A finite difference representation of Equations 3-88 and 3-85 may be 

 expressed in the form 



Qi'l = ^ [qi, + f (b,,,/, +b,,3/,) (kw^ cose):';^ 



(3-89) 



At 



where , 





n + 1 



G = 1 + 



{^I'A-'^ls/S (V/.+V3A) 



(3-90) 



(3-91) 



The value of G is greater than unity for most flow conditions except 

 for the case when the flow vanishes (Q = 0). The subscripts and super- 

 scripts i and n are used to denote discrete points in space and time, 

 respectively. A schematic of the grid system used is shown in Figure 3-58. 

 It is seen that S at the new time level (t + At) is first evaluated 

 based on the known values of Q, D, S, A and b(x) lying on triangle (1) 

 and subsequently followed by an evaluation of S at the new time level 

 based on known values lying on triangle (2). The solutions at successive 

 time levels are obtained by a marching process with Q evaluated at the 

 new time level for all integer steps along x; S is evaluated for all 

 mid-integer steps along x. Width as a function of x, b(x), is taken 

 constant for all t and the total depth D is permitted to vary with 

 time. Thus the cross-section area of the basin A is a function of 

 distance along the major axis of the basin and a function of time. 



The computational scheme is a combined boundary and initial value 

 problem. At each end of the basin, it is assumed that there is no flow 

 across the boundary, thus Q = at the boundary. The initial conditions 

 assumed are that Q = and S is uniform throughout the basin. 



The scheme requires that for numerical stability, the time increment 

 specified for successive calculations At be taken less than Ax/i/g^ax> 

 where Dmax is the maximum depth (d + S) anticipated in the basin during 

 passage of a storm system. (Abbott, 1966.) 



The restriction imposed by the criterion for numerical stability 

 results in a trade-off between the resolution obtained in the solution 

 and the number of calculations involved. Decreasing Ax gives better 

 resolution of the surge, but requires a smaller At and increases the 

 number of computational steps. It is important to choose Ax small 

 enough so that reasonable resolution of the surge is obtained, but large 

 enough to reduce the computations. The choice of a Ax depends on the 

 problem involved. 



3-132 



