Parallel Computational Fluid Dynamics:
Implementations and Results Using Parallel Computers
A. Ecer, J. Periaux, N. Satofuka and S. Taylor (Editors)
9 1995 Elsevier Science B.V. All rights reserved.
Parallel Computations of CFD Problems Using a New Fast Poisson Solver
Masanori Obata and Nobuyuki Satofuka
Department of Mechanical and System Engineering, Kyoto Institute of Technology
Matsugasaki, Sakyo-ku, Kyoto 606, JAPAN
A fast Poisson solver based on the rotated finite difference discretization is applied for
solving two-dimensional and three dimensional problems. The method is implemented for two-
dimensional problems on a cluster of workstations using PVM, and its convergence rate is
compared with that of the conventional SOR method. The result shows 6.5 times faster
convergence speed for two-dimensional test case with the same accuracy.
Many fluid dynamic problems of fundamental importance in the field of computational
fluid dynamics are governed by the elliptic partial differential equations (Poisson equation).
Especially, in the case of solving time dependent incompressible viscous flow problems, the
Poisson equation for either the stream function or the pressure have to be solved at each time
step. In such case, most part of computational time is spent in solving the Poisson equation.
Iterative methods are generally used as suitable approach to the solution of the Poisson equation.
The successive overrelaxation (SOR) method[ 1 ] is one of the most popular iterative method to
solve the Poisson equation and a number of modifications, e.g. line SOR, group SOR has been
proposed to accelerate the convergence. The group explicit iterative (GEl) method[2] is proposed
not only to improve the convergence rate but also to reduce the computational cost. The method
is easily adapted to architecture of recent advanced computers using vector and/or parallel
processing. The multigrid method[3] is used to accelerated further the convergence of those
iterative method.
In this paper, we propose a new efficient iterative method for solving the Poisson equation
based on the rotated finite difference approximation. In 2-D and 3-D problems, the convergence
history and the accuracy are compared with the SOR method with conventional finite difference
approximation. In 2-D case, the multigrid method is also applied for solving the Poisson equation.
The parallelization is can'ied out on a cluster of networked workstations. Finally, those Poisson
solvers are applied to solve 2-D driven cavity flow at Re= 100.
The Poisson equation for ~0 with the source termflx, y, z) in the three-dimensional Cartesian
coordinate is written as,
0"~ =
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
V2~i.j.k 1
--" "~.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
(1 -
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
+ 2~Oi,j,k+2-
2h.zf.j.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.

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