f * f^ = (15) 



3x 3y 



If partial derivatives in both the x- and y-directions are estimated using 

 central differences about the point i-l/2,j , then an approximate form of 

 Equation 15 can be written as 



Fl ' i 1 L Fl ''" 1 ' H ( Gi ' l " l ^ y Gl ' 1i '"' )-' i -">( Gl " H 2^ Gl "'" 1 ) = <16) 



The partial derivative 3G/3y at position i-l/2,j has been represented as a 

 weighted average of its values at locations i,j and 1— 1 , j . The value of 

 the weighting factor W used in RCPWAVE is 1.0. This choice implies that an 

 implicit solution of the equation is required. One additional approximation 

 is made by using the following weighted sum: 



to represent the variable F at position i,j . Here a is another weight- 

 ing parameter. The value of a is set to 0.167 in RCPWAVE. This "dissipa- 

 tive interface" (Abbott 1975) is used to enhance the stability of the numeri- 

 cal scheme. Substitution of Equation 17 into Equation 16 results in the 

 following expression: 



F. , .. = aF. , . + (1 - 2a) F. , + aF, . , 

 1-1 f J i»>1 i,J iJ-1 



+ Ax 



G i-1,>1 - G i-1 t , 1-l\ ( M) / G i,>1 " G i, ,1-1 

 2Ay I + U W; l 2Ay 



(18) 



The finite difference formulation of Equations 8 and 9, which will be de- 

 scribed next, is identical to that used by Perlin and Dean (1983). Their 

 choices for W and a were 1.0 and 0.25, respectively. 



21. The finite difference form of Equation 8 is derived by substituting 

 the following expressions for F and G into Equation 18: 



F = |vs| sine ( 19) 



13 



