Vu Tien Dung

VNU Joumal of Science, Mathematics - Physics 23 (2007) 201-209

Fully parallel methods for a class of linear partial differential-algebraic equations

Vu Tien Dung*

D epartm ent o f M athem atics, M echanics, ỉrựòrmatics, C ollege o f Science, VNU 334 N guyên Trai, Thanh Xuan, Hanoi, Vietnam

Received 30 N ovem ber 2007; received in revised fo rm 12 Decem ber 2007

A b s t r a c t . T h is note deals vvith tvvo fư lly parallel methods fo r so lv in g lin e a r partia l d iíĩe re n tia l- algebraic equations (P D A E s ) o f the form :

Aut + B A u = /( x , t) (1)

where A is a singular, symmetric and nonnegative matrix, while B is a symmetric positive define matrix. The stability and convergence of proposed methods are discussed. Some numerical

experim ents on h igh-perfo rm a nce com puters are also reported.

Keywords : DiíTerential-algebraic equation (DAE), partial diíĩerential-algebraic equation (PDAE),

nonnegative penciỉ o f m atrices, para lle l method

1. In tro d u c ỉio n

Recentỉy there has bcen a grow ing interest in the analysis and numerical solution o f PDAEs because o f their importance in various applications, such as plasm a physics, magneto hydro dynam ics, electrical, m echanical and chem ical engineering, etc...

A lthough the numerical solution for differential-algebraic equations (D A Es) and (PD A Es) has been studied intensively [ 1 ,2 ] , until novv we have not found any results on parallel m ethods for PDAEs.

This problem w ill be studied here for a special case.

T he paper is organized as follows. Section 2 deals w ith som e properties o f the so called nonnegative pencils o f m atrices. In Section 3 we describe tw o parallel m ethods for solving linear PDAEs, w hose coefficients found a nonnegative pencil o f matrices. T he solvability and convergence o f these m ethods are studied. Finally in section 4 some numerical exam ples are discussed.

2. Properties of nonnegativc pencils of matrices

In w hat follows we will consider a pencil o f matrices {^4, B } , w here A G R nxn is a singular, sym m etric and nonnegative m atrix vvith rank (j4) = r < Tí and B € R nXn is a sym m etric positive deíìne matrix. Such a pencil will be called shortly a nonnegative pencil.

We begin w ith the íollovving property o f nonnegative pencils.

• Tel.: 084-48686532.

E-mail: duzngvt@gmail.com

201

2 0 2 Vu Tien Dung

/

VNU Journal o f Science, Mathematics - Physics 23 (2007) 201-209

P ro p o sitio n 1. Any nonnegative pencil { A , B } can be reduced to the Kronecker-Weierstrass form { d ia g ( /r , On- r) , d ia g (D , / n - r ) } with a symmetric andpositive defìne matrix D e R r x r . Here Ir and On- r stand fo r the identity and zero matrices o f appropriate dimensions respectively.

Proof. T he sym m etric and nonnegative matrix A can be diagonalized by an orthogonal matrix u , such as ƯT A U = d i a g ( A i , A r , 0, ..0), vvhere Ai > Ả

2

> .. > Ar > 0 are positive eigenvalues o f A.

Defm e tw o m atrices s := d i a g ^ A i ) ^ , 1, ...1) and Ô : = S U T DUS. Clearly, Ỗ is also sym m etric and

positive

deíine. Morever, S U T A ư s = d ia g ( /r , O n_ r).

Now let

It is easy to veriíy that Bị and its Schur complement defm ed by B\ — B ìB ị 1 B$ are also symmetric and positive define. Putting

p . =

( l r

B 2B Ị l \ ^.

^ Ị B ì - B 2B ^ B z 0 \

P:= u V , J ; ^{B - { 0 B i )}

and

q

= (

b ị

'

b

3

/„ _ r )

we get Ế = P Ỗ Q and P d i a g ( / r , On-r)Q = d ia g ự r , O n-r). From the last relations it follows P ~ l Ỗ Q ~ l = Ổ and p ~ x d ia g ( /r , O n _r ) Q

_ 1

= d ia g ( /r , On-r ). Finally, letting p = d ia g ( /r , B ị X) we find P d i a g ( / r , On-r) = d ia g ( /r , 0 „ _ r ) and P ỗ - d ia g (D , In-r), where D D\ - B$

and D

t

= D > 0. T hus, there hold decom positions M A N = d ia g ( /r , On-r), M B N — d ia g ( 5 , In- r ) with nonsingular m atrices M := P P S U T and N := Ư S Q ~ l , vvhich was to be proved.

In \vhat follows, vve need two Toeplitz tridiagonal m atrices p and Q o f dim ension k X k, vvhere as a rule

k

is m uch greater than

TI.

(

2 _ 1

\ ị 4 - 1 \

- 1 2 - 1

- 1 4 - 1

p = * ; Q = *

• * • * •

- 1 2 - 1

- 1 4 - 1

2 )

^ - ! 4 /

Clearly, if D 6 R rx r is a symmetric and positive deíìne m atrix, then the K ronecker product p <g> D and <3 0 D are again sym m etric and positive diíìne. Let h > 0 be a positive parameter.

P ro p o sitio n 2. Let the pencil {A, B ) be nonnegative and ỉet M and N be two nonsingular matrices, such as M A N =

d i ă g ( I r ) O n - r ) ,

M B N = diag(£>, / n- r ) , where D r — D > 0. Then One can explicitily deỳìne two nonsingular maírices K and H, tranforming the pencil {Ik ® A, jf?p <s> B} to the corresponding Kronecker-Weierstrass form

{diag(//c, 0*;(n_ r )), d iag ( Ặ ĩ > , 4 ( „ - r ) ) } , (3) with symmetric and positive defìne matrix D.

Proof. A ccording to Proposition 1, the nonnegative pencil {Jfc <g) A, Ặ p ® B } can be reduced to the

canonical form (3).

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Physics 23 (2007) 201-209 203

Further, for the Toeplitz m atrix p , there exists an orthogonal m atrix u, such that ƯTP U := A = d i a g ( A i , A f c ) , vvhere Ai > A

2

> ... > > 0. Then, the matrix s ^{:= p}

⁰

^{I n- k} is diagonalized, (U 1 ® I n - r ) S ( U ®

/ „ - r )

= A <g> In -r = d ia g (A i/n_ r ,

A f c / „ _ f c ) .

Let M := I k ® _ M ; N :=

I k <g>N-,A := Ik <

8

> d ia g ( /r , On-dir)', B := Ặ p

0

diag(L>,

/ „ - r ) .

Clearly, M ự k ® = i4 and M ( & P ® B ) N = B.

Now w e define tw o special matrices

^{£ 1}

€ R rx n and E

²

e R (n - r )xr as

/ 1 0 0 0

\ ị

0 1 0 0

\

0 0 1 0 0 0 0 1

• » . ; E 2 = » « •

1 0 0 0 0 1 0 0

r-4

0

0

^

0 0 0 1

/

and put

6

:= & =

( ( £ 1

® /* ) T .

( £ 2

® /fc)T ) • Further, let D : = p ® D \ J X : = d i a g ( 4 r , ơ r ® Ịn-À:), h := diag(/A:r ,ỉ7 <g>/n_ r ) ; J

3

:= diag(/fcr , / l

- 1

®

/ n - r ) .

Finally, set /c :=

J \ A \ M \H N A i h h - We vviil show that / c and H transíòrm the pencil {Ik <

8

> A , jp P ® B } to the canonical form (3). Indeed, a sim ple calculation shows that £ ii

4 £ 2

= diag(/*;r , ơfc(n_fc)) and t i B ị ĩ = d i a g ( ^ Z ? , 5 ) ^ Futher , Kựk<S>A)H = A ị 2h h = d ia g (/fc r,0 * (n -r))- Similarly,

® B )H = M x B h h h = d ia g (^ jjD , h(n-k))- Thus the proposition

2

is com plete.

3. Fully p a ra lle l m etho d s fo r lin e a r PD A Es

In this section vve study the numerical solution o f the following initial boundary value problem s (IBVPs) for linear PDAEs:

A u t + B A u = f { x , t),

1 6

ÍỈ, í £ (0 ,1 ) , (5)

E u ( x , 0) = uo(x),

X

G n , (

⁶

)

u ( x t t ) = 0, X € d ữ , (7 )

d __

,where A u : = = { ^ (^ Ìi •••>Zd);0 < Xi < 1;» = l , d } ,

» =

1

*•

A, B, E are given n x n m atrices and the pencil {v4, B } is nonnegative. Further, u, f are vector íunctions, u, / : Q X [0,1] —♦ R n and the gi ven íìinction / (

X ,

t) is assum ed to be sufficiently smooth.

We propose tw o parallel m ethods for solving the IBVP (5)-(7) w here the parallelism vvill be períorm ed across both the problem and the method.

According to Proposition 1, there exist nonsingular matrices M, N such as M A N = d i a g ( /r , 0 „ _ r ) and M D N = d ia g ( jD ,/n_ r ), w here as above, r=rank (i4) and D = D T > 0. We will partition N ~ xu , M f and UQ into tw o parts, N ~ l u := (vT , w T)T \ M f := (F

ị

, F Ị ) T t uo := (

v q

,

w

Ị ) t , vvhere VũyV, F\ € K r and w0)w, F2 G R n~r . From (5) we get M A N § ị ( N ~ lu) + M B N A ( N ~ ì u) = M Ị , or eq u iv alen tly ,

Vt + D A v = F \ , and

A w = F2.

Further, as in D A E ’s case, the initical condition (

6

) cannot be given arbitrarily. It m ust satisíy some

so-calleđ hidden constraints. Indeed, suppose that the matrix E N is partitioned accorđingly to the

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Vu Tien Dung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 201-209

partition o f the vector

N ì u{x,

0) such that

E N = ịẸ }

where

E \ , E ị

are square matrices o f d im e n s io n

T X T

and ( n — r )

X

( n - r ) , respectively. F o r the sake o f s im p lic ity , w e assume that

Ẽ

2 = 0,

E ị —

0 a n d

E\

is non sin g ula r. T h e n c o n d itio n (6 ) c a n b e

revvritten

a s

E iv (x ,0 )

= ^ o (x ) and

E$v(x,

0 ) = tơ o (z )- F ro m the last re la tion s, it it cle a r th a t the va lu e

w (x,

0 ) w il l not p artic ip a te in fu rth e r c o m p u ta tio n s . Besides, the in itia l c o n d itio n l í o ( i ) = (U(f (:r) , tL> (f(x))7 satisfies a hidden c o n s tra in t

E

3

E ị lvo(x) = wQ(x)

^{( 8 )}

T h u s , IB V P (5 )-(7 ) is s p lit in to an IB V P fo r the p a ra b o lic equ a tion Vị + D & v = F\ ,

v(x, 0) = E ì

1

«o(x), X

6

í)

(9)

(10) (

11

)

v(x, t) =

0

, x e dQ, t € (

0

,

1

),

and a B V P fo r the e llip tic equation

A

w

⁼

F2

^{(1 2)}

w(x,t) = 0, X

⁶

dũt t e

( 0 , 1 ) , (1 3 ) A p a ra lle l fra c tio n a l step (P FS ) m ethod, proposed in [3 ] and developed in [4 ] , w il l be e x p lo ite d fo r s o lv in g the IB V P ( 9 ) - ( l l ) . F o r

this

purpose, w e firs t d is c re tiz e in the sp a tia l va ria b le

X =

( l i ,

...,Xd)

b y c h o o s in g a m esh size

h >

0 and a p p ro xim a te the p ro b le m in the discrete d om ain Q /, b y u sin g the second o rd e r centered d iíĩe re n c e ío rm u la . It leads to the O D E

(1 4 )

Vh{0) = vOh (15)

T h a n k s to t h e s y m m e t r y a n d p o .sitiv e d e f in ite n e s s o f D , in m a n y ca.ses, th e m a tr ix H is s y m m c tr ic and p o s itiv e d e íin e . F o r exam ple,

using

the m atrices

p

^and

Q

d eíìn ed b y (2 ) w e get

H

⁼

Ả P

^ộộ

D

( Q - I \

- I Q - I

in 1D

case (d =

1) and

H = ^ ĩL <8> D,

w here

L = for

2 D case

-I Q -I

- I Q

(d = ^2).

F u r th e r ,

suppose H can be split into the sum o f symmetric, painvise commutative and

p o s itiv e

semidefinite matrices Hk,

\

/

d

H = Y J Hk\ H Ị = H k >

0;

HỵHị = H i H M

= l, d ; k = i

( 1 6 )

We d is c re tiz e the tim e in te rv a l [0 ,1 ] w ith step T > 0 and a p p ly the PFS m e th o d [4 ];

Vu Tien Dung / VNU Jo u rn a l o f Science, M athem atics

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Physics 23 (2007) 201-209

205

A lg o r it h m PFS

Step 1. In itia liz e v ° ^{: =} Voh

Stcp 2. F or g ive n

m

> 0 and

V™

(an a p p ro x im a tio n o f

Vh(mr))

fin d

vm+1’k

b y s o lv in g (in p a ra lle l) systcms o f lin e a r equations

( / + y f W +1'‘ = ( / - y £

H i) v m + T~ F kíh

(17) j=

w here

Ffh

: = /,( ( /; + 1 / 2 ) r ) Step 3. C om pute

^ ' . 2 É -

fc=l

m +1’* + (1 - (1 8 )

d

N o tc that the lin c a r systems (1 7 ) can be solved b y any p a ra lle l ite ra tiv e m ethods [5 ,6 ,7 ,8 ,9 ]. N o w vve tu m to the B V P (1 2 )-( 13). F o r its s o lu tio n vve im p le m e n t the p a ra lle l s p littin g up (P S U ) m ethod, proposed b y T. L u , p. N e itta a n m a k i, and X .

c.

T a i [3 ].

D is c re tiz in g the B V P ( ] 2 )-( 13) One o bta in s a large-scale system o f lin e a r equations

Lw - g,

(1 9 )

vvhere

L

is a sym m etric p ositive d e fin e m atrix o f d im e n sio n

p

X

p,

vvhere

p

=

p(h)

d ep en d s on th e d is c re tiz a tio n param cter

h.

A s s u m c th a t

L

can be decom poscd in to the sum m o f s y m m e tric and

positive defmc matrices, vvhich commute with each other

7 7 1

L

^

L/ị — L ị

^

0; LfịL/j

=

L j L ị

)

z, J

= 1?

771. i = i

( ₂₀ )

T h e PSU m ethod consists o f the fo llo w in g steps:

A lg o r it h m PSU

Step 1. Choose an in itia liz a tio n

up

Step 2. S upposing

WJ

is k n o w n , w e co m p u te the ĩra c tio n a l stcp values

L iiẮ p + t t z=

f —

^{^}

L k w \ i

= 1 , ...,771. (2 1 )

k=2,k&

Step 3. F or chosen param eters set

m

w j + 1 _ ^Ịj_ + -|- (1 — £Jj-)uíJ .

»=1

(22)

N ote that fo r d iíĩe re n t

j

system (2 1 ) can be solved b y p a ra lle l processors.

T h e o re n i 3.

The PFS-PSƯ melhod (17)-(18),(2I)-(22)Ịor solving the IBVP (5)-(7) with a nonnegative pertcil

{.4 ,

B ) and the consistent initial condiiions (6), (8), is convergent.

Proof.

T h a n ks to the n o n n e g a tiv ity o f the p e n c il { A , B } , w e can s p lit the IB V P (5 )-( 7 ) in to the IB V P ( 9 ) - ( ] I ) and the B V P (1 2 )-(1 3 ). Theorem s 4.11 and 5.2 [4 ] ensure that the PFS m ethod in the s y m m e tric and c o m m u ta tiv e case is stable p ro vid e d r <

2{d

m a x | | i/ f c | | } _ 1 . M o re o v e r it is

\<k<d

convergent w ith g lo ba l e rro r

0 ( h

2 - f r 2). Further, a ccordin g to [3 ] the PSƯ m ethod is convergent. I f

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Vu Tien Dung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 201‘209

all the eig en v alu es o f the m atrix s : = ỊỊỊ £ L • 1L b elong to som e segm ent [a, b]y w here a > 0, then

1= 1

the a s y m p to tic rate o f the PSU vvith

Uj

^{= is w here}

p

^{: =}

cond(S)

^{< (}

b/a

^).

We end th is section b y c o n s id e rin g the IB V P (5 )-(7 ) in 1D case

(d

= 1). B csides, the m a trix

E

in (6 ) is supposed to be the id e n tity m a trix .

D is c re tin g the IB V P in the sp atia l v a ria b le w e get a system o f O D E s

( * * . « ) + ^ B [ u { x k+1, t ) - 2u ( x k , t ) + I t ( x fc_ i , í ) ] = f ( x k , t )

k = 1, M - 1, M = M ( h ) . P u t t i n g ư := ( ú [ , . . . , u Ị f)T ; F : = { f ĩ , f ĩ t - \ ) r . w h e r e u k : =

u(x*, í); /fc := /(zfc> í), we can revvrite the last system of equations as

(/m -1 ® + ^ ( P ® B)U = F, (23)

w h e re the m a trix p is d e te rm ine d b y (2 ). B y P ro p o s itio n 2 w e can fìn d n o n s in g u la r m atrices /<■ and / / tra n s fo rm in g the p e n c il { / a/ — 1 <8>>4, to the K ro n e cke r-W e ie rstra ss fo rm (3 ). M u ltip ly in g b o th sides o f (2 3 ) b y

K

and p u ttin g

H ~ l u =

(

V T , W T)T ; K F

= ( F 1r ,

F2r )T,

vvherc

V, F\

€

R(M~ì'> r

and

w

^{, Jp2} 6 R ( W - 1 )(n - r ), w e com e to the system

" + (24)

w = Ẽ2, (25)

w here as in P ro p o s itio n 2,

D

is a s y m m e tric and p o s itiv c d e íìn itc m a trix . N o tc that the boundary con- d itio n (7 ) has been in c lu d c d in E qu a tion (2 3 ). N o w le t

H~l (u

^{0 ( x i ) ,}

u Ị ị x M - ị ) ) 1

= (VoT ,

W Ị)'1

, w here Vo € R ( A /- 1 )r and ÍVo 6 R ( M _ 1 )(n _ r ) T h e n , the in itia l c o n d itio n (6 ), w ith

E = I

^becomes

y (

0

) = v0. ⁽

^{2 6}

⁾

M o re o ve r, th c in itia l c o n d itio n (6 ) in u st s a tis fy a h id d e n co n s tra in t

Wữ =

Í 2 ( 0 ) .

For s o lv in g the ỈV P (2 3 -2 6 ) in p a ra lle l, the PFS m ethod d escribcd above fo r the p ro b le m (1 4 )- (1 5 ) can be a p p lie d . We s h a ll not g ive the le n g th y deta ils.

4. Numerical experiment

C o n s id c r the b o u n d a ry - va lu e p ro ble m (5 )-(7 ) w ith the fo llo w in g data:

ị\ ^{0 0 \ / 2 - 0 . 5 l \}

n = 3 ;d = 2 M = 0 1 0 ; B = - 0 .5 1 0 ; E = I. (27)

\ 0 0 0) \ 1 0 1 /

T h e fu n c tio n

f ( x ,

í) is chosen such that the exact s o lu tio n o f the B V P (5 )-(7 ) is

u

= 103( í x i ( l -

x \ ) x ị { \

- X 2 ),

^{t x \ {}

1 - Xl)X2(l - X2),ÍX1X2(1 - X i)(l -

^X2 ) ) 1 U s in g n o n s in g u la r m atrices

/ 1 0 - 1 \ / 1 0 0 \

Aí = Ị 0 1 0 7V = 0 1 0 (28)

\ 0 0 1 / 0 ì )

Vu Tien Dung / VNU Journaỉ o f Science, Mathematics - Physìcs 23 (2007) 201-209

207

w e can s p lit the IB V P (5 )-(7 ) in to an IB V P fo r the p a ra b o lic equation

Vt - W{vXìXì + VX2X2) = F i ( x i ,X2,t)

v(x , 0) = 0 X

⁶

^{í ì (29)}

v(x, t) =

⁰

X

G ỡ í ì a n d

t

€ (0 , 1) and a B V P fo r the e llip tic equation

^ 1 2x2) -^2( ^ 1ì %2ĩ 0

tu(x, í) = 0 .

T h e PFS m ethod and PSU m ethod

[4,3]

are im p le m e n te d in

c

and

MPI

and executed on a L in u x C lu s te r 1350 w ith e ig h t c o m p u tin g nodes o f 51.2 G F Io p s. Each node contains tw o In te l X eo n dual core 3 .2 G H z , 2 G B Ram.

T h e ĩo llo v v in g table show s the dependence o f the e rro r o f the a p p ro xim a te s o lu tio n s on the num be r

N = ị

w h ile the ra tio p rem ains constant.

N r

ti*

^{16 24 30 40 50 6 0}

Residuaỉ

0 . 5 0 .0 0 0 3 4 5 0 .0 0 0 0 8 0 .0 0 0 0 3 8 0 .00 00 1 4 0 .0 0 0 0 6 0 9 0 .0 0 0 0 3 0 9

Residuaỉ

^{0 . 2} 0 .0 0 0 0 6 4 0 .0 0 0 0 1 6 0 .0 0 0 0 0 7 0 .0 0 0 0 0 2 0 . 0 0 0 0 0 1 0 .0 0 0 0 0 0 5

In vvhat fo llo \v s , w c stu d y the re la tio n bet\veen the to ta l (C P U ) tim e spent on the períb rm an ce o f a p ro gram , the spccdup and the e ữ ìc ie n c y o f th is períbrm ance. T h e specdup o f the p e ríb rm an ce is d efin e d as

s = Ts/Tp,

w h e re

Ts (Tp)

is serial e x e c u tio n tim e (p a ra lle l c x c c u tio n tim e ), resp ective ly.

T h e eíT iciency o f the p crfo rm a n ce is d cte rm in e d a

s E = s / p ,

vvhcre

p

is the n u m b e r o f processors.

T h e re sn lt o f an c x p c rim c n t w ith PFS m ethod fo r (2 9 ) is reported in the fo llo w in g table

Table 1. S peed up and E íĩic ien cy on C lu ste r 1350 w ith N = I 2 0 .

Processors

1 2 4 6 8 10

Toltal times(minutes)

252 126 62 4 3 37 32

Speedup

^{2 4 5 .8 6 .8 7.8}

Ẹffìciency

^{1 1 0 .9 7 0 .8 5 0 .7 8}

U s in g 2 processors o f C lu s te r 1350 and a p p ly in g the PSU m ethods to the B V P (3 0 ) w e observe that the to ta l tim e increascs to g c th e r vvith the g ro w th o f the n u m b e r

N = ị .

Table 2.

N

²⁴ 30 40

Toltal times(seconds)

120 180 300

F o r b e tte r convergcnce, w e use o th e r m ethods, such as the p a ra lle l Jacobi m ethod [5 ] , the p a ra lle l SOR R e d /B la c k [6 ,7 ,8 ,9 ] m ethod.

T h e p a ra lle l Jacobi m ethod and P arallel S O R R e d /B la c k m ethod are im p le m e n te d in

c

and M P I and executed on 1 node o f A I X C lu s te r 1600 o f 5 c o m p u tin g nodes, vvhose to ta l c o m p u tin g p o w e r is 2 40 G F lo ps. Each node co n ta in s 8 C P U P ow er4 6 4 b it R S IC 1.7G H z.

B e lo w are some results fo r p a ra lle l Jacobi m ethod and p a ra lle l S O R R e d /B la c k m ethod

T able 3. S peed up and Eflficiency on 1 node o f C lu ster 1600, N = 2 4 0 .

208 Vu Tien Dung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 201-209

Parallel Jacobi method

Processors 1 2 4 6 8

Toltal times(seconds) 937 484 232 191 155

Speedup 1.94 4.0 4.9 6.05

Effìciency 0.97 1 0.81 0.75

A lth o u g h t the p a ra lle l Jacobi m ethod converges íaste r than the PSU m ethods, it is ra re ly used as a p a ra lle l s o lv e r fo r e lip tic problem s.

T able 4. S peed up and E ffìcien cy on C lu ste r 1600 w ith N = 1 2 0 0 .

Parallel Red - Black SOR method

Processors 1 2 4 6 8

Toltal times(seconds) 275 154 83 64 54

Speedup 1.79 3.31 4.3 5.1

Effìciency 0.9 0.83 0.72 0.64

T h e n u m b e r o f ite ra tio n s needed fo r convergence and the to ta l tim e fo r the s e ria l c o m p u ta tio n o f Red - B la c k S O R and Jacobi m ethod are g ive n in the f o llo w in g tables.

T able 5. N u m b e r o f Iterations o f sequential R ed - B lack S O R and Jacobi m ethod.

N 60 120 180 240 300

SOR 284 565 836 1101 1351

Jacobi 10599 39680 86119 149311

T able 6. Total tim e s o f R ed - B lack S O R m ethod and Jacobi m ethod.

N 60 120 180 240 300

SOR(seconds) 1 2 7 12 19

Jacobi (seconds) 4 45 200 720

T h e Red - B la c k S O R m ethod is c le a rly the íastest o n c in term s o f s e ria l tim c and the n u m b e r o f ite ra tio n s.

T a b le 1,3,4 sh ow th a t w h e n the n u m b e r o f processors increases, the speedup increases. T h e actual speedup is s m a lle r than the ideal speedup because the c o m m u n ic a tio n cost is r đ a tiv e ly h ig h e r w h e n im p le m e n te d and executeđ on a L in u x C lu s te r 1350 and A I X C lu s te r 1600. F rom T a b le 1,3,4 it is cle a r th a t the m ore processors are used, the c o m m u n ic a tio n cost increases, and the e íĩic ie n c y decreases.

Acknovvlcdgements.

T h e a u th o r thanks P rof. D r. Pham K y A n h fo r su g g e stin g the co nsidered to p ic and fo r h e lfu l discussions. P a rtia lly supported b y the V N U ’ s K e y P ro je ct Q G T 05.10.

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