  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.
