## Numerical analysis for the solution of the work equation

Mathematically seen the solution of Eq. 2‑83 is a typical initial value problem. Based on a well-known water level at the profile i, the water level at the profile i +1 can be calculated. In relation to the water level the Eq. 2‑83 is non-linear, so that it can not be solved directly and therefore an iterative solution algorithm is required.

The best-known engineering methods to solve this equation is the **standard-step method of CHOW** (1959). Here initially the water level is estimated (**Predictor**), then the water level is adjusted by the calculation (**Corrector**) until Predictor and Corrector only differ with a negligible small error.

Better convergence is obtained, however, with differentiated procedures, such as the method of continuous interval generation (NAUDASCHER, 1992) or the Newton method.

The calculation can be done both with and against the flow direction. At supercritical flow it is always made in the direction of flow, as disturbances at the profile continue only in the direction of tail water. At subcritical flow calculation can be basically made in both directions. **However**, a calculation in the direction of flow is unfavourable, as in this case the estimated amount of hydrostatic water level z_{sp} at the profile i at the beginning still has effects on the last profile of the tail water in the calculated section. In contrast upstream, depending on the backwater situation, the water level at the profile i does not affect the flow situations at a certain profile in the upstream any more.