When a pseudo time method is selected, an advanced form of implicit under-relaxation is applied that adjusts the relaxation factor dynamically during the simulation according to the flow field behavior. After introducing the pseudo time method into the generic transport equation, the integral form of governing equation is of the following forms:

where denotes the pseudo time. Note that the pseudo time terms vanish and the original form of the equations is recovered when .

After discretizating the governing equation using the finite volume method, the algebraic form of the steady-state equation is obtained as

(23–101)

where denotes the value of at the previous iteration, and is the pseudo time step size that can be computed using the local or global time step method (as described in the sections that follow). For the local time step method, Equation 23–101 is extended to transient flow in a straightforward manner.

To learn how to apply under-relaxation using the pseudo time methods, refer to Performing Calculations with a Pseudo Time Method.