In general, this relationship between the entropy variables and the entropy adjoint becomes an approximation when the conservation law is discretized. However, this work shows that the relationship is mimicked discretely for entropy-stable discretizations based on summation-by-parts (SBP) operators and two-point entropy-conservative flux functions; for these discretizations, the discrete entropy variables exactly satisfy the discrete adjoint equation for a particular discretization of the entropy-balance functional. A detailed proof of this result is presented for first-order conservation laws that are semi-discretized using SBP operators with diagonal norm and boundary operators. Subsequently, generalizations to second-order conservation laws, temporal discretizations, and non-diagonal SBP operators are described. The result is verified for steady inviscid and viscous flows.