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The system of unsteady, three-dimensional, partial differential equations used to simulate the inviscid flow of air in chemical nonequilibrium is approximated by a set of factored, finite-volume difference operators where the effect of chemical production is also contained in the factorization. The method is similar to that of Rizzi and Bailey, except for the emphasis on vector-matrix re-formulation designed to be suitable for the special architecture of modern advanced computers (e.g., the Illiac IV, CDC 7600, or Star). The systematic application of the operators yields a second-order accurate numerical algorithm. The method is programmed in the vector Fortran-like language called CFD developed in the CFD Branch at the NASA Ames Research Center. Translators are available for this language which allows debugging on serial computers, but all results for the examples given were obtained from the Illiac. The problem described is a numerical simulation of the flow in the high temperature stagnation region of a reentering space shuttle orbiter flying at large angles of attack (~40°). Capability for treating arbitrary geometry in a flow containing subsonic, sonic, and supersonic regions is demonstrated by this method. The air chemistry is described by a five-reaction model which includes the three dissociation reactions for N2,02, and NO and the two rearrangement reactions involving NO. The vector-matrix formulation and the unique disk-memory mapping results in extremely efficient data management for the architecture of the Illiac and makes maximum use of the Illiac's "data crunching" capability. Comparative running times are given for the Illiac IV and the CDC 7600.