2. BASIC CONCEPT
The Poisson equation for ~0 with the source termflx, y, z) in the three-dimensional Cartesian
coordinate is written as,
v~o=-&v+-Sy,~ +~, ) f(x,y,z) (l)
We consider Eq.(1) as a model equation for applying the present methods.
2.1 Rotated finite difference approximation method
Equation (1) is discretized using the nine-point rotated finite difference approximation,
which is given by
--" "~.2 (~i-l,j-l,k-I "~" 0i+l,j-l.k-I at-
Oi-l,j+l,k-I q- Oi+l,j+l,k-I
+ 0,_,.j-,.,.+, + 0;+,.j-,.,+, + O,-l.j+,.k+, + O,+,.j+,.,+, - 8r =
where h denotes grid spacing, defined as,
h = zkr = Av = Az.
and i, j and k denote grid indices.
2.2 Successive overrelaxation method with rotated finite difference approximation
Equation (1) with the successive overrelaxation (SOR) method (RFD-SOR) is written by
using the rotated finite difference approximation, as follows,
~n+l ' ' ' 8 \'ffi-l,j-l.k-I "~- '?'i+l,j-l,k-I "+" Y'i-l,j+l.k-I "~ Y'i+l,j+l.k-I
COsoR )C~Tj k + tOSOR (~,,+l
r~,,+~ ~,,+~ t~,,+l
+ q~;'-Lj-,.k+, + OT+,.j-Lk+, + q;'-Lj+L*+, + O;'+,../+,.a.+, -- 4h2f.;.k )
where n denotes a number of iteration step. Since only diagonal points are used in this
discretization, the grid points are classified into four colors in 3-D. The computation on each
group of colored grid points can be carried out independently. Only a quarter of the grid points
in the computational domain are iterated until convergence. The solution of the remaining points
are obtained with seven-point two-dimensionally rotated finite difference approximation. For
example, r is obtained from x-y rotated approximation as,
O;.j.k+, = ~(~0;_,.j_,.,+, + #:+l.j-,.,-+~ + ~;-,.;+,.~.+, + O~,+,.j+,.,+l + 2q;.;.k
In two-dimensional case, the grid points are classified into two colors, as the same as red-
black ordering. After the solution of a group has converged, the remaining points are obtained
from conventional five-point finite difference approximation. The correction cycle algorithm is
used for the multigrid strategy.