---
title: "Finite Difference Equations"
canonical: "https://help.innovyze.com/space/xps/19663249/Finite+Difference+Equations"
format: markdown
---
Expressing [**Equation 13**](https://help-innovyze.atlassian.net/wiki/spaces/xps/pages/19661241) 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:

> Macro (anchor)

**Equation 18**:

![image](media://8a329517-7076-4d2d-b5ca-321d02c715ec)

Qt+Dt = Qt - g·k·Dt·|V<sub>t</sub>|Q<sub>t+ </sub><sub>D</sub><sub>t</sub> /**R**<sup>4/3</sup>

  - **V** (DA/Dt)·Dt + Q<sub>t+ </sub><sub>D</sub><sub>t</sub> ·[1/A<sub>up</sub>-1/A<sub>dn</sub>]/L

  - g·**A**·[(H<sub>dn</sub>-H<sub>up</sub>)/L]·Dt      (18)

Solving equation 18 for > Macro (inline-media-image)

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

> Macro (anchor)

**Equation 19**:

![image](media://8380ff96-22c6-416a-a652-01b52ed2f5ea)

> Macro (inline-media-image)

is the average area time derivative from time step n.