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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:

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Equation18
Equation18
Equation 18:

Image Modified 

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

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Equation19
Equation19
Equation 19:

Image Modified

Image Modifiedis the average area time derivative from time step n.

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pageEXTRAN Theory - Dynamic Wave Solution