Two different methods for the numerical simulation of steady, invisced, nonequilibrium reactive flow governed by Euler equations augmented by a 5-species 17-reaction finite-rate dissociation model are studied. The employed approximations are based on a conservative mixed finite-volume/finite- -element upwind formulation. The study concentrates on the efficiency of the pseudo-time integration methods as iterative algorithms. The iterative properties of the basic method are demonstrated by various computations of flow fields arount blunt bodies. In particular, the effect of varying the global Damkholer number (i.e. the size of the geometry ) is illustrated. A variant of the basic method is proposed. We know that for steady (external) flows with uniform free-stream, the total enthalpy is constant throughout the domain. Hence, an algorithm which conserves this quantity has been implemented in which the energy equation ( in differential form ) is not solved but replaced by its (algebraic) first integral. The extension of existing flux-vector splittings (FVS) to this context where one less PDE is solved and one algebraic constraint is enforced, is examined in Section 4 in which the FVS is demonstrated to be proper. Finally, the efficiency of the proposed new approach is assessed by numerical experiments for first and second order approximation schemes.