The general algorithm employed by the STAGE codes has been well documented by Rai  and Gundy-Burlet, et al.  and therefore only a brief description of recent modifications to the algorithm will be given here. The Euler/Navier-Stokes equations form the basis for the approach and are solved zonally in the turbomachinery domain. The Euler equations are applied in regions where viscous effects are negligible, while the Navier-Stokes equations are used in regions where viscous effects are important, such as near blade, vane, hub and tip surfaces. The two-dimensional algorithm has been modified to include stream tube contraction terms. Following the work of Rangwalla, et al. , the two-dimensional Navier-Stokes equations in cartesian coordinates with stream-tube contraction terms are
where h is the normalized area of the streamtube and
Both viscous and inviscid source terms are included in this formulation. Note that the equations reduce to the two-dimensional Navier-Stokes equations for the case where the stream-tube height does not vary. These terms model the midspan stream-tube contraction that occurs because of three-dimensional effects such as end-wall boundary layer growth, tip leakage effects and hub corner stall that cannot be directly captured by a two-dimensional algorithm.
For both the two- and three-dimensional algorithms, the governing equations are cast in the strong conservative form and are integrated using a fully implicit, third-order-accurate upwind-biased scheme. A Newton-Raphson subiteration scheme is used at each time step to solve the nonlinear finite difference equations which correspond to a fully implicit scheme. The kinematic eddy viscosity is computed using Sutherland's Law and the turbulent eddy viscosity is approximated using the Baldwin-Lomax turbulence model.