Defining F and G as follows, with the subscript indicating the previous time-step values:

(17–39)

we have the following discretized film mass and momentum equations, with subscript +1 representing the current time-step values, and the time step being used for film computations.

(17–40)

Film height and velocity are then computed as follows,

(17–41)

The above set of equations complete the explicit differencing scheme in which film height is computed first, based upon values of evaluated at the previous film time step; then film velocity is calculated using the latest film height and values of evaluated at the previous film time step.

With the explicit method, evaluations of and are done based upon the previous time step film height and velocity vector. In order to improve accuracy of this explicit method, a first-order implicit method is introduced in which and values are updated during an iterative loop within a film time step. This new method can be described as a predictor-corrector procedure. At the beginning, that is, the predictor step, the explicit scheme is used to compute film height and velocity vector,

Predictor:

(17–42)

The superscript indicates the first step in the iteration loop.

Corrector:

The latest film height and velocity vector are used to update and ; then film height and velocity are recomputed,

(17–43)

The superscript +1 and represent the current and the previous iterations, respectively. The iteration procedure ends with the following convergence criteria:

(17–44)

where represent each component of the velocity vector.

With the above discussed explicit and first-order implicit methods, time differencing is only first-order accurate. A second-order implicit method is introduced below. The iterative procedure is very similar to that used in the first-order implicit method, but two time-steps ( and –1) are used for time differencing in the predictor step.

Predictor:

(17–45)

Corrector:

(17–46)

The iteration procedure ends with the following convergence criteria:

(17–47)

where represent each component of the velocity vector.

The discretized expressions for film energy and passive scalar equations can be derived in a similar way as above.