below a specified tolerance value. The values of the variables found in the 

 terminal iteration cycle will be taken as the values of the variables for the 

 time step J+1 , and the computation will be advanced to time step J+2 . 



65. A significant feature of the system of Equations 22 is that the 

 matrix of coefficients has a maximum of four non-zero elements in any row, 

 because each equation involves at most four of the 2N unknowns. For a single 

 channel, the non-zero elements are banded around the main diagonal, giving a 

 band width of five. For multiple channels, the matrix band width is greater 

 and depends on the node niombering scheme. Nevertheless, the matrix will be 

 sparse and banded around the main diagonal. Band width can and should be 

 minimized by numbering nodes to minimize the differences between any two 

 junction node numbers. This property of the linear system can be used to 

 great advantage in devising an efficient solution method, and such a method 

 has been implemented in DYNLETl . 



66. Application of the Newton iteration procedure requires evaluation 

 of the coefficients of the linear system. The coefficients are the values of 

 the partial derivatives of the function F at each cycle of iteration. The 

 evaluation of the partial derivatives of the nonlinear algebraic system will 

 be considered in three parts: interior points, external boundary conditions, 

 and junction conditions. 



Interior equations 



67. The finite-difference equations arising from application of the 

 shallow-water equations to a channel segment located between nodes i and i+1 

 will be numbered F2i-i ^nd F2i, and are given as: 



^2i-i(^/' of' ^i*i' Oili) = i?2*-i 

 Fzi^zf' of. Zi^i, OL) = i?2i 



(25) 



Ay 



(Oi.i - <3i)e + ^ A.^i + q - <3r^*'Ay 



(26) 



27 



