large wave height would necessitate a small value of At . Although there are 

 calculation strategies to overcome this problem, it is inefficient to use an 

 explicit solution scheme to solve for shoreline position in a general case. 



173. Equation 1, of which Equation 24 is a special case, can also be 

 solved using an implicit scheme in which the new shoreline position depends on 

 values calculated on the old, as well as the new, time step. The main 

 advantage of the implicit scheme is that it is stable for very large values of 

 Rg . The disadvantages of the implicit solution scheme are that the program, 

 boundary conditions, and constraints become more complex, as compared with the 

 explicit scheme. These disadvantages are, however, not considered to be 

 major. 



174. An implicit solution scheme is used in GENESIS to solve Equa- 

 tion 1, as developed by Kraus and Harikai (1983) based on a method given by 

 Perlin and Dean (1978). Kraus and Harikai also showed for a specific example 

 that the magnitude of the stability parameter gives an indication of numerical 

 accuracy of the solution. Roughly speaking, for values of R s less than 10, 

 the numerical error equaled the magnitude of R s expressed as a percentage. 

 Above the value of 10, the error increased at a greater than linear rate with 

 Rg . GENESIS calculates the value of R s at each time step at each grid 

 point alongshore and determines the maximum value . If R s > 5 for any grid 

 point, a warning is issued. The implicit finite difference scheme is dis- 

 cussed further below. 



Grid System and Finite Difference Solution Scheme 



Staggered grid 



175. In GENESIS calculated quantities along the shoreline are dis- 

 cretized on a staggered grid in which shoreline positions y i are defined at 

 the center of the grid cells ("y-points") and transport rates Q A at the cell 

 walls ( "Q-points" ) , as shown in Figure 18. The left boundary Is located at 

 grid cell 1, and the right boundary is at cell N. In total there are N 

 values of the shoreline position, so the values of the initial shoreline 

 position must be given at N points. There are N+l values of the longshore 

 sand transport rate since N+l cell walls enclose the N cells; values of 



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