Stability Analysis and Structure Preserving Schemes for the Reactive Euler Equations with a New Equation of State

Abstract

This paper is concerned with the multi-dimensional reactive Euler equations under the new equation of state (EoS) recently proposed. We show that under this EoS the classical thermodynamic entropy is a strictly convex entropy function in general dimension. Based on this we further prove that the reactive Euler equations satisfy the stability conditions for hyperbolic relaxation systems, which guarantee the existence of zero relaxation limit. The eigen-decompositions of the Jacobian matrices in two and three dimensions are also provided. Moreover, we develop a positivity preserving and oscillation-free entropy stable discontinuous Galerkin scheme by adapting that for the EoS of ideal gas to the newly proposed one. A key step in doing so is to prove that the HLL (Harten-Lax-van Leer) flux is entropy stable, which is established by tactfully using a natural assumption on a function in the EoS. The high convergence orders stable entropy, no oscillation and positivity of the scheme are demonstrated with numerical examples.