Two Decades Old Entropy Stable Method for the Euler Equations Revisited

The objective of this paper is to prove for the first time that the entropy split scheme of Olsson and Oliger (Energy and maximum norm estimates for nonlinear conservation laws. RIACS Technical Report 94.01, 1994; Gerritsen and Olsson, J Comput Phys 129:245–262, 1996; Yee et al., J Comput Phys 162:33–81, 2000) is the recent definition of an entropy stable method for central differencing with SBP operators for both periodic and non-periodic boundary conditions for nonlinear Euler equations. The proof is to replace the spatial derivatives by summation-by-parts (SBP) difference operators in the entropy split form of the equations using the physical entropy of the Euler equations. The numerical boundary closure follows directly from the SBP operator. No additional numerical boundary procedure is required. In contrast, Tadmor-type entropy conserving schemes Tadmor (Acta Numer 12:451–512, 2003) using mathematical entropies do not naturally come with a numerical boundary closure and a generalized SBP operator has to be developed Roanocha (J Comput Phys 362:20–48, 2018).