The values VR, and A in equation 18 are weighted averages of the conduit upstream, middle and downstream values at time t and/or t+t. The values at time t+t are the values for the current iteration. At the first iteration they are equal to the previous time step’s values. Adn, Aup, Hdn, and Hup are respectively the conduit cross-sectional area and conduit depths at the upstream (up) and downstream (dn) nodes.V·(A/t) is the conduit average area time derivative and conduit average velocity. A is the change in cross-sectional area of the conduit between two time steps.

The basic unknowns in equation 12 and 18 are Qt+t, Hup and Hdn. The variables VR, and A in turn are all related to the center conduit Q and conduit end values of H. The equation relating the conduit flows to the node H is the continuity equation at a node:

Equation 21                 

Equations 13 and 21 can be solved iteratively to determine the discharge in each link and the head at each node at the end of a time-step t. The numerical integration of these two equations is accomplished by using an iterative solution using under-relaxation of the linearized momentum and linear continuity equation.

It should be noted that equation 13 has been linearized by:

  1. Using the product of Qt+t and Qt in the convective inertia term of the dynamic wave equation.
  2. Using equation 11 for calculating the value of Sf, which does not use the quadratic Q.