Entropy conservative high-order fluxes in the presence of boundaries

In this paper, we propose a novel development in the context of entropy stable finite-volume/finite-difference schemes. In the first part, we focus on the construction of high-order entropy conservative fluxes. Already in [LMR2002], the authors have generalized the second order accurate entropy conservative numerical fluxes proposed by Tadmor to high-order ($2p$) by a simple centered linear combination. We generalize this result additionally to non-centered flux combinations which is in particular favorable if non-periodic boundary conditions are needed. In the second part, a Lax-Wendroff theorem for the combination of these fluxes and the entropy dissipation steering from [Klein2022] is proven. In numerical simulations, we verify all of our theoretical findings.