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:
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),
is the average area time derivative from time step n.