to each coordinate direction independently in order to maximize grid resolu- 

 tion in areas of hydrodynamic importance and to minimize computational ".ells 

 in the far field. 



Numerical Method 



10. The alternating-direction-implicit (ADI) method has been used by 

 Leendertse (1970) and others to solve the two-dimensional equations of motion. 

 When the advective terms are included in the momentum equations (Equations 1-3) 

 the ADI method has encountered stability problems. Weare (1976) indicates 

 that the problem arises from approximating advective terms with one-sided 

 differences in time, and suggests the use of a centered scheme with three 

 time-levels. WIFM employs a centered stabilizing-correction (SC) scheme which 

 is second-order accurate in space and time, and boundary conditions can be 

 formulated to the same order accuracy. Details of the SC scheme can be found 

 in Butler (in preparation) and a general development is presented in the 

 following paragraphs. 



11. The linearized equations of motion can be written in matrix form 

 as: 



U + AU + BL =0 

 t X y 



(5) 



where 



n o d o o o d 



u , A=goo , B=ooo 



V o o o goo 



The SC scheme for solving Equation 5 is 



(1 + X ) U* = (1 - X - 2X ) U 

 X X y 



(1 + X ) u^"^-"- = u* + X u^"-"- 

 y y 



k-l 



(6) 



(7) 



where 



, 1 At ._!. , . 1 At „j 

 X 2 Ax X y 2 Ay y 



lA 



