A finite-volume method for the numerical computation of flows with non-equilibrium thermodynamics and chemistry is presented. Flux-splitting procedures are developed for the fully-coupled equations involving fluid dynamics, chemical production and thennodynamic relaxation processes. An analysis of the governing equations is performed, whereby theoretical conditions are developed under which the Euler equations retain the homogeneity property, F = AQ, where F is the flux-vector, Q is the vector of conservative dependent variables (including all the species densities and vibrational energies), and A is the Jacobian matrix ∂F/∂Q. This property, previously only shown to exist for thermally perfect gases, is utilized in the development of flux-split algorithms for non-equilibrium flows. These algorithms have been recently shown to give very accurate and robust solutions for high speed flows in chemical equilibrium. The analysis presented here develops new forms of flux-vector-split and flux-difference-split algorithms for flows with non-equilibrium thermodynamics and chemistry. The algorithms may be embodied in a fully-coupled, implicit, large-block structure, including all the species conservation and energy production equations. Several numerical examples are presented, including high-temperature shock tube and nozzle flows. The methodology is compared to other existing techniques, including spectral, implicit central-differenced and uncoupled procedures, and favorable comparisons are shown regarding accuracy, shock-capturing and convergence rates. © 1988.