Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations of Motion for One-Dimensional Unsteady Flow in Open Channels and Through Control Structures
If more than two branches are included at a junction, the bandwidth no longer remains constant. The simplest case is that of a three-branch stream system such as shown in figure 20. The NMCI (section 13.6) fragment for FEQ for this three-branch network is the following:
The junction is a simple junction in which the water-surface elevation is the same at all three nodes.
The user may specify the boundary node used to initiate the development of the Jacobian matrix. If node U1 is specified as the initial node, then the pattern of the Jacobian matrix is the following:
The subscripts on Q and y for the columns above correspond to the locations shown in figure 35. An X denotes an element that is nonzero, whereas those elements that are known to be zero are left blank in the matrix. The zeros shown in the matrix are initially zero but may become nonzero. The columns are labeled with the variable involved in that column. This pattern is almost banded. In this simple three-branch network, the equation in row 15 and the column under Q 8 modifies the banded pattern of the matrix. If branch 2 had more computational elements, the effect would have been more dramatic. With more branches in the network, the pattern of the matrix would have been a central band with vertical and horizontal spikes projecting out of the band at intervals. Such a matrix has been called a profile matrix or, more aptly, a skyline matrix because the pattern above the diagonal resembles the appearance of a city skyline. The pattern of explicit zeros has been placed in the matrix as an indication of how the matrix will be represented and solved. The discussion of the solution methods in section 9 indicates that only the nonzero and potentially nonzero parts of the matrix must be stored. The solution process is such that the known zero parts of the matrix will remain zero.