Abstract: A set of equations governing the fluid flow along a given S₁ stream surface is solved by the use of artificial compressibility and with a new iterative algorithm between the two variables stream function ψ and density ρ.The set of equations consists of a stream function equation and a new first-order partial differential equation for density derived from the continuity equation, momentum equation, and Bernoulli's equation.A strongly implicit approximate factorization algorithm (SI) of multidimensional partial differential equations is used in numerical solutions of this set of equations.By the method proposed in this paper, the problem of non-uniqueness of density in traditional stream function method is avoided.A number of transonic cascades are calculated and given herein to demonstrate the capability of the present method.