Piacsek 



and jj. is adjusted to obtain most rapid convergence. The remaining values of 

 r^ are defined modulo 8, i.e., rg = rj, rjo = t^^, etc. Each row iteration, Eq. 

 (30a), or column iteration, Eq. (30b), is implicit, in that all the values on the 

 respective row or column must be found simultaneously. Since there are only 

 three unknowns in each equation, the resulting coefficient matrix becomes tri- 

 diagonal: the only nonzero elements are on the diagonal and two adjacent lines. 

 For this type of matrix there is a special inversion algorithm that is very ef- 

 ficient; a detailed formulation is given by Varga (1962), p. 195, and Richtmeyer 

 and Morton (1967), p. 200. 



The ratio of the asymptotic rate of convergence for the overrelaxation to 

 that of the ADI method decreases as the number of gridpoints increases, so that 

 for large meshes the ADI method is far superior. This is particularly so in the 

 case of Neumann boundary conditions for which the overrelaxation is extremely 

 slow in converging. A detailed comparison of the two methods for Dirichlet and 

 Neumann boundary conditions and for various ratios of the grid spacings ax/az 

 may be found in Gary's (1967) article. 



When the boundaries in one or both of the directions are "f rictionless lids," 

 where ^ = ^ = 0, the Fourier inversion method becomes very suitable particu- 

 larly when the respective dimension is much shorter than the other. In this 

 method, ^j j and <f ■ j are both expanded in discrete Fourier series in the respec- 

 tive direction, say z: 



a 3 



"^ij " Z] ^i" ' ^'" (n^JA) , ^ij " Z] b." • sin (n77JA) 



(32) 



and the resulting j ordinary difference equations 



(a;;,l+a['.j-2ai")/A2 - n^^^aF^ = b." (33) 



are solved by the tri-diagonal algorithm mentioned above. Here practically all 

 the computation is spent in finding the b." from decomposing cf ^ ■ , and the lAij 

 from superposing the a^", which is ^^ip calculations. If j « i, the ratio of 

 computational work in this method to that of overrelaxation is -20/27 • Vj/l. 



Recently, two ingenious simplifications were introduced by Hockney (1965) 

 into this method. He noted that if a suitable number is chosen for J (such as 

 12, 24, 48), the symmetry in the sine functions may be used to reduce the com- 

 puting time to about a tenth of the above estimate. Furthermore, the two- 

 cyclic nature of the difference Eqs. (33) allows one to replace the original 

 equations involving all the points in the net to a set of ij/2 slightly more com- 

 plex equations involving only the points on the even lines of the mesh. The set 

 of revised equations could also be solved by the Fourier method. The final pro- 

 gram led to a solution time 1/10 that of the ADI and 1/60 that of the overrelaxa- 

 tion method, on a mesh size 48 x 48. 



On noting that the values of the vorticity on the boundary are actually not 

 needed to solve the finite-difference version of the Poisson equation, the 



776 



