where a , b , c , a , b , and c * are arbitrary constants for 

 p p p q q q 



regions p and q in the x- and y-directions, respectively, and a^ and a„ 

 are coordinates in the computational space. This transformation allows all 

 derivatives to be centered in the computational space. Many stability prob- 

 lems commonly occurring in variable grid schemes are eliminated when using 

 this transformation since the grid in real space varies smoothly and the co- 

 ordinates and their first derivatives are continuous at the boundaries between 

 regions. 









































_ 1 











T 













' 







Ay 



^1 



:Ax 



«i 









































\ 











Aa^ 



' 







■♦♦■ 



♦ 



Aa-i 



REAL SPACE COMPUTATIONAL SPACE 



Figure 2 . An example of variable grid 



9. The partial differential equations governing the different processes 

 are solved by finite difference integration on a grid of spatial points. A 

 right-handed coordinate system is used with the x-coordinate increasing in the 

 offshore direction and the y-coordinate increasing along the shoreline with 

 the ocean to the right. The partial derivative of an arbitrary variable s 

 in region p is 



9s 

 8x 



9s 



9a, 



(3) 



where 



P^ = 



9x_ 

 da. 



c -1 



= b c a, 

 P P 1 



(4) 



* For convenience, sjonbols and abbreviations are listed in the Notation 

 (Appendix A) . 



12 



