The authors present programming and computational techniques that exploit the capabilities of parallel supercomputers for a large-scale compositional simulation. A novel algorithm called sequential staging of tasks (SST) that can take full advantage of parallel-vector processing to speed up the solution of a large linear system is introduced. The effectiveness of SST is illustrated with results from computer experiments conducted on an IBM 3090-600E. It is demonstrated that compositional reservoir simulation jobs performed on parallel supercomputers can be done over 100 times faster than on conventional mainframe computers.>