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In this paper, we develop a fully conservative, positivity-preserving, andentropy-bounded discontinuous Galerkin scheme for simulating the chemicallyreacting, compressible Euler equations with complex thermodynamics. Theproposed formulation is an extension of the conservative, high-order numericalmethod previously developed by Johnson and Kercher [J. Comput. Phys., 423(2020), 109826] that maintains pressure equilibrium between adjacent elements.In this first part of our two-part paper, we focus on the one-dimensional case.Our methodology is rooted in the minimum entropy principle satisfied by entropysolutions to the multicomponent, compressible Euler equations, which was provedby Gouasmi et al. [ESAIM: Math. Model. Numer. Anal., 54 (2020), 373--389] fornonreacting flows. We first show that the minimum entropy principle holds inthe reacting case as well. Next, we introduce the ingredients required for thesolution to have nonnegative species concentrations, positive density, positivepressure, and bounded entropy. We also discuss how to retain the aforementionedability to preserve pressure equilibrium between elements. Operator splittingis employed to handle stiff chemical reactions. To guarantee satisfaction ofthe minimum entropy principle in the reaction step, we develop anentropy-stable discontinuous Galerkin method based on diagonal-normsummation-by-parts operators for solving ordinary differential equations. Thedeveloped formulation is used to compute canonical one-dimensional test cases,namely thermal-bubble advection, multicomponent shock-tube flow, and a movinghydrogen-oxygen detonation wave with detailed chemistry. We find that theenforcement of an entropy bound can considerably reduce the large-scalenonlinear instabilities that emerge when only the positivity property isenforced, to an even greater extent than in the monocomponent, caloricallyperfect case.

Positivity-preserving and entropy-bounded discontinuous Galerkin method  for the chemically reacting, compressible Euler equations. Part I: The  one-dimensional case