A positivity preserving and oscillation-free entropy stable discontinuous Galerkin scheme for the reactive Euler equations

Abstract

The reactive Euler equations are a basic model for fluid flows with chemical reactions. In this work, we construct a high order positivity preserving and oscillation-free entropy stable discontinuous Galerkin (DG) scheme for the reactive Euler equations. The main ingredients of the scheme include (i) entropy preserving and entropy stable fluxes to achieve entropy stability, (ii) artificial damping terms to restrain spurious oscillations near the shocks, and (iii) positivity preserving limiters to guarantee the positivity of solutions. These ingredients are compatible with each other so that our scheme simultaneously enjoys the properties of entropy stable, oscillation-free and positivity preserving. Another distinctive feature of our scheme is that it is entropy stable for both the thermodynamic and mathematical entropies. Numerical experiments validate the designed high convergence orders of the scheme and demonstrate its good performances for discontinuous problems.