Energy relaxation approximation for the compressible multicomponent flows in thermal nonequilibrium

This work concerns the numerical approximation with a finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy [Coquel and Perthame, \textit{SIAM J. Numer. Anal.}, 35 (1998), 2223--2249] to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver [Godunov, Math. Sbornik, 47 (1959), 271--306] and the HLL [Harten et al., SIAM Rev., 25 (1983), 35--61] and pressure relaxation based [Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkh\"auser, 2004] approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the scheme.