and 



£ * - (D B ! 2 D C ) [ H2C « COS ^ir] b (26) 



As Equation 24 is a diffusion- type equation, its stability properties are well 

 known. The numerical stability of the calculation scheme is governed by: 



At(e a + € 2 ) 



R s - (27) 



(Ax) 2 



The quantity R s is known as the Courant number in numerical methods; here it 

 is called the stability parameter. The finite difference form of Equation 24 

 shows that Ay -v At/ (Ax) 2 . 



171. Equation 24 can be solved by either an explicit or an implicit 

 solution scheme. If solved using an explicit scheme, the new shoreline 

 position for each of the calculation cells depends only on values calculated 

 at the previous time step. The main advantages of the explicit scheme are 

 easy programming, simple expression of boundary conditions, and shorter 

 computer run time for a single time step as compared with the implicit scheme. 

 A major disadvantage is, however, preservation of stability of the solution, 

 imposing a severe constraint on the longest possible calculation time step for 

 given values on model constants and parameters. If an explicit solution 

 scheme is used to solve the diffusion equation, the following condition must 

 be satisfied (Crank 1975): 



R s < 0.5 (28) 



172. If an explicit solution scheme is used and the value of R s 

 exceeds 0.5 at any point on the grid, the calculated shoreline will show an 

 unphysical oscillation that will grow in time if R s remains above 0.5, 

 alternating in direction at each grid point. The quantities e l and e z can 

 change greatly alongshore since they depend on the local wave conditions. 

 Assuming that the grid cell spacing is fixed by engineering requirements , a 



83 



