An implicit numerical scheme is developed to compute the transient one-dimensional flow of a two-phase mixture described by an unequal phase velocity (nonhomogeneous) model. This method is based on the modified FLASH-4 technique, but it suffers few of the drawbacks of other solution techniques recently used in the RELAP 4 and RETRAN computer codes. Significant advantages of the method are its consistency, stability, and ease of programming for complicated flow networks. The numerical scheme has been incorporated into a computer code and used to calculate three flow situations: a single-phase gas (ideal gas shock tube) a single-phase liquid (subcooled water blowdown) a two-phase blowdown (the Edwards experiment).