Expressing Equation 13 in fully implicit finite difference form (i.e., all Q values are at the t+Dt time step, or q=1:

Note: Henceforth all bold variables are the mean conduit values:

Equation 18:

 

Qt+Dt = Qt - g·k·Dt·|Vt|Qt+ Dt /R4/3

  - V (DA/Dt)·Dt + Qt+ Dt ·[1/Aup-1/Adn]/L

  - g·A·[(Hdn-Hup)/L]·Dt      (18)

Solving equation 18 for  1gives the final finite difference form of the fully implicit dynamic flow equation (excluding terms for Sc and Se),

Equation 19:

is the average area time derivative from time step n.