On strictly convex entropy functions for the reactive Euler equations

Abstract

This work is concerned with entropy functions of the reactive Euler equations describing inviscid compressible flow with chemical reactions. In our recent work (W. Zhao, Math. Comput. 91 (2022) 735–760.) we point out that for these equations as a hyperbolic system, the classical entropy function associated with the thermodynamic entropy is no longer strictly convex under the equation of state (EoS) for the ideal gas. In this work, we propose two strategies to address this issue. The first one is to correct the entropy function. Namely, we present a class of strictly convex entropy functions by adding an extra term to the classical one. Such strictly entropy functions contain that constructed in (W. Zhao, Math. Comput. 91 (2022) 735–760.) as a special case. The second strategy is to modify the EoS. We show that there exists a family of EoS (for the nonideal gas) such that the classical entropy function is strictly convex. Under these new EoS, the reactive Euler equations are proved to satisfy the Conservation-Dissipation Conditions for general hyperbolic relaxation systems, which guarantee the existence of zero relaxation limit. Additionally, an elegant eigen-system of the Jacobian matrix is derived for the reactive Euler equations under the proposed EoS. Numerical experiments demonstrate that the proposed EoS can also generate ZND detonations. Extension of the present results to high dimensions is direct.