Strictly convex entropy and entropy stable schemes for reactive Euler equations

This paper presents entropy analysis and entropy stable (ES) finite difference schemes for the reactive Euler equations with chemical reactions. For such equations we point out that the thermodynamic entropy is no longer strictly convex. To address this issue, we propose a strictly convex entropy function by adding an extra term to the thermodynamic entropy. Thanks to the strict convexity of the proposed entropy, the Roe-type dissipation operator in terms of the entropy variables can be formulated. Furthermore, we construct two sets of second-order entropy preserving (EP) numerical fluxes for the reactive Euler equations. Based on the EP fluxes and the Roe-type dissipation operators, high-order EP/ES fluxes are derived. Numerical experiments validate the designed accuracy and good performance of our schemes for smooth and discontinuous initial value problems. The entropy decrease of ES schemes is verified as well.