the nonlinear advective terms, a particular implicit scheme known as the 

 Stablizing Correction (SO scheme is used. The basic idea of this scheme is 

 as follows. The time level is indicated by a superscript r . The scheme 

 involves variables at three time levels. The values of the variables at time 

 levels r-1 and r are known from previous computations or prescribed 

 initial conditions. To advance the solution from time level r to the new 

 time level r+1 , an intermediate time level solution denoted by the super- 

 script * is introduced. Equations 34-36 are operated in a two-step proce- 

 dure. In the first step, the rectangular grid is swept in the x(a.. ) direc- 

 tion, advancing the solution from time level r to * . Next, the grid is 

 swept in the y( a ) direction, advancing the solution from time level * to 

 r+1 . The two sweeps together constitute a full time-step At . 



33. Before details of the double sweep technique are discussed, the 

 notation used for individual cells of the rectangular grid will be defined. 

 Let Ax and Ay denote the cell dimensions in real space in the x and y 

 directions, respectively. These dimensions may vary from cell to cell. Let 

 the corresponding dimensions in computational space be Aa and Aa 



These dimensions are the same for all the cells in the grid. Let m and 

 n denote indices corresponding to the center of an arbitrary cell (Fig- 

 ure 7). All the variables except the velocities U and V are defined at 

 the cell centers. Velocities U and V are defined at cell faces m+(l/2) 

 and n+(l/2) , respectively. In the x-sweep, the x-momentum equation is 

 centered about the cell face m+(l/2) , and the continuity equation is 



centered about the center of the cell. The two equations are solved, using in 



* r+1 — * 



the process the result U = U .At the end of this sweep, n and 



U r are known. Next the grid is swept in the y direction. In this sweep, 



the y-momentum equation is centered about the cell face n+(l/2) and the 



continuity equation about the cell center. Upon solving the two equations, 



the values n and V for each cell are obtained. Thus the two sweeps 



together complete the solution for n , U , and V 



34. Even though the SC scheme has been described so far in terms of the 

 x-, y-coordinate system for convenience, in reality the technique must be 

 applied to the equations of motion in the computational a, , a„ space. 

 After the application of the technique, the following finite difference 

 equations result. (Hereafter the bar over n is dropped for convenience.) 



25 



