Parallel Computational Fluid Dynamics '95

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.

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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.

1. INTRODUCTION

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.

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2. BASIC CONCEPT

The Poisson equation for ~0 with the source termflx, y, z) in the three-dimensional Cartesian

coordinate is written as,

02#

,92#

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

(2)

+ 0,_,.j-,.,.+, + 0;+,.j-,.,+, + O,-l.j+,.k+, + O,+,.j+,.,+, - 8r =

r

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

i.j.k

--"

(1 -

COsoR )C~Tj k + tOSOR (~,,+l

r~,,+~ ~,,+~ t~,,+l

(3)

+ 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,

1

O;.j.k+, = ~(~0;_,.j_,.,+, + #:+l.j-,.,-+~ + ~;-,.;+,.~.+, + O~,+,.j+,.,+l + 2q;.;.k

+ 2~Oi,j,k+2-

2h.zf.j.k ).

(4)

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.