Entropy conservative high-order fluxes in the presence of boundaries

TL;DR

This paper introduces a new method for constructing high-order entropy conservative fluxes that work well with boundaries, improving the stability and accuracy of finite-volume/finite-difference schemes.

Contribution

It generalizes existing high-order entropy conservative fluxes to non-centered combinations suitable for boundary conditions and proves a Lax-Wendroff theorem for these fluxes.

Findings

The proposed fluxes are verified through numerical simulations.

The method improves handling of boundary conditions in entropy stable schemes.

Theoretical proofs support the stability and accuracy of the new fluxes.

Abstract

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.