Equation (36) can be rewritten as 



(1 + U + V + Zl + Z2) yf J - (U)yf } .- (V)y"':} , 



- '"<]-! - '"<].! = 'A"A«E>,-J (37) 



where U = AtR. .S3. . 



■ 3 J 1 > J 



Zl = (ji) R, ,Const6, . 

 Z2 = (^)R. .ConstS. .,.• 



4.1 



Equation (37) is a weighted, centered scheme in which y-j^j is computed 

 using a weighting of itself and its four adjacent grid "neighbors". The 

 weighting factors (U, V, Zl, and 12) are functions of the wave climate, the 

 slope between contours, and the variables included in the original formula- 

 tion. An investigation of a small gridded system demonstrated that by writing 

 simultaneous equations, one for each yi,j, a banded matrix results. This 

 matrix can be solved by LEQTIB, one of the available routines from the 

 International Math and Statistics Library (IMSL). A schematic representation 

 of the matrix A which results from the matrix equation [A][y3 = [B] is 

 presented in Figure 5. In this schematic, the large zeros represent 

 triangular corner sections of all zeros and the 0...0 represents bands of 

 zeros, the number of which is dependent on the number of contours simulated 

 (the number of zero bands between either remote nonzero bands and the 

 tridiagonal nonzero bands equals two less than the number of contours modeled 

 (in both the upper and lower codiagonals of the matrix)). An inspection of 

 the subscripts in equation (29) yields the reason the zero bands are required. 

 The more j values (contours) used, the more y grids there are along any 

 perpendicular to shore. This causes zeros to appear in the matrix between 

 bands as the weighting factors await being used to operate on y""*"^ (i-1 ,j) 

 and y""*"^ (i+1 ,j). For this reason, the expense of simulating an increasing 

 number of contours is exponential. The LEQTIB routine, utilizes banded 

 storage and saves both storage and computation time; however, the routine has 

 no special way of handling the interior zero bands. One refinement which 

 would save computation time would be to develop an algorithm to solve and 

 store the matrix by taking advantage of these inner zero bands; however, it 

 is beyond the scope of this project. 



Of course, the matrix requires boundary values on longshore extrem- 

 ities and on both onshore and offshore boundaries. The longshore boundary 

 conditions are treated by modeling a sufficient stretch of shoreline so that 

 effects of a structure's presence are minimal. The y values along these 

 boundaries can therefore be fixed at their initial locations. In the 

 onshore-offshore direction, boundaries are treated quite differently. The 



25 



