The quantities 5 and 6 are centered difference operators and the super- 

 ^ X y 



script k counts time- levels. The starred quantities can be considered 

 intermediate values for variables at the (k+1) time-level. 



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

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

 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 equations 

 for the X- sweep is given by 



_1- (n* - ^-h + -2^^ 6^ (u*dW-ld) + ^ 6^ i.^-h) = (8) 



_i_ („* . ,1^-1) ,._^,^ (n* + n'-^ =0 (9) 



and the y-sweep by 



2A^ (n - n*) + 2^ \. (V a - V d) = (11) 



u''^^ = u* (12) 



__ (V _ V*) + 2f^ 6y (n - n ) = (13) 



13. Noting that v* in Equation 10 is only a function of previously 

 computed variables at the (k-1) time-level, its substitution into Equa- 

 tion 13 and the substitution of u* (Equation 12) into Equations 8 and 9 

 yield the simplified forms 



ih <""*'- "'"^ * ± \ <n' * n"-') . . , -, ,15, 



15 



