As a simple check, insertion of Q. 1 from Equation 10 into Equation 5 gives 

 the following desired result for the corrected shoreline position: 



* 



y ± = ys i (1D 



Again, this is the mathematical statement of the shoreline constraint. 



56. After Q. and y. are obtained, calculation moves to the next 

 grid point to determine y! 1 in similar manner. Calculation proceeds from 

 cell to cell in the downdrift direction along the seawall until either a plus 

 cell or the end of the seawall is encountered. If a plus cell is encountered 

 calculation of the shoreline position continues without necessity for correc- 

 tion until another minus point or the end of the seawall is encountered. 



57. On the other side of the original minus cell, where the transport 

 rate is in the negative x-direction, analogous corrections are made to trans- 

 port rates as described above, i.e., to Q. . This allows determination of 

 y! The calculation then proceeds from cell to cell in the downdrift 

 direction. 



58. The programs YSEXP and CORRE, discussed in Part V and listed in Ap- 

 pendix A, calculate shoreline change with the explicit numerical scheme for a 

 beach backed by a seawall. 



Implicit Numerical Model 



59. Compared to the straightforward development for the explicit 

 scheme, as presented in the previous subsection, representation of the seawall 

 constraint in an implicit numerical scheme is extraordinarily complex. In an 

 implicit scheme, values of the new Q i are solved for simultaneously, over 

 the whole grid, in terms of the old Q i and other quantities. Thus, in 

 checking to determine whether the seawall constraint has been violated, the 

 time level halfway between the old and new time levels is involved. In the 

 explicit method, transport rates of only those cells in contact with a seawall 

 need to be corrected; in the implicit scheme, correction of one cell will af- 

 fect all cells downdrift (whether in front of the seawall or not) and thus all 

 cells downdrift require correction. Correction of all downdrift cells in- 

 creases the complexity and execution time of the computation. 



60. As already discussed, the direction of sand transport must be 



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