177. The Crank-Nicholson implicit scheme is used (Crank 1975) in which 

 the derivative dQ/dx at each grid point is expressed as an equally weighted 

 average between the present time step and the next time step: 



3Qi -l 

 3x 2 



Qi+i - Qi Qi+i - Qi 



Ax Ax 



(29) 



Substitution of Equation 29 into Equation 1 and linearization of the wave 

 angles in Equation 2 in terms of dy/dx results in two systems of coupled 

 equations for the unknowns y{ and Q{ : 



y{ = B'(QJ - Qi +1 ) + y Ci (30) 



and 



Qi = Ei(y{ +1 - y{) + F ± (31) 



where 



B' = At/[2(D B +D' c )Ax] 



yc A = function of known quantities, including q' L and q A 



E L = function of the wave height, wave angle, and other 

 known quantities 



F i = function similar to E ± 



178. The so-called double-sweep algorithm is used to solve Equations 30 

 and 31. Details of the solution procedure are given in Kraus and Harikai 

 (1983), Hanson (1987), Hanson and Kraus (1986b), and Kraus (1988c). 



Lateral Boundary Conditions and Constraints 



179. GENESIS requires specification of values for Q at both boun- 

 daries, cell walls 1 and N+l , at each time step. The importance of the 

 lateral boundary conditions cannot be overemphasized, as calculated shoreline 

 positions on the interior of the grid depend directly upon them. The most 

 ideal lateral boundaries are the terminal points of littoral cells, for 



86 



