1 / * r -M 1 . /„r+l -v 



o~7T PI ~ n J + o 7 — 6 U d + 



2 At \ I Z\i Aa a 



u r_1 d r ) 



+ 7 S (v r ^J = at (n,m) 

 u Aa„ a„ \ / 



(38) 

 y Aa a ' 

 y 2 2 



In the above equations, a single bar represents a two-point average and a 

 double bar a four-point average. The difference operator 6 is defined as 



5 «. (Z> = V wo "V wo (39> 



i i+1/2 i-1/2 



for any variable Z . The definition may be extended to the operators 6 



and 6 x 



a. a. 



35. Equations 37 and 38 may be rearranged so that the unknown quantities 

 are to the left and the known quantities are to the right, as follows: 



where 



— r+1 



-an* + a . ._U , ._ + a . n* , = B , ._ (40) 



m n,m m+1/2 n,m+l/2 m+1 n,m+l m+1/2 



-a . . U + n* + a , ,„U r , .. = A (41) 



m-1/2 n,m-l/2 n,m m+1/2 n, m+1/2 m 



% = a m+l = /p \ BAt Aa" (i,2) 



'm+1/2 



2 Ate < 



1 + 



orb | n, m+1/2 



m+1/2 — r 



n, m+1/2 



27 



(43) 



