which the basic assumptions of shoreline change modeling approximate condi- 

 tions at the site. Good numerical accuracy does not necessarily imply good 

 physical accuracy. For a rapid numerical solution, the time step should be as 

 large as possible. On the other hand, the numerical and physical accuracy 

 will obviously be improved if the time step is small, since changes in the 

 wave conditions and changes in the shoreline position itself (which feed back 

 to modify the breaking waves) will be better represented. Similarly, use of 

 many small grid cells will provide more detail or improved numerical accuracy 

 in the shoreline change calculation than use of fewer but longer cells, but 

 the calculation time will increase as the number of cells increases. 



Numerical stability 



170. The allowable grid spacing and time step of a finite difference 

 numerical solution of a partial differential equation such as Equation 1 

 depend on the type of solution scheme. Under certain idealized conditions, 

 Equation 1 can be reduced to a simpler form to examine the dependence of the 

 solution on the time and space steps. The main assumption needed is that the 

 angle bs in Equation 2 is small. In this case, sin20 bs - 20 bs . By Equa- 

 tion 17, 8 hs = 9 b - dy/dx , since the inverse tangent can be replaced by its 

 argument if the argument is small. The derivative of Q with respect to x 

 is required (Equation 1 or Equation 23) and, under the small -angle approxima- 

 tion, 3Q/3x a, d(20 bs )/3x -v 23 2 y/dx 2 , if it is assumed that 6 h does not 

 change with x . After some algebraic manipulation, Equation 1 (or Equation 

 23 rewritten as a partial differential equation) can be expressed as (Kraus 

 and Harikai 1983): 



£=(< 1 + e 2 )^4- (24) 



ax 2 



at 



where 



2K i 



dTTXT (h2c s^ (25) 



82 



