preserved when correction of the transport rate is made to satisfy the seawall 

 constraint. Since, in general, the transport direction can reverse along a 

 beach, in an implicit scheme the transport rate must be solved for twice, 

 starting independently from each of the two lateral boundaries. This doubles 

 the number of calculations performed, even if no corrections are required, and 

 greatly reduces the speed advantage the implicit method normally holds over 

 the explicit solution method. Kraus and Harikai (1983) discuss and compare 

 the relative efficiencies of the explicit and implicit numerical schemes for 

 the shoreline model without inclusion of the seawall constraint. A similar 

 comparison of relative efficiency, including operation of the seawall con- 

 straint, is given in the examples discussed in Part IV. 



61. The finite difference equations in an implicit scheme will be de- 

 rived for calculating shoreline change in the presence of a seawall. The grid 

 and notation are the same as those used in the explicit scheme, described in 

 the previous subsection. As the starting point, Equation 1 is rewritten to 

 give equal weight to present and future values: 



at 2 \d ax D' ax / v ; 



In finite difference form, Equation 12 becomes 



y! = B' (Q! - Q! ,) + yc. (13) 



J i i i+1 J l 



where 



yc. = y. + B (Q. - Q. +1 ) (14) 



The quantity yc. can be interpreted as the shoreline position midway between 

 y- and y! ; it is known since it only contains values at the present time 

 step and input data. The quantity B' = At/ (2D' Ax) differs from the un- 

 primed version in that it contains the depth of closure at the new time step, 

 which can be calculated from the new wave conditions. 



62. It is possible to solve Equation 13 by an iterative procedure be- 

 tween the y! and the Q! , as done for example, by Le Mehaute and Soldate 

 (1978). A computationally faster approach is to express the Q! in terms 

 of the y! through linearization of Equation 2. Such a linearization is 



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