(1 + X ) U* = (1 - A - 2A ) U^"-^ (11) 



X X y 



V+1 lf-1 



(1 + X ) U = U* + X U (12) 



y y 



where 



1 ^t . - 

 = -T -7— A6 



2 Ax I 



X =^f B6 

 y 2 Ay y 



The quantities 6 and 6 are centered difference operators, and the super- 

 script k indicates time-level. The starred quantities can be considered 

 intermediate values between the k and k+1 time-levels. 



16. The first step in the SC procedure computationally sweeps the grid 

 in the x-direction, with the second step sweeping in the y-direction. Com- 

 pleting both sweeps constitutes a full time-step, advancing the solution from 

 the k time-level to the k+1 time-level. The form of the difference equa- 

 tions for the x-sweep is given by 



2^ (^* - ^"""'^ -^ 2k ^x (-*d+u^"'d) + ^ S ^"''"''^ = ' ^''^ 



_i_ (,* _ ,^-1) + _|_ 5^ (,* + ,k-l) = (14) 



2At (-* - - ^ + fe 'y (^ ^ = ^''^ 



and the y-sweep by 



1 , k+1 *N , 1 . / 1^+1 J k-1,- - .,.. 



u'^"^^ = u* (17) 



18 



