d. Solve Equation 16 for i = N to 1 , in descending order. This 

 step is the second sweep through the grid. 



e. Substitute the Q! into Equation 13 to obtain the new shore- 

 line positions, y! . 



64. The shoreline positions thus obtained at each time step must be 

 compared with the position of the seawall to determine if the seawall con- 

 straint was violated. If so, then the shoreline position and associated 

 transport rates must be corrected. In general, when making corrections to 

 satisfy the seawall constraint, it is necessary to calculate the Q! in 

 ascending order, as well as descending order, so that transport corrections 

 can be made in either direction. The above procedure must be repeated by 

 using a recurrence relation similar to Equation 16, but which allows calcula- 

 tion of Q! from the boundary condition at i = 1 . This relation has the 

 form: 



Q! = PP! Q! , + RR! (20) 



l l 1-1 l 



The quantities PP! and RR! depend on PP! . and RR! . , respectively. 

 These quantities are defined similarly to EE! and FF! in Equations 17 and 

 18, and will not be written here. Expressions for these quantities and their 

 solution scheme can be found in program YSIMP, discussed in Part V and listed 

 in Appendix A. 



65. The time evolution of y! in the implicit scheme is shown pictori- 

 ally in Figure 8a. For comparison, the analogous picture for the explicit 

 scheme is given in Figure 8b. The shoreline positions y^ are assumed to be 

 the same in both cases. In the implicit scheme, it is seen that both the pre- 

 sent values (time level n) and the future values of Q (time level n + 1), 

 entering through 3y/3t in Equation 12, are used to calculate the shoreline 

 change from y. to y! . Since the shoreline change rates 3y/3t 



are constant during the time increment At , the shoreline change over a time 

 step is a straight line. 



66. The shoreline position midway (in time) between y. and y! was 

 previously denoted as yc, . It is seen that y-points lie on a straight line 

 between two adjacent yc-points. Hence, yc-points represent possible extremes 

 in shoreline position. The important implication of this is that in the im- 

 plicit scheme the seawall constraint must be formulated in terms of the yc^ 

 and not the y^ . 



27 



