Entropy-stable in- and outflow boundary conditions for the compressible Euler equations

We propose general inflow and outflow boundary conditions for the Euler equations and prove that they are both linearly well-posed and lead to entropy-bounded solutions. Furthermore, we provide numerical boundary fluxes that enforce these boundary conditions and prove entropy stability for (entropy-stable) finite-volume schemes. The method is generalisable to most entropy-stable summation-by-parts schemes. Finally, we demonstrate their performance in various flow regimes using a second-order accurate entropy-stable finite-volume scheme. • Linearly well-posed and entropy-bounded boundary conditions for inflows, outflows and far-field boundaries. • Entropy-stable numerical boundary fluxes enforcing the proposed boundary conditions. • Numerical examples that corroborate the mathematical and numerical properties of the proposed boundary conditions.