Panel Discussion 



Orszag (1968) and Williams (1967). The vorticity approach has been used suc- 

 cessfully by Aziz and Heliums (1967). 



SOLUTION OF POISSON'S EQUATION 



ON THREE-DIMENSIONAL GRIDS ' W 



The standard iterative techniques that have been developed for the two- 

 dimensional Poisson equation, such as the successive over -relaxation (SOR) and 

 the alternating-direction implicit (ADI) cannot be carried over directly to three 

 dimensions. Two alternative approaches are being employed at present to solve 

 (2) with 3p Bn given or (4) with '^j given on the boundaries. 



(A) One regards the Poisson equation as the steady- state version of the 

 time-dependent parabolic diffusion equation ,- 



d^j. . - 



. • ^= V^^ + ^, , ■ (5) 



in which ^. is a known source function, and uses any of the techniques devel- 

 oped for iterating (5) in three space dimensions. Among the successful tech- 

 niques that have been used by the author with good success are the DuFort- 

 Frankel (1953), Douglas-Rachford (1956), Douglas (1962), and Saul'ev (1957) 

 schemes. In any of these methods, if the spectrum of the initial error is known 

 in advance, one can choose a sequence of time steps such that each extinguishes 

 a particular harmonic. By repeating this sequence several times one can ob- 

 tain very good convergence; e.g., on a 10^ mesh, four sweeps of a five-time- 

 step sequence resulted in decreasing the error by a factor of 10"^ . 



(B) If the boundary conditions are either periodic or of the "dynamically 

 free" kind (no stress and no normal velocity) at one or two pairs of opposing 

 boundaries, one can expand both the components ^^ and ^. parallel to these 

 surfaces in a sine or a sine/cosine series, as the case may be. Equation (4) 

 may in general be reduced to a system of n^ ordinary differential equations of 

 the type 



m n 



- (m^ + n^) 7^2 • a^_^ = b„„ , (6) 



where the a^^ and the h„„ are the Fourier coefficients of ^ and ^ in the x-y 

 expansion. The finite-difference version of (6) can be solved easily by a special 

 algorithm devised for tri-diagonal matrices (see Varga, p. 195). 



Most of the computer time in approach b is spent in finding the coefficients 

 b„n and superimposing the a^^ to find .a. For Fourier synthesis of functions 

 with complex values, a very efficient algorithm exists if the number of grid 

 points has a particular value, say, n = 2" , as shown by Cooley and Tukey (1965). 

 This approach, however, has as yet no known counterpart in the case of real 

 functions (e.g., sines alone). Hockney (1965) devised a related method utilizing 

 the symmetry of the sine functions and a cyclic reduction technique on grids of 

 size N = 3-2", but his technique was applied to two dimensions only. Studies are 

 being made to extend the method to three dimensions. 



1617 



