Resolution of the water surface profile equation

The water surface profile equation [11] is solved by Newton’s method. The initial value is taken equal to:

Zio =

In our case, the function f(Z) does not have a continuous first derivative, because the geometry of the computational section is only known at certain points. In order to avoid any adverse effects on the computation convergence, it is limited by a bisection algorithm.

Starting bisection algorithm values are:

Zmin = Zci
Zmax = Hj + DH(Zci)

In a Newton iteration l two cases can occur:

1) The calculated value Zil is in the interval (Zmin, Zmax).

  • This value is retained
  • if f(Zil) > 0 then Zmin = Zil
  • if f(Zil) < 0 then Zmax = Zil

since f is a decreasing function.

2) The calculated value Zil is not in the interval (Zmin, Zmax). Replace Newton’s iteration by a bisection algorithm:

Zil =

Similarly, if one detects a breakdown of the Newton algorithm due to the non continuity of the derivative of f(Zi), one also adopts the bisection algorithm.