New flux-split algorithms are developed for high velocity, high-temperature flow situations, when finite-rate chemistry and non-equilibrium thermodynamics greatly affect the physics of the problem. Two flux-vector-split algorithms, of the Steger-Warming and of the Van Leer type, and one flux-difference-split algorithm of the Roe type are established and utilized for the accurate numerical simulation of flows with dissociation, ionization, and combustion phenomena. Several thermodynamic models are used, including a simplified vibrational non-equilibrium model and an equilibrium model based upon refined statistical mechanics properties. The framework provided is flexible enough to accommodate virtually any chemical model and a wide range of non-equilibrium, multi-temperature thermodynamic models. A theoretical study of the main features of flows with free electrons, for conditions that require the use of two translational temperatures in the thermal model, is developed. Interesting and unexpected results are obtained, because acoustic wave speeds of the symmetric form u±α no longer appear. A simple but powerful asymptotic analysis is developed which allows the establishment of the fundamental gas-dynamic properties of flows with multiple translational temperatures. The new algorithms developed demonstrate their accuracy and robustness for challenging flow problems. The influence of several assumptions on the chemical and thermal behavior of the flows is investigated, and a comparison with results obtained using different numerical approaches, in particular spectral methods, is provided, and proves to be favorable to the present techniques. Other calculations in one and two space dimensions indicate large sensitivities with respect to chemical and thermodynamic modeling. The algorithms developed are of sufficient generality to begin to examine these effects in detail. Preliminary numerical simulations are performed using elementary modeling of transport phenomena. / Ph. D.