Node Equations

In nodes a continuity equation covering the surrounding h- and q-grid points is set up. the node with three reaches connected to it gives a close look at the grid points surrounding a node.

Figure: Node with three reaches connected to it.

The finite difference approximation of the continuity equation around the node is given by:

\[ \frac{H^{n+1} - H^{n}}{\Delta t}A_{fl} = q_{I}^{n+\frac{1}{2}} - q_{O}^{n+\frac{1}{2}} \Rightarrow \frac{H^{n+1} - H^{n}}{\Delta t}A_{fl} = \frac{1}{2}\left(q_{A,n-1}^{n} + q_{B,n-1}^{n} - q_{C,2}^{n}\right) + \frac{1}{2}\left(q_{A,n-1}^{n+1} + q_{B,n-1}^{n+1} - q_{C,2}^{n+1}\right) \]

Equation: Node continuity equation, time-centred between inflow and outflow contributions.

where

q_ISum of inflows

q_OSum of outflows

Dt   Time step