In order to find a simpler numerical method than that given in sec$.12-4($Chow$).$

$1-$Derive the equation of $GVF$ for variable discharge in a form similar to the direct step

method $($Sec$.10-3$ Chow$)$ and the standard step method $($Sec$\mathrm{.}10-4$ Chow$).$ See also

Vennard section $10\mathrm{.}9$

$2-$Indicate which of the two methods is preferable for numerical integration.

$3-$How do you account for the energy loss due to turbulent mixing of the added water and

water in the channel?

$4-$Can we integrate Eq$\mathrm{.}12\mathrm{.}4$ using the trapezoidal rule, i$.$e$.$

$y_{i+1}=y_i+x\frac{y_i^{\text{'}}+y_{i+1}^{\text{'}}}{2}$