Solution of the Finite-Difference Equations 



61. The finite -difference equations based on the shallow-water equa- 

 tions as given by the system of Equations 12 and 13, together with the 

 boundary conditions given by Equations 16 and 17 and the junction conditions 

 given by Equations 19 and 21, constitute a system of 2N nonlinear algebraic 

 equations in 2N unknowns, where N is the total number of nodes. In this 

 system of equations, the values of the variables at time step t^ are known and 

 may be treated as constants. The unknowns consist of all the variables with 

 superscript (J+1) . Because the number of equations is equal to the number of 

 unknowns, the system is determinate. 



52. For convenience, let the entire system of finite -difference 

 equations, consisting of the external boundary conditions, interior nodes, and 

 junction conditions, be represented by the system of Equations 22. 



F21-1 (z,>^ei^^^z,.,^*^!?,.,J^^) = o 



^2i (z^J^\0>\z^J*\Q^J*') =0 (22) 



•^2Ar-i ' ^2N-i ' QiN-i I ^2w / Qzn ) = 



■''2*' (^2W-1/ !?2W-1» ^2W iQzS I- 



63. Routine methods for the solution of nonlinear systems do not exist. 

 For the present model, the generalized Newton iteration method is applied to 

 solve the nonlinear equations. The equation system involves 2W unknowns, but 

 each equation contains a maximum of four unknowns , which can be of great 

 advantage in devising efficient computational schemes. 



64. Let J?2i-i ^^^ ^i be the residual at the k*'^ cycle of the system of 

 Equations 12 and 13 corresponding to F2i-i and ¥2^.. Then, according to the 

 generalized Newton iteration method, the residuals and partial derivatives are 



25 



