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The parallelization of a NASA Navier-Stokes simulation code is carefully and systematically investigated. Based on numerical, dependence and partition analysis, three different levels of parallelization have been proposed and implemented. The low level parallelization is at the kernel level. A previously developed parallel tridiagonal solver is used for concurrent processing. The middle level parallelization solves the multiple right hand sides of the tridiagonal systems concurrently. The high level parallelization is at the level of time step iterations. A PVM implementation of the parallelization has been accomplished. Different communication patterns and different PVM communication calls have been examined for best performance. The parallelized code has been tested on two parallel platforms, a cluster of workstations available at Louisiana State University and the IBM SP2 parallel computer available at Cornell Supercomputing Center. Experimental results confirm our analytical findings: the three level parallelization is feasible and effective. A linear speedup is measured. In addition to parallelization, through profiling and performance analysis, the original sequential code is optimized as well.