investigation of implicit methods for solution of the
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Brigham Young UniversityBYU ScholarsArchive
All Theses and Dissertations
1968-5
Investigation of Implicit Methods for Solution ofthe Fourier EquationKjell Steinar GundersenBrigham Young University - Provo
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BYU ScholarsArchive CitationGundersen, Kjell Steinar, "Investigation of Implicit Methods for Solution of the Fourier Equation" (1968). All Theses and Dissertations.7121.https://scholarsarchive.byu.edu/etd/7121
C l o ^ ^ L
£<l f r
INVESTIGATION OF
FOR SOLUTION OF TE
IMPLICIT METHODS
:s fourisr equation
yV
A Thesis
P resen ted to the
Department o f Mechanical .Engineering
Brigham Young U n iv e rs i ty
In P a r t i a l F u l f i l lm e n t
of the Requirements f o r the Degree
Master of Science
by
K je l l S te in a r Gundersen
May j 19 6 S
This t h e s i s , by K j e l l S t e i n a r Gandersen, i s accepted in
i t s p re s e n t form by the Department c f Mechanical Engineering
of Brigham Young U n iv e r s i t y as s a t i s f y i n g th e t h e s i s r e q u i r e
ment f o r th e degree of Master of Sc ience .
i i
To E l le n
ACKNOWLEDGMENTS
The au tho r wishes to express h i s s i n c e r e s t a p p r e c i a t i o n
to Dr. Howard S. Heaton f o r h i s pe rso n a l h e lp and sugges t ions
th roughout t h i s p r o je c t and f o r h i s w i l l i n g n e s s to share of
h i s i n s i g h t i n to the f i e l d of numerical methods i n h ea t
t r a n s f e r .
The a u t h o r ' s w ife , E l le n , a l s o dese rves a s p e c i a l word
of a p p r e c i a t i o n . Besides being a good mother and w ife she
has been a co n s tan t source of encouragement and support and
has done a l l the typ ing connected w i th t h i s p r o j e c t .
TABLE OF CONTENTS
DEDICATION
ACKNOWLEDGMENTS ..................... '............................................ iv
LIST OF FIGURES ........................ ; ................. v i i
NOMENCLATURE ......................................................................................... v i i i :
C hapter
I . INTRODUCTION ................................................................................ 1
P rev io u s Work
I I . STATEMENT OF THE PROBLEM......... ........................................... 4
I I I . DEVELOPING THE BACKWARD DIFFERENCE METHOD ............... 6
Summary of Node Equations
IV. DISCUSSION OF METHODS OF SOLUTION................................ 14
Gauss E lim in a tio n Method G au ss-S e id e l Method C ho lesk i D ecom position Method G auss-Jordan E lim in a tio n Method Booy Method
V. DISCUSSION OF RESULTS ..................................................... 23
Computer Time E rro r
T ru n ca tio n E rro r Round-Off E rro rO ptim ising Computer Time and Accuracy
VI. CONCLUTIONS .................................................. 41
APPENDIX A THE GAUSS ELIMINATION METHOD ....................... 43
APPENDIX B THE GAUSS-SEIDEL METHOD......... ........................... 53
APPENDIX C THE BOOY METHOD ................................. ' ..................... 63
v
Page
i i i
iv
v i i
v i i i :
iJL
4
6
14
23
41
4-3
53
63
Page
APPENDIX D THE GAUSS-JORDAN ELIMINATION METHOD ............ 80
APPENDIX E THE CHOLESKI DECOMPOSITION METHOD ................ 85
BIBLIOGRAPHY ................. 101
ABSTRACT ....................... . .................................. 104
v i
LIST OF FIGURES
F ig u re Page
1. R ec tangu la r P la te Problem .................................................. 4
2. P la t e w ith Superimposed Grid ......................................... 6
3. Numbering of Nodes ........................................................... 12
4 . Temperature Response f o r Node of C o n s id e ra t io n . . 25
5. Number of O perations v s . Number of Nodes ............... 26
6. Computer Time v s . Number of Nodes fo r =1/16 . . . 27
7 . Computer Time vs. Number of Nodes fo r =2 ............. 28
8. T ru n ca tio n E rro r v s . Time f o r D i f f e r e n t ' s . . . . 33
9 . E rro r Curves fo r Choleski and Gauss Methods . . . . 34
10. E rro r Curves fo r G auss-Seide l I t e r a t i o n ................ 34
11. E rro r Curves fo r Booy Method ......................................... 35
12. O ptim izing fo r Booy Method ......................................... 39
13. O ptim izing fo r Gauss E lim in a tio n Method ............ 39
14. Optim izing fo r Cholesfci Method ............. 40
15. Optim izing f o r G auss-Seide l Method ....................... 40
16. R ec tangu la r P la te w ith Convective Boundaries . . . 65
v i i
4
6
12
25
26
2?
28
3334
34
35
39
39
40
40
65
NOMENCLAT CHE
AA Thermal d i f f u s i v i t y In th e y - d i r e c t i o n
AB Thermal d i f f u s i v i t y i n th e x - d i r e c t i o n
c S p e c i f i c h ea t of th e m a t e r i a l i n th e p l a t e
DX H o r izo n ta l s id e of r e c t a n g u la r p l a t e
DY V e r t i c a l s id e of r e c t a n g u la r p l a t e
KA Convective h e a t t r a n s f e r c o e f f i c i e n t on th e l e f t s id e
HB Convective h ea t t r a n s f e r c o e f f i c i e n t on th e lower s id e
HC Convective h e a t t r a n s f e r c o e f f i c i e n t on th e r i g h t s id e
HD Convective h ea t t r a n s f e r c o e f f i c i e n t on th e upper s id e
i , j , k In d ic e s in th e x - d i r e c t i o n , y - d i r e c t i o n , time
KA Thermal c o n d u c t iv i ty i n th e y - d i r e c t i o n
KB Thermal c o n d u c t iv i ty i n the x - d i r e c t i o n
m A rb i t r a ry node in y - d i r e c t i o n
n A r b i t r a r y node in x - d i r e c t i o n
NA Number of nodes i n th e y - d i r e c t i o n
NB Number of nodes i n the x - d i r e c t i o n
T Temperature
TCA Temperature of f l u i d on the l e f t boundary
TCB Temperature of f l u i d on the lower boundary
TCC Temperature of f l u i d on th e r i g h t boundary
TCD Temperature of f l u i d on the upper boundary
At Time increment
Ax Grid spacing in x - d i r e c t i o n
v i i i
Grid spacing i n y - d i r e c t i o n
Densi ty of m a te r i a l in th e p l a t e
Ay
s
ix
CHAPTER I
INTRODUCTION
In r e c e n t yea rs th e re has been an ex te n s iv e development
of f i n i t e d i f f e r e n c e techn iques fo r s o l u t i o n of th e t r a n s i e n t
h e a t conduction equa t ion due to the a v a i l a b i l i t y of h ig h
speed d i g i t a l computers. I t i s th e purpose o f t h i s paper to
i n v e s t i g a t e and compare s e v e r a l i m p l i c i t methods fo r the
s o lu t io n of th e F o u r ie r equa t ion with r eg a rd s to t r u n c a t i o n
e r r o r , ro u n d -o f f e r r o r and computer time which probably a re
among the most im portan t f a c t o r s to be cons idered in choosing
a method.
The methods to be i n v e s t ig a te d in c lu d e an ex ten s io n of
a method r e c e n t l y developed by M. L. Booy ( 3 ) , and th e more
c l a s s i c a l backward d i f f e r e n c e method. The l a t t e r w i l l u t i
l i z e G auss-Se ide l i t e r a t i o n , Gaussian e l im in a t io n , and the
Choleski decomposit ion methods to so lve a system of s im ul
taneous eq u a t io n s . Although th e method by Booy u t i l i z e s a
backward time d e r i v a t i v e i t d i f f e r s from th e c l a s s i c a l back
ward d i f f e r e n c e method in t h a t the f i n i t e d i f f e r e n c e equa t ions
a r e no t s e t up i n a p en ta -d iagona l form.
The G auss-Se ide l method i s widely used in th e s o lu t io n
of hea t f low problems. Most of the l a r g e s c a l e i m p l i c i t
computer programs p r e s e n t ly i n use u t i l i z e i t e r a t i o n as a
method of s o l u t i o n , fo r i n s t a n c e Anderson ( 1 ) . The Booy
1
2
method was developed r e c e n t ly and published i n r e f e r e n c e ( 3) .
The au tho r of. t h i s paper claims t h a t the method i s s im pler
and more a c c u ra te than i t e r a t i v e p rocedu res . The Gauss e l i
m ina t ion method i s th e most commonly used method in s t r e s s
and s t r u c t u r e problems (2 1 ) . Among o th e r s i t i s being used
by Wilson (19). and the Rohm and Haas Company (2 1 ) . The
Choles.ki decomposit ion method i s being used i n a program
named SAMIS, v / r i t t e n by P h i lco Ford and i n COSMOS which i s
being used by the Boeing Co. (21 ) . ■ I t i s a l s o w r i t t e n f o r
the H ercu les Powder Co. by Dr. H .C h r i s t i a n s e n (21) .
To the a u t h o r ' s knowledge, the l a t t e r two methods a re
no t being used in th e s o lu t io n of hea t f low problems. How
ev e r , s in ce th e s e methods have been used s u c c e s s f u l l y in
s t r e s s and s t r u c t u r e problems to so lve th e same type of pro
blem t h a t o f ten occurs i n hea t flow a n a l y s i s (symmetric band-
m a t r i c e s ) a comparison w ith th e i t e r a t i o n and Booy methods
seems to be j u s t i f i e d . Also t h i s paper w i l l d i s c u s s th e so
l u t i o n of a two dimensional problem, whereas most comparisons
have been made fo r th e one dimensional ca se .
P rev ious Y/ork
There i s ex ten s iv e d i s c u s s io n on th e v a r io u s i m p l i c i t
and e x p l i c i t methods in th e l i t e r a t u r e . R ich tm yer•(15) l i s t s
13 d i f f e r e n t f i n i t e d i f f e r e n c e schemes rang ing from th e pure
i m p l i c i t to th e pure e x p l i c i t form. In deve lop ing th e back
ward d i f f e r e n c e method f o r so lv ing complex t r a n s i e n t h ea t
t r a n s f e r problems, Anderson e t . a l . (2) has a s h o r t compari
son between the Gauss e l im in a t io n method and th e a c c e le r a t e d
G auss -Se ide l i t e r a t i o n method, g iv ing s e v e r a l su g g es t io n s fo r
a c c e l e r a t i n g th e i t e r a t i o n s . Anderson (1) has a d i s c u s s io n
of p re se n t s p e c i a l purpose and g en e ra l purpose e x p l i c i t and
i m p l i c i t - t y p e programs inc lud ing a d i s c u s s io n on the Gauss-
S e id e l method and the most common l a rg e s c a le h e a t t r a n s f e r
programs. In a r e c e n t paper Gay e t . a l . (7) compare the ex
p l i c i t , the backward d i f f e r e n c e , and th e Crank-Nicolson me
th o d s , th e l a t t e r two u t i l i z i n g two d i f f e r e n t types of i t e r
a t i o n methods to o b ta in s o lu t io n of the s e t of s im ultaneous
e q u a t io n s . Gauraer (6) compares the forward d i f f e r e n c e , back
ward d i f f e r e n c e and mid d i f f e r e n c e methods f o r th e one dimen
s i o n a l case w ith convective and r a d i a t i n g boundary c o n d i t io n s .
CHAPTER I I
STATEMENT OF THE PROBLEM
The p a r t i a l d i f f e r e n t i a l equa t ion governing h ea t conduct
ion i n two dimensions w ith o r th o t r o p ic p r o p e r t i e s , p r i n c i p a l
axes a l ig n ed w i th th e co o rd in a te axes , and no i n t e r n a l hea t
g e n e ra t io n i s g iven by:
The problem t h a t w i l l be solved in t h i s paper i s the r e c t a n g
u l a r p l a t e problem shown in F ig . 1.
F i g . l . —Rectangular P l a t e Problem.
4
3
r DY
r e # ,HR
0
rc c> H c
*x-DXT dB jH B
kn
KB
r c o , H o
0
The convec t ive boundary cond i t ions a re as fo l lo w s :
1. IVca- tc o , y , t ) l KA*-KB%M1- -* b x j x=0
2. fTCB-T (x , 0, t )1 H3=-KA-~-- )L J * y / y=0
3 . [TCC-TO.X,y,t|HA=KB-|| ) x=DX
4. [TCD-T(x,DY,tllB=KA-^-| \/ ysDY
The i n i t i a l c o n d i t io n i s :
j 5. T (x ,y ,C )= f (x ,y )
A FORTRAN computer program was w r i t t e n f o r each method g iv ing
th e s o l u t i o n to the r e c t a n g u la r p l a t e problem w i th the
fo l low ing c h a r a c t e r i s t i c s :
1. O r th o t ro p ic p r o p e r t i e s ( a r b i t r a r y c o n d u c t i v i t i e s in
the two co o rd in a te d i r e c t i o n s ) .
2. D i f f e r e n t uniform f l u i d tem pera tu re on each f a c e .
3. D i f f e r e n t uniform hea t t r a n s f e r c o e f f i c i e n t on each
f a c e .
4. Uniform g r id spacing but d i f f e r e n t i n the x- and y-
d i r e c t i o n s .
For th e sake of un ifo rm ity in the comparisons, the fo l low ing
case was chosen:
1. KA-KB
2. DX=DY
3. Two a d jace n t boundar ies i n s u l a t e d .
4 . Two ad ja c e n t f l u i d tem pera tu res equal to 100°.
5. I n i t i a l tem pera ture d i s t r i b u t i o n equal to 0° .
CHAPTER I I I
. DEVELOPING THE BACKWARD DIFFERENCE METHOD
I n th e case of r e c ta n g u la r p l a t e w i th convec t ive bound
a ry co n d i t io n s as given In F i g . 2 we can superimpose a g r id
on th e domain. We then 'approx im ate th e h e a t t r a n s f e r in the
p l a t e w i th th e h ea t t r a n s f e r between the v a r io u s nodes, i . e .
we r e p la c e the o r i g i n a l p a r t i a l d i f f e r e n t i a l equa t ion with a
s e t of one or more f i n i t e d i f f e r e n c e equa t ions which have to
be so lved .
F i g . 2 . —P la te w i th Superimposed Grid
6
1 2 3 n-1 n n-1 NB i
3
NA
ra-.l
m
m-1
2
1
7
Considering th e i n t e r i o r node (n,m) i n F i g . 2 the second space
d e r i v a t i v e s a t time k can. be approximated by:
2frb x 2 /
n,m,k
rp x'? — PT= i i k l l l . a i s z l a l n 5 1 b__fllffij.fe
(AX) 2
£ t_' i ? )n.m, k
_ ^n,m+l , k*^n , m - l ,k~2^n.,m,k (^y) 2
The time d e r i v a t i v e can be approximated a s :
_Tn,m,k~Tn ,m ,k - l 4 t /n ,m ,k -----A t -----------
In s u b s t i t u t i n g th e s e ex p ress io n s back in to the p a r t i a l
d i f f e r e n t i a l equa t ion (1) one p i l l no te t h a t while th e space
d e r i v a t i v e s a r e approximated a t t ime k, the d e r i v a t i v e w ithv
r e s p e c t to time i s a c t u a l l y approximated a t t im e:
- fc-i
To compensate f o r t h i s we can express th e space d e r i v a t i v e s
as a weighted average between time k and t ime k -1 . S e t t i n g
th e f r a c t i o n of space d e r i v a t i v e a t time k equa l to <f> , g ives
(assuming KA-K3, &x=Ay, and 6 = :
^n ,m,k“Th , r a ,k - l=® [ ^ Tn + l sm,k+Tn - l ,m ,k ~ 2Tn,m,k+ i n,m+l,k+
Tn ,m - l , k ”2Tn,m ,k^f ^Tn + l ,m ,k - l +^ n - l , m , k - l “ 2Tn , m , k - l +
Tn ,m + l ,k - l +Tn , m - l , k - l “ 2Tn,m,k-l^j ^
This method has been r e f e r r e d to by C ra n d a l l , Richtmyer (15)
and o th e r s . Two s p e c i a l cases a r e ob ta ined when <|> =1 andcj>=0.
S e t t i n g <J) “0 i n (2) r e s u l t s i n an eq u a t io n where Tn ^mj^
can be solved f o r e x p l i c i t l y i n terms of tem pera tu res a t time
k -1 . The method i s c a l l e d th e pure ly e x p l i c i t or the f o r
ward d i f f e r e n c e method s ince the time d e r i v a t i v e i s ev a lu
a ted a t T. m i, i o r in o th e r words forward from th e p o in t ofn ni ^c o n s id e r a t i o n ^njiri}u p
s e t t i n g <p s i i n (2) r e s u l t s i n an e q u a t io n where one
known tem pera tu re a t time k-1 i s expressed i m p l i c i t l y i n
terms of s e v e r a l unknown tem peratures a t t ime k. A s e t of
s im ul taneous , l i n e a r , equa t ions w i l l r e s u l t . So lv ing the
s e t w i l l y i e l d the d e s i re d tem pera tu re d i s t r i b u t i o n . This
method i s c a l l e d th e pure ly i m p l i c i t or the backward d i f f e r
ence method s in c e the time d e r i v a t i v e i s eva lua ted backward
from th e p o in t of c o n s id e ra t io n Tn jm j^.
At <J> =f an a r i t h m e t i c average of th e space d e r i v a t i v e s
a t t im es k-1 and k i s obta ined and th e method i s a p p ro p r i
a t e l y c a l l e d th e m id -d i f fe re n ce or the Crank-Nicolson method.
I t should a l s o be noted t h a t any value o f O ' w i l l r e s u l t in
a system of equa t ions t h a t have to be solved s im u l taneous ly .
Both th e e x p l i c i t and the i m p l i c i t methods have t h e i r
advan tages . The forward d i f f e r e n c e method i s easy to s e t up
and i t i s claimed t h a t i t has an advantage over the backward
method in the computation time per time s t e p . However, s t a
b i l i t y f o r th e two dimensional forward method i s only assured
when©> which pu ts a severe r e s t r i c t i o n on the s i z e of the
time increm en t . The backward method i s inconven ien t i n t h a t
i t r e q u i r e s a s o lu t io n of s imultaneous eq u a t io n s . However,
more freedom i n choice of v a r i a b l e s i s ob ta ined s in ce the
method i s s t a b l e f o r a l l va lues of & . A lso , s teady s t a t e
9
s o lu t i o n s a re obtained immediately by using an i n f i n i t e time
s t e p . The m id -d i f fe re n ce method r e q u i r e s a s o l u t i o n of sim
u l taneous equa t ions and i s s t a b l e , but f r e q u e n t ly develops
o s c i l l a t i o n s when a sudden change in boundary co n d i t io n s
occu rs .
This paper d ea ls p r im a r i ly with the pu re ly i m p l i c i t or
i . e . the backward d i f f e r e n c e method. For a r e c t a n g u la r p l a t e
problem we w i l l ge t nine c h a r a c t e r i s t i c f i n i t e d i f f e r e n c e
eq u a t io n s , one fo r each corner node, one f o r each boundary,
and one f o r th e i n t e r n a l nodes . In the fo l lo w in g th e se
equa t ions a re developed and put i n a form convenien t f o r
s o lu t io n on a d i g i t a l computer.
The therm al d i f f u s i v i t i e s in the x - and y - d i r e c t i o n s
a r e de f ined a s :
Making an energy balance on an i n t e r n a l node using the
lumped c a p ac i tan c e approach v/e ge t :
where Tnjm jj. i n d i c a t e s th e tem pera tu re a t node (n,m) a t time
k. This equ a t io n can be reduced t o :
Summary of Node Equations
AB=KB/$c and AA=KA/$c
A t(A B )r., fa x 2
10
- A t(AA) A t( AA) __m . .Ay2 1n , r a - l , k “ Ay2 Tn,m+A,k- i n ,m ,k - l
Making an energy ba lance on node (1 ,1 ) we ge t co r re sp o n d in g ly :
i . k - Ti . i . k * TCA- Ti , i , k * TCB- !ri a , k =A.y__ 4.x__ 1 1
KAax/ 2 KBAy/2 HAAj72 HBax/2
which can be reduced t o :
,AA4t , AB4t . HAAt , HB4t . l >rp , ABAt™' CIv2 ' + ^ 1 " + “H E + ~ 5 E + a )Ti * i , k + 1 5 ^ 2 , 1 , k
+ AA^Xp = - Tl , l , k - 1 . HAAt HBAtT0BAy2 1 ’ 2 ’ lc 2 S04XTCA- Jc4yICB
S im i l a r ly we get f o r th e fo l low ing boundary nodes:
NODE <1,NA): - ( M | l + m ± t M g t ^ ) T 1>MA)k ♦Ayc
*AAMrp, A , + ABAt Tl,NA, k - l HAAtA/2 Tl , S A - l , k A.**’ T2«HA>k =--------- 2 --------- Jc4x~
rA np /wi-p i > - / AAAt ABAt HBAt HoAt\mNODE (NB,1) : - ( — 2- - - - g - - ^
•TCA - HDAtm P r\ JcAy
AB41 Ax
Awn iiAAtm , 1 , k— 1 HCAtmpp HBAtrriprp2 TN B ~ l , l ,k - . p TKB,2,k = -------- V - ---- --- *ca£TGC ? £ a ^ CB
Tt
* y f cAx So Ay
NODS (NB,NA): - ( AAAt x ABAt M HCAt ^ HDAtAy4 Ax_ g o E * gcAy ANSjNA, k *■
THB,NA,k-l“ JC4X-
AAAtm , ABAtm n , TNB.NA,k-l HCAtTPp~~2 T^ j N A - l , k + ^ 2 -TKB-l,NA,k = -------------- ~ ^ f e TCC ~
HDAt~SC Ay
:TGD
Making th e energy ba lan ce on a node on th e lower boundary we
g e t :
11
- ( 2 -AAt f Ay2
2ABAt2AxLa ly ^ oKBAt
cAy * * ^ 9 Tn - l , l , k f
♦ ^ Tn , 2 , k - ' I n . l . b - X - 2^ TCB
Correspondingly we get f o r a node on th e upper boundary:
/(-.AA^t , ^ABAt 0HDAt ■> \m , , ABAtm .• ( ^ r f 2r 3 - *■ 2 jH y + 1>I n,N A ,k *■ — g -T n -l.N A .k +
«AAAtn ,HBAtn
Ay* Axc
jiBAtrn 0AAAtm rn oHDAtmpT-.Ax2~Tn+'1,NA,k 2Ay®”Tn,NA‘*1 ,k = ” Tn’NA’ k" 1 ~ fcAyTCD
The energy ba lance on a boundary node on th e l e f t hand bound
a ry g iv e s :
/ oAAAt oABAt oHaAt n \m , AAAtm i. AAAtm+ 2 - 2 - * - + D T 1>mjlc * - y T 1)m- l , k * 7 2 ™1 , m+1 , kAy*
^ oABAtj Ax
oHAAtn2 -2 ,m ,k = - T l , « ,k - 1 - 2P S TCA
S im i l a r ly we ge t f o r a node on the r i g h t hand boundary:
r -,AAAt ,- k2 — 5- +A y-
oABAtAX2
oHCAt^?CAX ^NB,m,k
AAAtmA y 2 ■NB j m+1, k +
AAA t rnK
O AB At mk - % B ,m ,k - l - 2p ^ j I C C
Given th e tem pera tu re d i s t r i b u t i o n a t t ime k-1 the s o lu
t i o n a t t ime k i s ob ta ined by so lv ing s im u l taneous ly the s e t
of equa t ions r e s u l t i n g from making th e h e a t ba lance on each
node.
By numbering the nodes from 1 to N as in d i c a t e d i n F i g . 3
th e s e t of equations can be transformed in to m a t r ix form which
then can be solved by using i t e r a t i o n , Gauss e l im in a t io n , e t c .
N-NE N
NB+1
1 2 3 4 MB
S e t t i n g :
F i g . 3*—Numbering of Nodes
A1 s HAAtgCAX
A2 = | l ^ t $ cA y A3 HCAt
* f C AX
A 4 - HDAt $ c Ay
THA =A y ^
THB ABAt" a x 2
- (THA-TKB-A1-A2-0.5 ) = 9 j
-(THA-THB-A2-A3-0.5) = 0 ;•
“ (2THA-2THB-1)
-(THA-THE-Al-A4-i) = © ■:
-(THA-TKB-A3-A4>i) = © 9
- ( 2THA-2THB-2A2-1) - Q 2
-(2THA-2THB-2A1-1) = © 4
-(2THA-2THB-2A3-1) = © 5
- ( 2THA-2TKB-2A4-1) = © 3
Using th e se s i m p l i f i c a t i o n s , the equa t ions a r e shown below
i n m a tr ix n o t a t i o n ap p l ied to a square p l a t e w ith n in e nodes:
e, THB 0 THA 0 0 0 0 0" -T]_/2-Al(TCA) -A2(TCB)
THB ©;l THB 0 2THA 0 0 0 0 T2 -T2-2A2(TCB)
0 THB ©* 0 0 THA 0 0 0 T3 -T3/2-A3(TCC)-A2(TCB)
THA 0 0 &H 2THB 0 THA 0 0 T4 -T4-2A1(TCA)
0 THA 0 THB Osr THB 0 THA 0 t 5 -T50 0 THA 0 2THB ©<. 0 0 T.EP. t 6 -T6-2A3(TCC)
0 0 0 THA 0 0 ©7 THB 0 T7 -T7/2-Al(TCA)-A4(TCD)
0 0 0 0 2THA 0 THB CD THE t 8 -Tg-2A4(TCD)
__0 0 0 0 0 THA 0 THB ©<J To -T9/2-A3(TCG)-A4(TCD)
13
I t i s noted t h a t th e n ine nodes are the fewest number of
nodes p o s s ib le i n order to r e p r e s e n t a l l the c h a r a c t e r i s t i c
d i f f e r e n c e equa t ions with equation. 5 r e p r e s e n t in g the only
i n t e r n a l node. As seen from th e c o e f f i c i e n t m a t r ix i t has a
d ia g o n a l band which can be made symmetric by m u l t ip ly in g the
i n t e r n a l equa t ions by a f a c t o r of 2.
I t should a l s o be noted t h a t the width o f th e band de
pends e n t i r e l y on the number of nodes i n the x - d i r e c t i o n , the
width being equal to 2(i\7B ) - l .
CHAPTER. IV
DISCUSSION OF METHODS OF SOLUTION
Gauss E l im ina t ion Method
The Gauss e l im in a t io n method i s developed in Appendix A,
p p . 43-53. As noted p rev ious ly t h i s method i s commonly used
i n s t r e s s a n a ly s i s and s t r u c t u r e problems, and most o f t e n
ap p l ied to symmetric m a tr ices where the non-zero elements
a r e co n cen t ra ted in a d iagona l band. In many hea t f low pro
blems the c o e f f i c i e n t m a tr ix i s a l so i n a s i m i l a r band form.
The number of c a l c u l a t i o n s can then be reduced by working
only w i th the elements w i th in the band. I t i s noted t h a t
th e i n d ic e s i n equations I , I I and I I I p .4? can run from n+1
to n+Mb-1 where Mb i s the width of txhe band. Furthermore i f
th e c o e f f i c i e n t m a tr ix i s used i n a symmetric form, we
observe t h a t
a i 3=a? i
The number of c a l c u l a t i o n s can then be f u r t h e r reduced by
le av in g the i unchanged in equa t ion I I , but by l e t t i n g j run
from i , i + 1 , ......... , n+Mb-1.
The s tandard Gauss e l im in a t io n working on a. f u l l m a tr ix
r e q u i r e s a number o f numerical o p e ra t io n s which i s equal to
n.3/34n2-n /3
o p e r a t io n s , n e g le c t in g a d d i t io n and s u b t r a c t i o n which a re
14
15
much f a s t e r than m u l t i p l i c a t i o n and d i v i s i o n (1 1 ) . For a
band m a t r ix , however, the number of o p e ra t io n s i s p r o p o r t i o
n a l to n(Mb)2 . Also the computer s to rag e can be cu t down from
n2 to n(Mb) (a l though t h i s was not done i n th e programs r e
por ted h e r e ) .
In many h ea t flow problems the a b s o lu te value of the
d iagona l term in th e c o e f f i c i e n t m a t r ix w i l l be the dominat
ing term in t h a t p a r t i c u l a r row. R e fe r r in g to an a r b i t r a r y
equa t ion of e l im in a t io n (see p #44) suppose th e e r r o r i n the
numerator i s £, . Accordingly the e r r o r i n th e unknown solved
f o r i n the k ' t h row i s approxim ate ly : .
Now i f
e r r o r (Xfc) ^ £
> I , th e magnitude of th e e r r o r in i s l e s s
than | £ ( (9) p. 104.
One major d isadvan tage w ith th e Gauss method i s t h a t i t
i s no t s e l f - c o r r e c t i n g , i . e . the ro u n d -o f f e r r o r w i l l accumu
l a t e p ro p o r t i o n a l to th e number of o p e r a t io n s , and the values
of th e v a r i a b l e s w i l l tend to be s u c c e s s iv e ly more i n a c c u r a t e
as the b a c k s u b s t i t u t i o n co n t in u es .
Gauss-Seidel Methodi
The G auss-Se ide l i t e r a t i v e method i s de r ived in Appendix
B p p . 53-62. The i t e r a t i o n can be d isc o n t in u e d by e i t h e r
sp e c i fy in g the number o f i t e r a t i o n s or by having a c e r t a i n
accuracy , e p s i lo n , s p e c i f i e d . However, i t should be noted
t h a t the magnitude of ep s i lo n does no t n e c e s s a r i l y sp e c i fy
th e e r r o r which may e x i s t in the tem p era tu re s , as t h i s i s a l s o
a f u n c t io n of th e r a t e of convergence. The f a s t e r th e r a t e
16
of convergence, the g r e a t e r the accuracy f o r a given e p s i lo n ,
because e p s i lo n i s checked a g a in s t the d i f f e r e n c e between two
su c ce s s iv e i t e r a t i o n s , not between the exac t s o l u t i o n and the
l a s t i t e r a t i o n .
One d isadvan tage w ith th e method i s t h a t i t does not
always converge. There a re s e v e ra l t e s t s f o r convergence
( see (14) p p . 56-61) and one s u f f i c i e n t , but n o t n ecessa ry
t e s t i s very convenient f o r most h ea t flow problems. Con
vergence i s assured i f :
a i i J ^
t h a t i s , i f the magnitude of each d iag o n a l element i s g r e a t e r
th an the sum of th e magnitudes of a l l th e o f f - d ia g o n a l e l e -
£j= i ii j i —l , 2 , . . . • , n
ments i n a row.
One major advantage with the G auss -Se ide l method i s t h a t
j t i s s e l f - c o r r e c t i n g . The e r r o r does not c a r r y over from
one i t e r a t i o n to an o th e r , but i s a f u n c t io n o f th e l a s t i t e r
a t i o n only which in c lu d es n^ com puta t ional o p e r a t i o n s . Also
advantage can be taken of a l l th e zeroes in t h e c o e f f i c i e n t
m a t r ix by no t inc lud ing them in the i t e r a t i o n .
The i n i t i a l so lu t io n v ec to r can be chosen a r b i t r a r y or
c a l c u l a t e d in some manner. With dominant d ia g o n a l e lem ents ,
one method i s to c a l c u l a t e T. a s :
% -Tia 11
One common way to improve the G auss-Se ide l i s to a c c e l e
r a t e the method using an o v e r - r e l a x a t i o n f a c t o r to a c c e l e r a t e
17
th e convergence. The p r i n c i p l e behind the a c c e l e r a t i o n i s
simply to improve the i n i t i a l guess to a va lue c l o s e r to the
a c t u a l s o l u t i o n , and i n t h i s way save computer time by
c u t t in g down th e number of o p e r a t io n s . One d isad v an tag e i s
t h a t th e a c c e l e r a t i o n r o u t in e s i n some cases may r e q u i r e more
o p e ra t io n s than what i s gained by cutting down th e number of
i t e r a t i o n s . Anderson (2) g ives two methods of a c c e l e r a t i o n .
In t h i s paper th e i n i t i a l guess i s assumed to be the temper
a t u r e s a t th e old t im e . This method, which a c t u a l l y i s a
form of a c c e l e r a t i o n , i s very s a t i s f a c t o r y f o r small time
s t e p s . For a d d i t i o n a l in fo rm a t io n see (14) p p . 144-156.
Cholesk i Decomposition Method
The Choleski decomposit ion method i s de r iv ed in Appendix
E, p p . 85-100. Like th e Gauss e l im in a t io n method i t i s p re
s e n t ly mainly being used in s t r u c t u r e and s t r e s s programs.
The number of o p e ra t io n s i s equal t o :
m u l t i p l i c . and d i v i s i o n s + n square ro o t so 2 3
The main advantage o f th e method i s the r e l a t i v e l y few oper
a t i o n s compared to Gauss e l im in a t io n and G auss -S e ide l . The
number of o p e ra t io n s can be a d d i t i o n a l l y cut down by tak ing
i n t o account band and symmetry c o n d i t io n s . However, l i k e
a l l e l im in a t io n methods, the ro u n d -o f f e r r o r w i l l accumulate
i n th e answers w ith in c re a s in g number of o p e r a t io n s . Another
d isad v an tag e i s t h a t th e Choleski method only can be app l ied
t o a symmetric and p o s i t i v e d e f i n i t e c o e f f i c i e n t m a tr ix , see
(1 8 ) , p .124 . I f th e m a t r ix i s not symmetric i t can ,be t r a n s -
18
formed by p rem u l t ip iy in g by I t s t r a n s p o s e . Most h ea t flow
problems a re not n a t u r a l l y s e t up in a p o s i t i v e d e f i n i t e form,
so t h i s i s one th in g t h a t must be checked b e fo re the method
i s a p p l i e d .
For a d d i t i o n a l d i s c u s s io n on the Choleski method see
(1 0 ) , p p .270-277, and (11) , -p.36 .
Gauss-Jordan E l im in a t io n Method
The Gauss-Jordan e l im in a t io n method i s de r ived i n App
endix D, p p .80-84. I t i s a m o d i f ic a t io n of th e c l a s s i c a l
Gauss e l im in a t io n with much the same advantages and d i s a d
v an tag es . The number of o p e ra t io n s i s equa l to :
+ nr - G ops.2 2
o r , i . e . somewhat l a r g e r number of o p e ra t io n s than th e Gauss
method. However, i t i s claimed t h a t a l i t t l e b i t i s gained
i n accuracy because of th e way the e l im in a t io n i s performed.
From the Appendix i t i s noted t h a t dur ing the r e d u c t io n i t
i s p o s s i b le t h a t a-Q may take on a zero v a lu e . I f t h i s
occu rs , th e f i r s t row' can be in te rchanged w ith ano ther row ■
which has a f i r s t element not equal to ze ro . In te rch an g in g
th e rows w i l l not a l t e r the o rde r of the unknown in th e f i n a l
column m a t r ix .
In t h i s paper th e method i s used to so lve a system of
HA s im ultaneous equa t ions i n the Booy method. For f u r t h e r
d i s c u s s io n on the method see James, e t . a l . (1 2 ) .
1 •*>J-9
Booy Method
The Booy method fo r the r e c t a n g u la r p l a t e problem i s
de r ived in Appendix G, p p .63- 79. I t i s q u i t e s im i l a r to the
backward d i f f e r e n c e method i n t h a t i t i s pu re ly i m p l i c i t
u t i l i z i n g a backward time s t e p . However, i t ' s s o lu t io n i s
based ori the forward e l im in a t io n , forward s u b s t i t u t i o n p r in
c i p l e . The tem pera tu res a t i = l and j=l,NA a r e l i n e a r fu n c
t i o n s of the tem pera ture of the f l u i d a t the l e f t - h a n d
boundary. The tem peratures a t i=2 and j =l,NA a r e l i n e a r
f u n c t io n s of the tem pera tures a t i = l , and so on. Solving f o r
tem pera tu re v e c to r s a t co n s tan t i and j = l , M and s u b s t i t u t i n g
w i l l r e s u l t in a s e r i e s of r e c u r s io n m a t r i c e s e v e n tu a l ly
y i e ld in g the tem pera ture of th e f l u i d on th e r ig h t -h a n d s id e
as a f u n c t io n of tem pera tures a t i = l and f l u i d tem pera tu re a t
t h e l e f t - h a n d boundary. Accordingly the tem p era tu res a t i= l
can be determined, by solving a s e t of NA sim ultaneous equa
t i o n s . The r e c u r s io n m a t r ic e s a re then used to f in d the
tem pera tu res a t the r e s t of the i - s t a t i o n s .
This p r i n c i p l e i s much l i k e t h a t of th e t r i - d i a g o n a l
s o l u t i o n , see I saacson (11) . To apply t h i s s o l u t i o n , th e
c o e f f i c i e n t m a t r ix has to be i n the form of t h r e e non-zero
d ia g o n a l s . Considering th e p en ta -d iag o n a l m a t r ix on page 12,
t h i s can be transformed in to a t r i d i a g o n a l form by a simple
m a t r ix p a r t i t i o n i n g . The normal procedure i s to e l im in a te
T^ from the f i r s t equa t ion , T2 from th e second, e t c . , always
so lv in g f o r the h ig h e s t order of the unknown. E ven tua l ly the
l a s t equa t ion can be solved f o r Tn . B a c k - s u b s t i t u t i o n w i l l
20
then y i e ld th e unknown tem pera tu res i n r e v e r s e o rd e r . How-'
eve r , in s t e a d of e l im in a t in g in th e f i r s t eq u a t io n , T2
could be e l im in a ted and so on u n t i l Tj_ i s found from the
l a s t eq u a t io n . Back s u b s t i t u t i o n would th en y i e l d T2 > T3 ,
e t c . This p r i n c i p l e of s o lu t io n would then be i n d e n t i c a l to
t h a t of the Booy method. Also, i n performing th e m a t r ix p a r
t i t i o n i n g , th e r e s u l t i n g subsystem w i l l be reduced i n s iz e
which i s the s t r e n g t h of the Booy method.
The main advantage w i th the Booy method seems to be the
f a c t t h a t in s t e a d of so lv ing a s e t o f n s im ultaneous equa
t i o n s we only so lve a system of NA equa t ions and then apply
a m a tr ix m u l t i p l i c a t i o n r o u t in e to f in d th e t o t a l tem pera tu re
d i s t r i b u t i o n . This may be p a r t i c u l a r l y advantageous when the
problem has most o f i t s nodes along th e x - d i r e c t i o n . As to
th e o th e r methods d iscu ssed i n t h i s c h a p te r , i t w i l l be a
d isadvan tage to have many nodes i n th e x - d i r e c t i o n because
th e bandwidth i s a f u n c t io n only of the number of nodes in
th e x - d i r e c t i o n . ?rhen th e p l a t e has a l a r g e r number of nodes
i n th e y - d i r e c t i o n and few i n the x - d i r e c t i o n , e x a c t l y th e
o p p o s i te i s t r u e . The o th e r methods w i l l be more advantage
ous because th e band w id th w i l l d ec rease and ac co rd in g ly a l s o
the number of o p e r a t io n s . Also the Booy method w i l l have an
in c reased number of s im ultaneous equa t ions to so lv e . The
s to ra g e requirem ent i s l e s s because of th e sm a l le r system of
equa t ions t h a t has to be so lved .
The method i s somewhat inconven ien t to s e t up. Since
th e problem i s solved by cons ider ing tem pera tu res in column
v ec to r s f o r c o n s ta n t i ’s , t h r e e d i f f e r e n t s e t of f i n i t e
21
d if fe rence equa t ions i n m a tr ix form have to be s e t up com
pared to one f o r the o the r methods., one a t the l e f t boundary,
one f o r the i n t e r i o r nodes, and one f o r the r ig h t -h a n d bound
a r y . There i s a lso l e s s freedom i n choosing th e boundary con
d i t i o n s . An in s u l a t e d boundary on the r ig h t -h a n d s id e fo r
in s tance , -would normally j u s t y i e ld a h ea t t r a n s f e r c o e f f i
c i e n t equal to ze ro . However, a t the r ig h t -h a n d boundary
the h e a t t r a n s f e r c o e f f i c i e n t appears i n th e denominator i n
th e Booy method. Accordingly , a d e r i v a t i v e w i l l have to be
eva lua ted and s e t equal to ze ro . In th e problem i n t h i s
pape r , the l e f t and the lower boundar ies were taken to be
i n s u l a t e d , thus avoiding the problem of d iv id in g by ze ro .
I t was found t h a t the elements of the r e c u r s io n m a t r ic e s
E^ may a t t a i n l a rg e a b s o lu te values w i th in c r e a s in g number of
nodes i n th e marching d i r e c t i o n . This r e q u i r e s inc reased
p r e c i s io n to avoid u ns tab le s o lu t io n . As an example f o r the
problem with two ad jacen t boundar ies i n s u l a t e d and th e o th e r
two a t 100°, elements of the order of 10-^ r e s u l t e d . The
i n s t a b i l i t y i s normally in troduced a t th e nodes f a r t h e s t from
th e s t a r t i n g po in t and then spreads i n a d i r e c t i o n o p p o s i te
t h a t of th e marching d i r e c t i o n . From eq u a t io n s ( 1 ) , ( 2 ) , and
(3) p.66 i t can be seen t h a t the numerical v a lu es of th e e l e -
— ments of E^ depend l a r g e l y on the num erica l s i z e of th e temp
e r a t u r e o f the f l u i d s a t the boundar ie s .
However, b es id es problems with l a r g e numerical v a lu e s ,
th e s t a b i l i t y co n d i t io n s a re a l s o dependent upon th e number
o f s i g n i f i c a n t f i g u r e s . When the boundary tem pera tu res were
22
decreased by a f a c t o r of 10, the method s t i l l became un
s t a b l e f o r small time increm ents . The reason i s t h a t during
th e c a l c u l a t i o n of the tem pera tu res , numbers t h a t only d i f f e r
a t a s i g n i f i c a n t f igu re -w hich exceeds the s in g l e p r e c i s io n
c a p a c i ty a r e su b t r a c te d and added. This problem resembles
an i n i t i a l va lue problem mentioned by Richtmyer (15 ) , in
which he concludes t h a t the method i s of l im i t e d va lue .
CHAPTER V
DISCUSSION OF RESULTS
For th e sake of comparison a l l th e runs were made w ith
th e fo l low ing param ete rs :
DX = DY - 1”
KA = KB = 1 B/sec i n F
HA * HB » 0 B/sec in^ F
HC = HD = 1 B/sec in2 F
TCA ® TCB = 100° F
TCC = TCD = 0° F
AA = AB - 1 sq. i n / s e c
NA = NB
Runs where made w ith 16, 49, and 100 nodes and a l l compari
sons made a f t e r 0 .2 seconds. The tem pera tu res were compared
a t th e same p o in t , node 4 f o r N=l6, node 17 f o r N=49, and
node 34 f o r N=100. The exact tem pera tu re a t t h i s po in t a f t e r
0 .2 seconds w i th th e parameters s p e c i f i e d was ob ta ined
from t a b l e s of s o lu t io n s fo r one d im ensiona l problems (20) .
By apply ing th e p r i n c i p l e of th e product s o l u t i o n , t h e s o lu
t i o n of the two-dimensional problem was o b ta in ed . For a
d e r i v a t i o n of the product s o lu t io n method, see Carslaw and
Jaeg e r ( 5) , pp .33-35 . Only one i t e r p o l a t i o n had to be made
i n th e t a b l e , and t h i s was done by f i t t i n g a second o rde r
23
24
parabo la to th r e e g iven p o in t s . The tem pera tu re according
to th e t a b l e was:
T ( s tan d a rd ) ~ 15.4207°
F i g . 4 p.25 shows th e tem pera tu re versus time response f o r the
node of c o n s id e r a t i o n . A ll comparisons were made a f t e r 0 .2
seconds, when th e tem pera tu re has reached about 15$ of the
s teady s t a t e va lve as i n d ic a t e d in th e f i g u r e .
A s u b ro u t in e , CLOX, was in c o rp o ra te d i n th e program in
o rder to e v a lu a te th e computer time d i f f e r e n c e per time s tep
f o r each method.
Computer Time
The computer time f o r a p a r t i c u l a r method and a p a r t i c u
l a r computer i s mainly a f u n c t io n of the number of a r i t h
m etic o p e ra t io n s (here def ined as m u l t i p l i c a t i o n s and d i v i
s i o n s ) , bu t a l s o of n ec essa ry l o g i c , number o f a d d i t io n s and
s u b t r a c t i o n s , e t c . F i g . 5 p .26 shows th e number of o p e ra t io n s
versus th e number o f nodes, and F i g . 6 p .2? and F i g . 7 p .28
show the computer time per t ime s te p fo r© =1/16 and © =2
r e s p e c t i v e l y . In F i g . 5 i t should be noted t h a t the o p e ra t io n s
f o r the G auss-Se ide l method i s given per i t e r a t i o n . This
i n d i c a t e s how many i t e r a t i o n s a r e r eq u i red f o r convergence
f o r th e G auss-Se ide l method to have a t o t a l number of oper
a t i o n s equal to th e number of o p e ra t io n s f o r th e o th e r methods.
From the f i g u r e i t can be seen t h a t a t 40 nodes fo r i n s t a n c e ,
th e G auss-Seide l method must have 5 i t e r a t i o n s or l e s s in
o rder f o r i t to be f a s t e r than th e Choleski method. From the
Tem
pera
ture
2?
1 2 3 4Time, sec .
F i g . 4 . —Temperature Response f o r Node o f C o n s id e ra t io n
Steady S t a t e100 '
80-
40-
0
Num
ber
of
Ope
rati
ons
26 '
F i g . 5 . —Number of Operations vs . Number of Nodes
3(104)
2(104)
104
010 20 30 40 50
Number of Nodes
T o p s . / i t e r a t i o n )
G auss-Se ide l
Choleski / ( o p s . / / time s tap )
Gauss E l im in a t io n ( o p s . / t im e s tep)
27
GaussG auss-Seide lCholeskiBooy
20 40 60 80 100
150-
100-
5 0 -
0
• -v©nOvOX0-pCO0S•HEH
0Ph0E
•HEH
0-P3Q.aoo
F i g . 6—Computer Time vs . Number of Nodes for© =1/16
Number of Nodes
Com
pute
r Ti
me
per
Tim
e S
tep
x 60
se
c.
2.8
800
600
400
200
020 40 60 80 100
F i g . 7 . — Computer Time vs. Number of Nodes f o r ^ -2Number of Nodes
29
graphs i t i s obvious, t h a t the number of o p e ra t io n s to a g r e a t
e x ten t i n d i c a t e s th e time per time, s t e p . For 6 =2 the
Choleski method proves to be th e f a s t e s t method; about twice
as f a s t as th e Gauss e l im in a t io n method and s e v e r a l t imes as
f a s t as th e i t e r a t i o n . The Bcoy method, however, has ap p ro x i
mate ly th e same computation t imes as th e C ho lesk i .
For 0 * 1 / 1 6 i t i s noted t h a t the Gauss e l im in a t io n method
i s by f a r . t h e s low est . The Choleski method i s f a s t e s t up to
about 70 nodes w h erea f te r th e Gauss-Seide l i s f a s t e s t .
In h e ren t i n the i t e r a t i o n method i s th e f a c t t h a t th e c lo s e r
to the a c t u a l answer the i n i t i a l guess i s , the fewer i t e r a
t i o n s a re r eq u i red f o r a s p e c i f i e d accu racy . The sm al le r 9 i s , the sm a l le r the time s tep i s , and th e c l o s e r the temper
a t u r e s a t time k+1 w i l l be equal to those a t time k, and
acco rd in g ly fewer i t e r a t i o n s a re needed f o r convergence. I t
should a l s o be noted t h a t the computer time per time s tep
was taken as the one a t time equal to 0 .2 s e c . when the
tem pera tu re was 15$ of th e s teady s t a t e v a lu e . However, f o r
t imes c l o s e r to zero th e re i s a g r e a t e r change in tem pera tu re
per time s te p and acco rd ing ly computer t imes were ap p rec iab ly
h ig h e r than the one p lo t t e d on the graph. I n f a c t the t o t a l
computer time f o r the Gauss-Seide l method up to time equal
to 0 .2 seconds i s probably g r e a t e r than th e t o t a l time fo r
Cholesk i or Booy. I t should a l so be noted t h a t f o r a l l values
of 1/16 the computation t ime f o r G auss-Se ide l i s h ighe r
than those of the Choleski and Booy methods f o r a l l g r id
sp ac in g s . -
30
Another i n t e r e s t i n g obse rva t ion i s t h a t due to the
i n t e r n a l lo g ic of the computer, the computer t ime per time
s te p f o r a p a r t i c u l a r g r id spacing w i l l i n c r e a s e i f
r each es a c e r t a i n l e v e l . For© =1/16 and © =1/4 and w ith 49
nodes the computer t i n e w i l l be 58/60 seconds. However,
fo r© =1 and©=2 and the same number of nodes , th e time jumpes
t o 98/60 seconds.
E rro r
Numerical s o lu t io n s of p a r t i a l d i f f e r e n t i a l equa t ions
a r e s u b je c t to d i f f e r e n t types of e r r o r , th e most im por tan t
of which a re t r u n c a t i o n and round-o ff e r r o r . The t o t a l e r r o r
i s de f ined as the d i f f e r e n c e between th e exac t and the ca lc u
l a t e d v a lu es . I t i s agreement in the l i t e r a t u r e t h a t i n most
a p p l i c a t i o n s the t r u n c a t io n e r r o r has a much g r e a t e r i n f l u
ence on the t o t a l e r r o r than does numerical o r ro u n d -o f f
e r r o r . However, the Booy method in t h i s paper dem onstra tes
th e in f lu e n c e round-o ff e r r o r can have.
The t r u n c a t i o n e r r o r i s due to the approxim ation of the
p a r t i a l d i f f e r e n t i a l equa t ion with th e f i n i t e d i f f e r e n c e
ex p re s s io n s . The t r u n c a t io n e r r o r de te rm ines the convergence
to th e exact s o l u t i o n . The ro u n d -c f f e r r o r i s due to the
chopping o f f of a number a f t e r a c e r t a i n number of s i g n i f i
cant f i g u r e s . Thus I f an i n f i n i t e number of decimals were
c a r r i e d along i n th e c a l c u l a t i o n s , the ro u n d -o f f e r r o r would
van ish . In p r a c t i c a l a p p l i c a t i o n the ro u n d -o f f e r r o r can be
decreased by using double p r e c i s io n which, however, g r e a t ly
in c r e a s e s the computer t ime.
31
T runca t ion Error
An ex p res s io n of the t r u n c a t io n e r r o r f o r the one dimen
s io n a l case i s given in Schneider (16) and Richtmyer (1 5 ) .
For th e two dimensional case expanding Tn,m,k-1> ^n-l ,m,k>
Tn-l ,m,k> Tn ,m - l ,k , Tn ,m - l ,k i n T a y l o r ' s s e r i e s about Tn>fflyk
g iv e s :
T n - l ,m ,k Is n ,m,k"Ax^ T/ ^ x )n ,m ,k4,.........+ax4/2 4 (^ f / d x 4 ) n>ni}k+ . . .
Tn ,m - l ,k =Tn}rn,k-Ay ^ T / ^ y ) n , mji£+ . . . . .+Ay4/2 4 ( b 4T/a>y4) n , m, k+ . . .
I 'n -1, m, k=^ n , m, k+AX /^x) n , m, k+.........+Ax4/24 (&4T /^>x4 ) n , m, k+ • • •
^n ,m +l,k=^n ,m ,k+Ay$T /& y )njm>k+ .........M y4/24(&4T/^y4 ) njmjk+. • •
^n ,m ,k+ l”^h .^m ,k+^ i^T /^ i)n ,m ,k+ .........+ A t - /2 (^ ^ T /^ t^ )njmjk+. • • •
The f i n i t e d i f f e r e n c e approximation can now be r e w r i t t e n by
s u b s t i t u t i n g th e s e r i e s expansions. Assuming-uniform
c o n d u c t iv i ty :
(™n, m, k- l “Tn , % k^ t =c / n , m, k+£ ^ ^ ) n , m, k* • •]
Assuming Ax=Ay:
^Tn - 1, m, ktTn*1, m, k*'Tn,m-1, k ^ n , mvl, k”4xn , m, / ' =
( i aI / i x 2) I, t ,Il)li4.Ax2/ 1 2 ^ 4T / ix 4 ) n)In. lc+ . . . J + (■& 2T /i>y2) n , m:a k+
Ay4/ 1 2 [ ( i ^ y 4 ) n>1Citf........... ] (2)
D ef in ing th e t r u n c a t i o n e r r o r as th e d i f f e r e n c e between the
l e f t - h a n d s id e s of equations (1) and (2) and assuming t h a t
T i s an exact s o lu t io n , i . e . :
A(& T/bt)-b2T/5x2- b 2T/6y2«0
g iv e s , i f we n e g l e c t h ig h e r o rde r te rms:
T runca t ion Error= A O t / 2) (d2T / d t 2)-Ax2/1 2 ( ^ 4? / ^ 4 )-Ay2/1 2 ( ^ 4T / V b
S u b s t i t u t i n g : ^ 2T / ^ t 2=l/dk2^ 4T/^Jc4+b4T/by^j
o r : T runca t ion H rro r=(A t /2 - (A 2x/12) ) (d4T/&x4+b4T/£>y4 )
I t i s seen t h a t the t r u n c a t i o n e r r o r i s minimized when:
At/2-dUAx2/12)=0
o r : A tA A x 2»i /6= ©
The t r u n c a t i o n e r r o r can now be c a l c u l a t e d by using a f i n i t e
d i f f e r e n c e approxim ation f o r ^ 4t / d x 4 a n d ^ T / ^ t . Of course
th e t r u n c a t i o n e r r o r can only be approximated t h i s way.
F i r s t of a l l t h e r e i s the i n i t i a l approxim ation b:/ only
cons ide r ing a few terms in the T a y l o r ' s s e r i e s . Second,
th e r e w i l l be an e r r o r i n approximating the t r u n c a t i o n e r r o r .
The r e s u l t s a r e shown i n F i g . 8 p .33 where t r u n c a t i o n e r r o r i s
p lo t t e d versus time f o r v a r io u s 0 ' s . The r e s u l t s seem to
v e r i f y t h a t th e t r u n c a t i o n e r r o r i s indeed minimized f o r& =1 /6 . A ll th e methods a r e in f lu en ced to the same degree by
t r u n c a t io n e r r o r s in c e they a l l use the same f i n i t e d i f f e r e n c e
approx im ations .
Round-Off E r r o r .
The e f f e c t of ro u n d -o f f e r r o r i s d i f f i c u l t to p r e d i c t
and many of th e p r e s e n t t h e o r i e s a re very l im i te d in a p p l i
c a t i o n . According to Hammond (8) " . . . A t p re se n t time th e r e
i s no r e a l l y s a t i s f a c t o r y t h e o r y , . . . " However, the l a r g e r
th e number of a r i t h m e t r i c o p e r a t io n s , th e g r e a t e r w i l l be the
Tru
ncat
ion
Err
or
33
F i g . 8 . —Trunca t ion E rro r vs. Time f o r D i f f e r e n t <91s
10
1
.1
.01
0
-.01
- . 1
-1
5.55 11.11 \ 16.66 22.22 \ \ Time x 100, sec
$ =1/16
O = 1/6
O =1/4
e=±9 =2
Err
or
x 10
0,
degr
ees
Err
or x
100
, de
gree
s34
F i g . 9 . —E rro r Curves f o r Choleski and Gauss Methods
F i g . 1 0 .—E rro r Curves f o r G auss-Se ide l I t e r a t i o n
F i g . 1 1 .—Erro r Curves f o r Booy Method
moot-iM0)oc*
oof-twuouflm
36
e f f e c t o f round ing . Rounding and t r u n c a t io n w i l l be i n t e r
r e la t e d to a c e r t a i n e x t e n t , but n e g le c t in g t h i s i n t e r r e l a t e d
e f f e c t i t should be v a l id to say th a t the t o t a l error w i l l be
ap p rox im ate ly a l in e a r fu n c t io n o f rounding and t r u n c a t io n .
In t h i s paper th e t o t a l error i s the d i f f e r e n c e between
1 5 .4 2 0 7 ° and th e v a lu e c a lc u la t e d by th e com puter. I t should
be noted th a t th e va lu e 15 .4207 i s n o t e x a c t , but depends on
th e accu racy o f th e t a b le s used and th e i n t e r p o l a t i o n made.
The t o t a l error as a fu n c t io n o f Q i s shown in F i g s . 9>
10 and 11 on p p .34-35* However, even a t 100 nodes th e accu
r a c i e s f o r th e C h o lesk i and th e Gauss e l im in a t io n methods
a re th e same t o th e th ir d and fo u r th d e c im a l . C on siderin g
th e g r e a t d i f f e r e n c e in th e number o f o p e r a t io n s , t h i s should
i n d i c a t e th a t the e f f e c t o f rounding i s n e g l i g i b l e . The
Booy method, how ever, i s g r e a t ly in f lu e n c e d by r o u n d -o f f error
b e in g u n s ta b le f o r 1=49 a n d © = l / l 6 and f o r N-100 and©£q|*,
Comparing th e e rro r o f th e Booy method w ith th e o th e r m ethods,
i t a l s o seems th a t th e r e s u l t s at© =1 and N=100 are i n f l u
enced by round ing , which d e c r e a se s w ith in c r e a s in g © .
Comparing th e t o t a l e r r o r s o f th e methods shows th a t
th e C h o lesk i d e c o m p o sit io n method and th e Gauss e l im in a t io n
method are i d e n t i c a l as fa r a s a ccu ra cy . The G a u s s -S e id e l
method i s somewhat l e s s a c c u r a te , p a r t i c u l a r l y f o r la r g e N1s .
T h is could o f co u r se be c o rr ec ted by d e c r e a s in g e p s i l o n which
would a g a in in c r e a s e th e number o f o p e r a t io n s and a c c o r d in g ly
th e computer t im e . The Booy method i s very a c c u r a te fo r sm e ll
N 's
37
Making a comparison between th e t r u n c a t i o n e r r o r and the
t o t a l e r r o r shows t h a t f o r 16 nodes th e r e i s a minimum e r r o r
a t 6 approxim ate ly equal to 1 /6 fo r a l l the methods. This
i s to he expected s ince i t i s the value of Q which minimizes
t h e t r u n c a t i o n e r r o r . This r e l a t i o n s h i p i s no t obvious fo r
l a r g e r number of nodes. One reason f o r t h i s i s t h a t th e
e f f e c t of the' t r u n c a t io n e r r o r d ec rease s w i th d ec re a s in g © .
From the t r u n c a t i o n e r ro r curve i t i s noted t h a t the e f f e c t
of t r u n c a t i o n w i l l dec rease w ith time f o r a g iven © . How
eve r , b e fo re th e tem pera tu re - t im e curve s t a r t s to l e v e l o f f
towards th e s teady s t a t e va lue , the t o t a l e r r o r w i l l c a r ry
over from time k to time k+1 and not d ec re a se w i th time l i k e
t r u n c a t i o n e r r o r . I t can a l s o be noted from th e curves t h a t
f o r small N 's , the e r ro r in c re a s e s g r e a t l y w i t h # , whereas
f o r N=49 and N=100, the e r r o r w i l l i n c r e a s e w i th in c re a s in g
© , but to a much l e s s e r degree .
Optimizing Computer Time and Accuracy
In a l l numerical work i t i s d e s i r a b l e to have the
r e s u l t s as a c c u ra te as p o s s i b l e . However, i f th e accuracy
i s improved by f o r in s ta n c e d e c r e a s i n g © , in c reased computer
t ime w i l l i n e v i t a b l y fo l low . In most a p p l i c a t i o n s the
computer time has to be l im i te d because of economical r e a so n s .
The q u es t io n i s then how much can be s a c r i f i c e d in accuracy
i n o rde r to cu t down the computer t im e, depending on the
p a r t i c u l a r problem.
One way to do t h i s i s to p lo t the f u n c t io n :
y=a(Tot. computer t im e ) + b ( to t a l e r r o r )
38
v e r s u s © , where a and b a re f a c t o r s whose va lues depend on
th e importance of computer t ime and t o t a l e r r o r fo r the
p a r t i c u l a r problem. To get a common, b a s i s fo r comparison
the p l o t s of computer time vs . © and e r r o r v s . & a re normal
iz e d , the e r r o r w i th r e s p e c t t o the l a r g e s t e r r o r a t ©=2 and
th e computer time with r e s p e c t to th e l a r g e s t t ime a t ©=1/16
choosing b equal to 1-a and 0£a£l w i l l r e s u l t i n a l l va lues
of y being between 0 and 1. The e r r o r w i l l in c re a s e w i th
in c re a s in g © and th e computer t ime w i l l d e c re a s e . This
a s s u re s t h a t y w i l l be minimized f o r a c e r t a i n value of Q .
T h is © w i l l acco rd in g ly g ive an optimized s o l u t i o n of the
method f o r th e par t icu lar va lue of a .
F ig s . 12-14 on. p p .39-4-0 show y v e r s u s © fo r the fo u r
methods, w i th a=-g-, and N-l6 . Based on th e s e param eters the
Choleski method i s b e s t w i th y=0.3> then Booy w ith y=0.33,
Gauss e l im in a t io n with y=0.38 and Gauss-SeideT w ith y - 4 .2 .
39
F i g . 1 2 .—Optimizing 0 f o r Booy Method
F i g . 1 3 .—O ptim iz ing© f o r Gauss-Slim. Method
40
F i g . 1 4 .—O ptim iz ing© f o r Choi .Ilethod
F ig . 1 5 .—Optimizing Q f o r Gauss-Seidel Ilethod
CHAPTER VI
CONCLUSIONS
The behav ior of s e v e ra l i m p l i c i t f i n i t e d i f f e r e n c e
methods in the s o lu t io n of a two-dimensional h e a t flow
problem w ith convec t ive boundary co n d i t io n s has been d i s c u s s e d .
In l i g h t of th e in fo rm at ion ob ta ined , s e v e ra l conc lus ions
may be drawn,
1. The computer time per time s te p i s a f u n c t io n of the
number of ar i thrae tr ic o p e r a t io n s . In most cases the Choleski
decomposit ion method i s f a s t e s t with the Booy method a l s o
being f a s t e r th an Gauss e l im in a t io n and G auss-Se ide l i t e r
a t i o n . However, f o r small time s tep s the G auss-Se ide l method
w i l l improve co n s id e rab le compared to the o th e r methods.
2. The t r u n c a t i o n e r r o r I s the same fo r a l l methods
because they a r e a l l using the same f i n i t e d i f f e r e n c e approx
im a t io n s . Also, the t r u n c a t i o n e r r o r can be minimized by
using a 0 = 1 / 6 . This was p a r t l y v e r i f i e d by th e r e s u l t s .
3 . The ro u n d -o f f e r r o r does not have any s i g n i f i c a n t
e f f e c t r e l a t i v e l y e a r ly i n th e t r a n s i e n t s o l u t i o n on the
Cho lesk i , Gauss e l im in a t io n , and Gauss-Seide l i t e r a t i o n
methods. This can be concluded because a l though the Gauss
e l im in a t io n and Choleski methods d i f f e r by a l a r g e number of
o p e ra t io n s t h e i r a c c u ra c ie s a r e the same. The Booy method
4 1
42
has s t a b i l i t y problems f o r small time s te p s and l a rg e number
of nodes . However., once s t a b l e , i t i s r e l a t i v e l y a c c u r a te .
As f a r as o v e r - a l l accuracy the .G auss e l im in a t io n and Choleski
decomposit ion a r e the most a c c u r a t e .
4 . The Choleski decomposit ion method i s f a s t e r and a t
l e a s t as a c c u ra te as t h e o th e r methods. The Gauss method i s
a c c u r a te , but' r e l a t i v e l y slow, whereas the Booy method i s
l im i te d i n a p p l i c a t i o n because of s t a b i l i t y problems.
5. An optimum O can be ob ta ined f o r a p a r t i c u l a r p ro
blem by a s s ig n in g weight f a c t o r s to accuracy and computer
t im e . The "b e s t" method can then be found f o r th e p a r t i c u l a r
problem simply by comparing th e optimum© 1 s of the v a r io u s
methods.
43
APPENDIX A
THE GAUSS ELIMINATION METHOD
THE GAUSS ELIMINATION METHOD
The system of equa t ions le ad in g to th e s o l u t i o n of the
backward d i f f e r e n c e method can be w r i t t e n i n th e form:
* a l l Tl +a12T2+.................. +alNTH = b l
a 21Tl +a22T2+.................. +a2NTN “ b2
aNlTl +aN2T2+ ....... ...........+aM % = bN
Using the Gauss e l im in a t io n method the f i r s t s t e p i s to
so lve the f i r s t eq u a t io n f o r T]_:
Tl=b l / a l l “ ^a 12^a l l ^ T2’' (a 13 /a n ) T3~ ...............- ( a i N/ a n ) T N
S u b s t i t u t i n g t h i s back in to th e remaining equa t ions of the
o r i g i n a l s e t and e l im in a t in g T1? a modif ied s e t of equa t ions
i s o b ta in e d :
a 22T2*a23T3+
a 32T2*a33T3*
+a2 # f = b2
+a3NTN = b3
aN2T2+aN3T3f '+aNNTN = bN
where:
44
4 8
a i r s i r Ai i /Au
bi =bi - a i l bl / a U
. •. • • •, i\
1 = 2 , 3 , ............ ,W
A s i m i l a r procedure can now be used to so lve f o r Tp in
th e primed system. Back s u b s t i t u t i o n w i l l f u r t h e r reduce th e
o r i g i n a l system. The e l im in a t io n o f Tn w i l l then y i e l d :
V v > * jj=n+l,n+2, ......... ,N ( I )
i , j= n + l ,n + 2 , . . . . . . N ( I D
i=n+l ,n+2 , • ......... N ( i l l )
equa t ions i s reduced to one equa t ion which can be solved
d i r e c t l y f o r T,^:T bN - lA.N-li N=DN /c-NN
The r e s t of th e unknowns a re then determined by back s u b s t i
t u t i o n i n t o equa t ion I where the necessa ry c o e f f i c i e n t s a re
determined from I I and I I I .
A computer program l i s t i n g of the backward d i f f e r e n c e
method w ith th e Gauss e l im in a t io n method of s o l u t i o n i s shown
on pp .47-52. The Gauss e l im in a t io n method i s conta ined in
th e su b ro u t in e ’'MUSS** and the flow c h a r t i s shown on p .4 6 .
The nomenclature in the flow c h a r t corresponds to t h a t of the
computer program.
Having ap p l ied t h i s procedure .N-l t im es , the o r i g i n a l s e t of
Read A,B,N,NB,JP,M
Eand width MB«2(NB)*1
K~1
r - i + i
Genera te A ( I , J )
I-Nil
Generate D(K)
NN-IP--MB-2
Compute X(Njl)I=IP
Generate H(IC,J).
T-■IP
KK=N-1
Compute X(XX,1)
KK=iK-1
+
46
SJOB THS750 MOD. GAUSS K. GUNDERSEN 7DIMENSION! W(101.1) .NAME (13) .0(101. 1).B(101.1).A(1G1.101)
C----- THIS SECTION READS AND WRITES INPUT OATA30 11=0
CALL CLOX(II)12=11-11 11 = 1 ILOGICAL ZEROREAD(5.10) (NAME(M).M*l.13)READ(5.11) NA.NB.IFREQREAD(5.12) DA.D8.DELT.TMAXREAD(5.12) AA.A3.FKA.FKBREAD(5.12) TCA.TCB.TCC.TCO.HA.HD.HC.HDREAD(5.13) ZERO
10 FORMAT(13A6)11 FORMAT(3110)12 FORMAT!8E10.0)13 FORMAT(LI)
N=NA*NBIF(.NOT.ZERO) GO TO 3 DO 1 1=1.N
1 W( I . 1 )=0.GO TO 4
3 READ(5.12) (W (I .1).I=1.N )4 WRITE(6.20) (NAME(M).M=1,13)
20 FORMAT(1H1.13A6)WRITE{6 * 21) NA.NB.DA.DB.DELT
21 FORMAT(1H0» 3HNA = »13 »11X,3HNB=,13.11X .3HDA=.E10.3.4X.3H0B=.E 10.3. 4X 1,5H0ELT=»E10.3)WRITE(6.22) AA.AB.FKA.FKB.TMAX WRITE(6 * 23) HA.HB.HC.HD.TCA.TCB.TCC.TCD
23 FORMAT(1H , 3HHA=E 1 0.3.4X ,3HHE3= .El 0.3.4X , 3HHC= . E10.3.4X, 3HHD=. E10.3 2.4X,4HTCA=,F7.1,4X,4HTC8=.F7.1.4X ,4HTCC=*F7.1.4X»4HTCD=*F7•1)
2 2 FORMAT( 1H . 3 H A A = » E 1 0 . 3 , 4 X . 3HAB= . E l 0 . 3 , 4 X , 3 HKA=. E l 0 . 3 , 4 X . 3 H K B = . E 1 0 • 3 3 . 4 X . 5HTMAX=. E 1 0 • 3 )WRITE!6.24)
24 FORMAT*1H0.42HTEMPERATURES PRINTED IN FORMAT— (N) TEMP--) 99 FORMAT*1H ,1OX,5HTIME=,E12.5,16HI NT. TIME DIFF.=,I12)
C----- THIS SECTION GENERATES NECCESSARY PARAMETERSICO = l TIME=0.0 ANA=NA BNB =NBY=DA/(ANA-1•)X=DB/*BNB— 1•)THA=DELT*AA/*Y**2.)THB-DELT*AB/(X**2.)RHOC=FKA/AAAl=HA*DELT/*RHOC*X)A2=HB*DELT/*RH0C*Y)A3 = HC*0ELTy'*RH0C*X)A4=HD*DELT/*RHOC*Y)N=NA*NBN3=NB**NA-l>+l N2=N-1 N4=NB-1 N6=N3+1 N 8=1+NB N 9 = N 3—NB N10=2*NB Nil=N—NB DO 45 1=1,N
45 O*I,1)=0.D*1♦1)=-Al*TCA— A2*TCB 00 41 I=2 * N4
41 D*I,1)=-2.*A2*TCB0(NB * 1)=-A3*TCC-A2*TCB D*N,1)=-A3*TCC-A4*TCD O (N 3»1)=-A1 *TCA-A4*TCO DO 42 I=N6,N2
42 0*I.1)=-2.*A4*TCD DO 43 I=N8•N9 »NB
■Pco
43 DtI»1 ) =—2.*A1*TCA DO 44 I=N10»N11.N8
44 Dt I . 1 )=-2.*A3*TCCCALL MATOUTt D * N « 1*101*1.0.0.0.0)CALL CLOX(I I)12=11-11WRITEi6*99) TIME.12 11 = 11CALL MATOUT(W.N.l.101*1.0.0.0.0)TI ME=TIME+OELT
51 CONTINUEC----- THIS SECTION GENERATES THE PENTA-DIAGONAL MATRIX
DO 9 1=1.N DO 9 J=1•N
9 A(I.J )=0.DO ICO I=1.N
100 A(I.I)=-t2.*THA+2.*THB+l.)A{1.1 )=-(THA+THB + Al + A2+0.5>A(N .N ) =— (THA + THB+A3+A4+ 0•5)A(NO «N B )=-t THA+TH8 + A2+A3+0.5)A(N3*N3)=-(THA+THB+A1+A4+0.5)N 1=N—NB DO 110 1=1.N1 N 13=I+NB
110 A d ,N13)=THA N 12=NB+1 DO 120 I=N12.N N 14=I—NB
120 At I»N 14 > =THA NN = 0DO 130 1=1.N2 NN=NN+1IF(NN.EQ.NB) GO TO 14 At I , 1 + 1)=THB GO TO 130
14 NN=0
vO
130 CONTINUE KK = 0DO 15 1=2,N KK=KK+1IFCKK.EO.NB) GO TO 16 A ( I * I— 1)=THB GO TO 15
16 KK = 0 IS CONTINUE
00 210 I=2•N4 N 5=I+NBA(I, I)=-(2.*THA+2.*THB+l.+2.*A2>
210 A (I,N5)=2,*THA DO 220 I=N6 * N2 N 7=1—NBA (I, I )=— (2, *THA+2,*THB+1•+2,*A4)
220 A (I,N7)=2.*THADO 230 I -=N8 * N9 » NBA(I,I)=-(2,*THA+2.*THB+l.+2,*A1)
230 A( I,1+1)=2•*THBDO 240 I=N10 * NX 1 ,NB A (I• I )=-(2.*THA+2.*THB+l.+2.*A3>
240 A (I» I— 1)=2.*THBC----- THIS SECTION GENERATES THE CORRESPONDING COLUMN MATRIX
DO 46 1=1,N 46 Q(I * 1 )*D(I•1)-W(I,1)
B(l.l)=D(l,l)-vm,l>/2.8(NB * 1)=D(NB * 1) — W (NO * 1)/2•B(N3,l)=D(N3,l)-W(N3,l)/2.8(N,1 )=D(N,1)-W(N.l )/2.
C----- THIS SECTION SOLVES THE SYSTEM USING MOOIFIED GAUSS ELIMINATIONCALL MAUSS(A •101*101 ,D »101 •W * i 01» N,NB)CALL CLOX(H)12=11-111 1=1 IIF(TIME-TMAX) 40,47,47
'Ji.O
48 IF(I CO•EQ•IFREQ)GO TO 49 ICO=ICO+lTIME=TIME+DELT GO TO 51
49 WRITEl6,99) TIME.12CALL MATOUTtW.N.l.101.1 .0.0.0.0)ICO = lTIME=TIME+DELT GO TO 51
47 CONTINUE GO TO 30
333 STOP ENO
$ IBFTC MAUSS DECKSUBROUTINE MAUSS(A •I A •JA.B.IB•X ,I X ,N.NB)DIMENSION A {IA.JA).B{IB•1)»X(IX.1).H (101.101).D(101)
C----- MAUSS SOLVES THE SYSTEM USING MODIFIED GAUSS ELIMINATIONMB=2*NB+1 M=N-1 IP = 2DO 6 K=1.M D ( l< ) =B I K . 1 ) /A ! K . K )NN= IP +MB—2 IF(NN.GT.N) GO TO 10 GO TO 1 1
' 10 NN=N11 CONTINUE
DO 16 I=IP,NN 16 B(I,1)=B<I,1)-A<I,K)*D(K)
00 5 J=IP.NN H(K,J)=A(K,J)/A(K,K)DO 5 I=IP.NNA (I . J ) = A(I.J)—A{I»K / *H(K .J )
5 CONTINUE6 IP=IP+1
X {N » 1 )=B(N,1)/A(N.N)
vnH-*
GO TO 12
JP=N KK=N
a s u m =o *KK=KK—1 NN=JP+MB-2 IFINN.GT.N)GO TO 13
12 NN=N13 CONTINUE
DO 9 J=JP.NN9 SUM=SUM+H(KK.J)*X(J.l)
JP=JP-1X (KK•1)=D(K K )— SUM IF(KK.NE.l) GO TO 8 RETURN END
ro
53
APPENDIX B
THE GAUSS-SEIDBI. METHOD
THE GAUSS-SSIDSL METHOD
The system of equat ions to be solved s im u l tan eo u s ly i n
th e backward d i f f e r e n c e method can be w r i t t e n i n th e form:
a l l Tl +a12T2+............. +aln Tn=bl
a21Tl +a22T2+ * *....................... +a2nTn=b2
an l Tl +ab2T2 * .........................+annTn=bn
The G auss-Se ide l method i s a l i n e a r , i t e r a t i v e process
f o r approximating th e s o l u t i o n of th e n s e t of s im ultaneous
e q u a t io n s . The b a s ic p rocedure of the method i s to assume
an a r b i t r a r y i n i t i a l s o l u t i o n v e c to r . The f i r s t equa t ion i s
then solved f o r T- , the second f o r T2 > e t c . A new va lue f o r
T- i s c a l c u l a t e d usj.ng th e i n i t i a l g u esses . This p rocess i s
r e p e a te d , r e p l a c in g the old T^ va lues by the newly c a l c u l a t e d
ones.
Applying t h i s procedure to the system of equa t ions above
and assuming an i n i t i a l v e c to r of T-^, T j ^ , ............»Tn,k: g iv e s :
m . b j - a j j g g t r ......... ..l , k - l ® 11
54
55
i - l n
Ti , k + l
The process i s repea ted u n t i l T j ^ + i s a t i s f i e s a c e r t a i n
accuracy check, i . e . t i l l the a b s o lu te value of the d i f f e r
ence between T^ and i s l e s s than some p red esc r ib ed
.value,, say, e p s i l o n .
A computer program l i s t i n g of backward d i f f e r e n c e method
u t i l i z i n g the Gauss-Seidel method i s shown on p p .57-62. The
G auss-Seide l method i s d esc r ibed in the s u b ro u t in e UGAS3I!I
and th e flow c h a r t of the sub ro u t in e i s shown on p.56.
56
Guess i n i t i a l s o lu t io n vec to r
$ JOB THS750 ITERATION K. GUNDERSEN 7DIMENSION W{100 *1).NAME(13)•D (100•1)•B (100•1),A<100•100)
c----- THIS SECTION READS AND WRITES INPUT DATA30 11=0
CALL CLOXCII)12=11-11 11 = 1 ILOGICAL ZEROREAD( 5 »10) (NAME(M),M=l*13)READ(5*11) NA.NBtIFREQREAD(5,12) DA.DD.DELT.TMAX.EPSREAD(5,12) AA.AB.FKA.FK8READ(5*12) TCA,TCB,TCC,TCD.HA,HB,HC»HOREAD(5*13) ZERO
10 FORMAT!13A6)11 FORMAT(3110)12 FORMAT(8E10.0)13 FORMAT(LI)
N=NA#NQIF(.NOT.ZERO) GO TO 3 DO 1 1 = 1,N
1 W ( I * 1 ) =0.GO TO 4
3 READ(5,12) (W (I,1)•I=1,N)4 WRITE(6,20) (NAME(M ),M=1,13)
20 FORMAT(1H1*13A6)WRITE(6,21) NA,NB,DA,DB,DELT
21 FORMAT(1H0,3HNA=,I3.11X ,3HN0=,13,11X ,3HDA=»E10. 3.4X»3HDB=,E10.3, 4X 1,5HDELT=,E10.3)WRITE(6,22) AA.AB.FKA.FKB.TMAX WRITE(6,23) HA.HB.HC,HD,TCA,TCB,TCC,TCD
23 FORMAT(1H ,3HHA=E10.3,4X,3HHB=,E10.3,4X,3HHC=,E10«3•4X•3HHD=,E10.3 2,4X* 4HTCA=,F7.1,4X,4HTCB=,F7.1.4X,4HTCC=,F7.1,4X.4HTCD=«F7.1)
22 FORMAT ( IH , 3HAA = , E1 0.3,4 X , 3H AB= ,E10.3,4X, 3Hi< A= , El 0.3,4X , 3HKB=» E 10 • 33•4 X •5HTMAX = * E 10.3}WRITE(6,24)
M l'O
24 FORMATCIH0.42HTEMPERATURES PRINTED IN FORMAT— (N) TEMP--)99 FORMAT{1H •1 OX•5HTIME = • £12•5.16HI NT• TIME DIFF.=.I12)
C----- THIS SECTION GENERATES NECCESSARY PARAMETERSICO = l TIME=0.0 ANA =NA BNB=NBY =DA/(ANA— 1•)X=OB/(BNB— 1 • )THA=DELT*AA/(Y**2. )THB=DELT*AB/(X**2. )RHOC=FKA/AAAl=HA*DELT/(RHOC*X>A2=HB*DELT/(RH0C*Y>A3=HC*D£LT/<RHOC*X)A4=HD*DELT/(RH0C*Y)n =n a *n b00 9 1=1.N
C----- THIS SECTION GENERATES THE PENTA-D!AGONAL MATRIXDO 9 J=l.N
9 All.J)=C.DO 100 1=1.N
100 A (I * I )=— {2**THA+2»*THB + 1•>A{1.1)=— (THA+THB+A1+A2+0.5)A (N » N )=— (THA + THB +A3 + A4 + 0.5)A (NB * NB ) =-(THA+THB+A2+A3+0.5)N3=NB*INA-1)+lA(N3»N3)=— {THA+THB+A1+A4+0.5)N1=N-NB DO 110 I-~1.N1 N 13 = I+NB
110 A (I» N 13 ) =THA N 12 =NB+1 DO 120 I=N12.N N 14=I—NB
120 A {I *N 14)=THA
VST.CX>
GO TO 14
N2=N-1 NN = 0DO 130 1=1.N2 NN=NN+1 IF(NN.EQ.N0 )A tI * 1 + 1)=THB GO TO 130
14 N N = 0 130 CONTINUE
KK = 0DO 15 1=2.N KK=KK+1 IF(KK.EO.NB) GO TO 16 A (I,I— 1)=THB GO TO 15
16 KK = 015 CONTINUE
N4=NB-1DO 210 1=2.N4 N 5=I+NBAt I.I)=-(2.*THA + 2.*THB+l.+2.*A2)
210 At I,N5)=2.*THA N6=N3+100 220 I=N6.N2 N 7=I—NBAt I.I>=-(2.*THA+2.*THB+l.+2.*A4)
220 At I»N7)=2.*THA N8=1+NB N 9=N 3—NBDO 230 I=N8.N9,NBAt I.I)=— t 2•*THA + 2.*THB+1»+2.*Ai)
230 At I.1 + 1)=2.*THB N 10 = 2*NB N i l = N - N BDO 240 I=N10.N11.NBAt I.I)=-t2.*THA + 2.*TH8+1.+2.*A3)
VJlvD
240 At I•1-1)=2.*THBC----- THIS SECTION GENERATES THE CORRESPONDING COLUMN MATRIX
DO 45 1=1.N45 D M * 1 )=0.
DC 1.1)=-Al*TCA-A2*TCB DO 41 1=2.N4
41 DC I»1)=-2.*A2*TCB DCNB.1)=-A3*TCC-A2*TCB DCN,1>=-A3*TCC-A4*TCD DCN3,1)=-Al*TCA-A4*TCD DO 42 I=N6.N2
42 DC I * 1)=-2.*A4*TCD DO 43 I=N8,N9.NB
43 D M • 1 )=-2.*Al*TCA DO 44 I=N10.N 11 • N8
44 DC I.1>=-2.*A3*TCCCALL MATOUT(D.N . 1.100.1.0.0.0*0)CALL CLOXtI I)12=11-11WRITEC6.99) TIME,12 11=11CALL MATOUTCW.N.l.100.1.0.0.0.0)TIME=TIME+DELT
51 CONTINUEDO 46 1=1.N
46 B(I.1)=D tI.1)— W 11 • 1)B C1.1)=D(1.1) — Wtl.l)/2.B CNB , 1 ) =D { NB » 1 ) — W (NB • 1 ) /2 •BCN3,1)=DCN3,1)-WtN3.1)/2.BIN, 1 )=DCN, 1 >-VK CN.l ) /2.
C----- THIS SECTION SOLVES THE SYSTEM USING GAUSS-SEIDEL ITERATIONCALL GA SEICA.100.100,B,100.M.100.NB » N ,EPS)CALL CLOXCI I)12=11-11 I 1 = 1 IIFtTIME-TMAX) 48,47,47
ONo
48 IF(ICO.EQ.IFREQlGO TO 49 ICO=1CO+1TIME=TIME+DELT GO TO 51
49 WRIT£(6,99) TIME.12CALL MATOUTIW.N.I.100*1.O.G.O.G)ICO = lTIME=TIME+DELT GO TO 51
47 CONTINUE GO TO 30
333 STOP END
iIBFTC CASE I DECKSUBROUTINE CASE ItA•I A •JA*B .IB.T »IT.NB,N ,EPS) DIMENSION A(IA•JA). B < I O » 1) * T (IT*1)» X(100 »1)
C----- GASEI SOLVES THE SYSTEM USING MOOIFIED GAUSS SEIDELDO 18 I = 1 ♦ N
18 X(I.1)=T(I.1)23 N4=N—NB 17 J=1
N 1 —NB +1 N6=N1
6 SUM = 0•DO 5 1=1.N1IF( I .EQ.J) GO TO 5SUM=SUM+T(I»1)*A(J»I)
5 CONTINUET(J,1> = (B(J»1) — SUM)/A(J * J)J=J+1 N1=N1+1 N3=J-1IF(M3.EQ.N6) GO TO 7 GO TO 6
7 N2 = 28 SUM=0«
On
DO 9 I —N2*N1 IF ( I«EQ•J ) GO TO 9 SUM=SUM+T(I.1)*A(J.I)
• 9 CONTINUET(J. 1) = CQ(J.1)-SUM)/A<J,J)J=J + 1 N2=N2+1 N 1=N 1 + 1 N 3 = J— 1IF(N3.EQ.N4) GO TO 19 GO TO 8
19 N1=N1-1 10 SUM = 0 •
DO 11 I=N2.N1IF{I.EO.J) GO TO 11SUM=SUM+T(I.1)*A(J.I)
1 1 CONTINUET(J,1)=(B(J,1)— SUM)/A(J •J)J=J + 1 N2=N2+1 N 3 = J— 1IF(N3.EQ.N> GO TO 12 GO TO 10
12 CONTINUEDO 13 1=1,N DIFF=X(I*1)— T(I*1)IF{ABStDIFF).GT.EPS) GO TO 14
13 CONTINUE GO TO 16
14 CONTINUEDO 15 1=1,N
15 XCI, 1 )=T(1,1)GO TO 17
16 RETURN END
ON
63
APPENDIX C
THE BGOY METHOD
THE BODY METHOD*
The c l a s s i c a l backward d i f f e r e n c e method was developed
f o r a r e c t a n g u la r p l a t e problem w ith convec t ive boundary
c o n d i t io n s i n Chapter I I I . In. developing th e Booy method,
the energy ba lance f o r each node i s made as b e fo r e . However,
now the f i n i t e d i f f e r e n c e equa t ion i s solved w i th r e s p e c t to
th e tem pera tu re of th e node f a r t h e s t along i n th e marching
d i r e c t i o n and the tem pera tu res f o r i= c o n s ta n t and j=l,NA
considered as a column v e c to r . R e fe r r in g to F i g . 16 th e f i
n i t e d i f f e r e n c e equa t ions f o r a l l th e nodes on th e l e f t - h a n d
boundary as developed in the backward d i f f e r e n c e method can
be r e w r i t t e n i n th e form:
N0D3 ( 1 , 1 ) : T2 ) J A = l / IH B(0 ,T1 )1 )k-(lHA)T1) i t l i r 4Tl i l > k . r
A1(TCA)-A2(TCB))
NODE (1,M): T2>m, k =
(THA)Ti)m+i jk --(2Al)TCA)
NODE (1,NA): T2>NA>k = l /T H B (^ T 1#M >kr(THA)T1>NA»i j k -
«T1 , NA, k-1“A1(TCA)" A 4 (TCE° )
The l e f t s id e of th e s e equa t ions c o n s t i t u t e the s e t of
*This method was developed by M.L. Booy f o r P o i s s o n ’s equa t ion w i th homogeneous boundary c o n d i t io n s . (See r e f . ( 3 ) ) . I n t h i s paper th e method i s de r ived fo r the F o u r ie r conduct i o n equ a t io n ap p l ied to the two dimensional r e c t a n g u la r p l a t e problem w i th convec t ive boundary c o n d i t i o n s .
64
65
tem pera tu res a t i~2 fo r j=l,MA. In g e n e ra l , we can express
th e tem pera tu res a t .the boundary a t time k s im u l taneous ly as
a column v e c to r :
Tl ,k =
1 2 3 n-1 n n+1 NB
T ( l , l )
T(1,M)
T(1,NA) k
F i g . 1 6 .—Rectangular P l a t e w i th Convective Boundaries
J
NA
m+1
m
m-1
2
1
66
The s e t o f equa t ions can no if be r e w r i t t e n i n m a tr ix form:
THA-THB-AI-A2-? . -TKA. ................ Q.......................0
T2 , k =l
THB THE
-TEA 2THA-2TK3-2A1-1 -THA 0 , 2THB 2THB 2THB '
>0
■ 0 -THA THA-THB-Al-A4-j- THB THB
T l , k
1__2THB * i , k - l
A2THBtgb"thbtca
AlTHB•TCA
A im p ATHB
A4-T&0AI m f-1 a THB"THB
(1)
For is.n and j=l,NA we can w r i t e th e f i n i t e d i f f e r e n c e equa
t i o n s a s :
NODE ( n , 1) :
NODE (n ,m ) :
N0DE(n,NA):
T n - l , l * k = 1/THB (©*.^ n , i , k*(2THA)Tn? 2 , k“Tn , 1 , k - 1“
2A2(TCB)-(THB)Tn_1>1}k)
Tn - l , m , k = l/THB(%Tn>m5k-(THA)Tn?m_1>k-
^n,m, jk~^TKB^Tn- l ,m ,k^
Tn - 1 , NA, k= l/^HB ( , NA, k" ( 2THA) Tn ? jfA- ]_} m-
Tn ,NA,k-l-^HB)Tn _ljNA)k-2A4(TCD))
Using the same m a t r ix n o t a t i o n as. def ined p re v io u s ly , th e
system of equa t ions a t i=n can be w r i t t e n a s :
67
^ n - l , k :
2THA-2THB-2A2-1 -2THA 0 *THB THB
-THA 2THA-2.THB-1 -THATHB THB THB u
0 ............................. -2THA 2THA-•2THB-2A4-1THB THB
rn ,k
2A2m.^thbtcb
0
2A4;m-,y.THBiLU
- Tn , k - l (2)
W rit ing th e f i n i t e d i f f e r e n c e equa t ions f o r th e nodes on the
boundary on the r ig h t -h an d s id e g iv e s :
HODS TCC =
(THB)TjtB_tl j x,k~A2(TCB))
NODE (SB,m): TOC - l / 2« a . T KB)ln)K- ( T » ) T MBtI, , 1 )ll.-TJ(B!|affe. 1-
(THA)Tjjg>m- l Jk“ (^HB)Tji}3 _^> m j )
NODE (NB,NA): TCC = 1/A3(©»TWB,.NA}k- (TKA)TNB, NA- 1 , k“
*% B, NA, k - 1 " (THB % B -1, NA, k~PA C TCD))
M atr ix n o t a t i o n g ives :
TCC
'TCC
7HA-THB-A2-A3-P" -THA A3 . A3
o........... ............0
-THA 2THA-2THB-2A3-1 2A3 2A3
-THA o •• 2A3
• • • •* • Q
-THA TH A-THB - A - A4-|rA3 A3 J
TNB,k
I t i s seen from equa t ions 1, 2, and 3 t h a t they a l l a r e of
th e fo rm :
T i * l , k * BT1 >k- r T 1. 1 )k-cT1)!c. 1-B (4)
where r and c a re co n s tan ts and B a square NAxNA m a t r ix and
D a NAxl column m a tr ix . B, I), r , and c a re known a l though
d i f f e r e n t f o r each o f th e t h r e e cases 1 , 2 , and 3 .
Equation 4 expresses f i +p jk as a l i n e a r r e c u r s io n r e l a
t i o n s h i p between Ti_]_jk and Tj_jk _p. or o th e r words
th e tem pera tu re a t s t a t i o n i+1 fo r j=l,NA i s a f u n c t io n of
th e tem pera tu res a t s t a t i o n s i and i - 1 f o r j =l,NA. According
ly we can a l s o say t h a t the tem pera tu re T i j k =^ ( ^ i - l , k j
^ i - l , k - l » ^ i -2 ,k ^ wkere T in d i c a t e s a l i n e a r f u n c t i o n a l r e
l a t i o n s h i p . This can be repea ted f o r and so on. In
t h i s manner we can move in the marching d i r e c t i o n from the
l e f t boundary to th e r i g h t u n t i l we e v e n tu a l ly a r e ab le to
express the elements of i n terms of e lements of T-|)k
and T0?k and c o n s ta n t s , where To?k t ie c o l ^111*1 m a tr ix
whose elements a r e equal to the f l u i d tem pera tu re on. the l e f t
Defining Z ( j ) = T ( l , j ) to d i s t i n g u i s h th e tem pera tu res
a t i = l from those a t o ther l o c a t i o n s an-e lem ent of Tj_>k
would become:
in
i s
(3)
68
69
T ( i , j ) = e i ( j , l ) Z ( l ) i - e i ( 3 > 2 ) 2 ( 2 )+ . . . . . +TCA
where th e ej_'s a r e c o n s t a n t . ..
Def in ing Z(NA-1)-1 we can w r i t e t h i s i n m a tr ix n o t a t i o n :
Z(l)Z(2)
e (1 ,1 ) e±( l , 2 ) . . . . . . e i d j N A ) eiCljNA+l)
Ti ,k =61(2 ,1) ei ( 2 , 2 ) . . . . . . e i (2 ,H A ) ei (2,NA+l)
e i (NA,i) ei (NA,2). . ej_(NA,NA+l
The tem pera tu r e v ec to r i s now i n the form:
Z(NA)
Z(NA-l)
f i j k = S i z (5)
M atrix S i i s an augmented KAx(NA+l) m a t r ix , v ec to r Z an
augmented v ec to r w i th Z(NA+1)=1. For i = l we g e t :
^ l , k " E1 zBut s in ce we a l r e a d y know t h a t we must d e f in e Ei a s :
Si =
1 0 .........
0 1 0 . . . . . 0
0
S im i l a r ly we know t h a t :
?0,k=
TCA
TCA
0 ___ . . . 0 TCA
Ml
O il
0 ____ _ . 0 TCA
0 .............. 0 TCA
Accordingly Eq•j must be def ined a s :
70
The r e c u r s io n equa t ion (1) can. now be w r i t t e n i n the form:
it= ( S ^ i - r i b i _ i - c ± i ^ j - D ) Z - i ^ i . ^ Z (6)
o r : Ei +i=EEi - r3 i - r cTi , k - i " Dwhere now i s of the augmented form:
0 . . . . . . . . 0 T ( i , l )
T ( i ,2 )- 0 . . . . . . . . oTi , k - 1=
0 ___ T(i,NA)
and D i s of the augmented form:
0 ...................... 0 D(1 j 1)
0 ................ 0 D (2 ,1)D=
0 0 D(NA,1)
Both and D a r e now of t h e -order-HA x (NA-1) .
Repeated a p p l i c a t i o n of ( 6) w i l l y i e ld a s e r i e s of
m a t r i c e s S^. The p rocess i s repea ted u n t i l an ex p res s io n
f o r corresponding to the f l u i d s id e of th e r i g h t boun
dary i s o b ta in ed . Then we have:
' TNB+l=ENB+lZ=TCCwhich can be w r i t t e n in expanded form a s :
eNB+l^1 *1 • • • • eN B + l^ * ^ ^
eN B + l^A’ ^^ eNB*l(^,NA)
Z(l)
Z(NA)
T CC- ejjg+ p (1 j NA+-1)
TCc- e 1B+i ( M , N A+1)
(7)
Equations (7) a r e now a s e t of l i n e a r l y independent equa t ions
which can be solved s im ultaneously to y ie ld th e s o lu t io n of
71
Z. Back s u b s t i t u t i o n of Z i n t o equa t ion (5) w i l l then y ie ld
th e d e s i re d tem pera tu re d i s t r i b u t i o n .
A flow c h a r t of the Booy method i s shown on page 72
and a program l i s t i n g on pp. 73- 79, th e nomenclature of the
flow' c h a r t corresponding to t h a t of the computer program.
72
SJOB THS750 BOOY K. GUNDERSEN 7DIMENSION B (11 *11),D<11 .11) ,E1(11.11) .E2<11,11),V(11,1),W3(11.11)* IF1(11.11).F 2 (11,11) ,F3(11 .11)*WD(11,11),W(ll,ll)•02( 11.11 ),B1( 11, 1 21 ).B2(11,11 ),NAME(I3),E(11.11.1 1 )
C---— THIS SECTION READS AND WRITES THE INPUT DATA30 11=0
CALL CLOX(II)12=11-11 11 = 11LOGICAL ZEROREAD(5,10) (NAME(M),M=1.13)READ(5.11) NA.NB.IFREO READ(5.12) DA.DB.DELT.TMAX READ(5*12) AA.AB.FKA,FKBREAD(5 »12) TCA.TCB.TCC.TCD »HA »HB »HC »HD READ(5*13) ZERO
10 FORMAT(13A6)11 FORMAT(3110)12 FORMATt8E10.0)13 FORMAT(LI)
IF(.NOT.ZERO) GO TO 3 N4=NB+1 DO 1 N=1,NA DO 1 M=2,N4
1 W(N.M)~0.GO TO 4
3 CALL MATIN(W ,NA,NB.25.40.2)4 WRITE(6.20) (NAME(M),M=1.13)
20 FORMAT!1H1.13A6)WR1TE(6,21) NA.NB.DA.DB.DELT
21 FORMAT{ 1H0•3HNA = .13,11X ,3HNB=,I 3.11X,3HDA=,E10•3,4X,3H0B=,E 10«3,4X 1,5HDELT =»E10.3)WRITE(6,22) AA.AB.FKA.FKBWRITE(6,23) HA, HQ, HC , HD,TCA.TCB.TCC,TCD
23 FORMAT(1H ,3HHA=E10.3.4X,3HHB=,E10•3,4X.3HHC=.E 10.3.4X.3HHD=,E1C.3 2.4X,4HTCA=,F7.1,4X.4HTCB=,F7.1,4X,4HTCC=,F7.1.4X,4HTCD=,F7.1}
-<3
22 FORMAT(1H •3HAA = .El 0.3.4X,3HAB=,E10•3•AX.3HKA=•El 0•3,4X•3HKB=.E10• 33)
WRITE(6•24)24 FORMATl1H0,74HTEMPERATURES PRINTED IN MATRIX FORM STARTING IN THE
4L0WER LEFT-HAND CORNER)99 FORMAT(1H •1 OX,5HTIME =,E12.5.i6HINT. TIME DIFF.=»I12)
C----- THIS SECTION GENERATES PARAMETERS BASED ON THE INPUT DATAIC0 = 1 T IM E = 0 •ANA =NA BNB=NBY=DA/tANA-1.)X=DB/tBN8— 1•)THA=DELT*AA/tY**2.)THB=DELT*AB/tX**2.)RHOC=FKA/AA Al=HA*DELT/tRHOC* X)A2=HB*DELT/(RH0C*Y)A3=DELT*HC/(RH0C*X)A4=HD*OELT/tRHOC*Y)
C----- THIS SECTION PERFORMS THE FORWARD ELIMINATIONDO 41 1=1,NA DO 41 J = 1»NA BtI * J)=0•B2tI,J)=0.
41 Bit I•J ) =0.DO 42 1=1.NABtI•I)=t2.*THB+2.*THA+l.)/THBB2<I,I)=(2.*THA+2.*THB+1.+2.*A3)/(2.*A3)
42 BttI,I)=(2.*THA+2.*THB+2.*A1+1.)/(2.#THB)B t 1.1)=t2.*THA+2.*THB+1.+2.*A2)/THBB 2 t1 * 1)= t THA+THB+A2+A3+0.5)/A3 Bit 1 , 1) = t THA + THB + A1+A2 + 0.5)/THB 8(NA » N A ) = (2•* THA + 2•*THB +1• +2 •*A4)/THB 8 2 t NA * N A )= tTHA+ THB+A3 +A4 + 0•5)/A3 B 11 NA,NA) = tTHA+THB + A1 + A4 + 0.5)/THB
■vj4
Nl=NA-l 00 43 1=2.N1 8(1.14-1) =-(THA/THB)82(1.1+1)=-(THA/(2.*A3>)
43 Bit I.1 + 1>=-(THA/(2.*THB))DO 44 1=2.NAB (I»I— 1)=-<THA/THB)B2(I.1-1)=-(THA/(2.*A3))
44 Bl(I.1-1)=-(THA/(2.*THB>)B (1.2)=—2.* THA/THBB 2(1.2)=— (THA/A3)B K 1,2 ) =-( THA/THB)B(NA.NA-1)=-2.*THA/THB B2(NA » NA— 1)=— (THA/A3)BKNA.NA-l ) =-( THA/THB)N2=NA+1 DO 47 1=1.NA DO 47 J=1.N2
47 0(I,J)=0.00 48 1=1.NA DO 48 J = 1.NA
48 E 1(I.J)=0.DO 49 1=1.NA
49 El(I,N2)=TCA*Al/THB DO 50 1=1.NADO 50 J=1*N2
50 E2(I.J)=0.DO 51 1=1.NA
51 E2(I.I)=1.CALL CLOX(II)12=11-11WRITE(6.99) TIME.121 1 = 11CALL MATOUT(W.NA.N8.il,11.0.0.0.0) T IME=TIME+DELT
71 CONTINUE
VII
D(1.N2)=-A2*TCB/THB D(NA.N2)=-A4*TCD/THB 00 52 1=1.NA DO 52 J=1,N2 Fit r . j y s f n i . j )F2CI.J)=E2<I•J)02(I.J)=D(I.J)DO 53 J=1.NAD2(J,N2)=D<J,N2)-W(J.2)/(2.«THB)CALL RECUR(B1.F2.F1.D2.F3.11.NA.N2)D(1,N2)=-2.*A2*TCB/THBQ (NA » N 2 )=—2 ® *A4 * TCD/THBDO a 1=1.NADO 8 J=1,N2F 1 ( I.J )=F2(1.J)F2<I,J)=F3(I.J)E(1»I»J)-F3(I»J)D2(I.J)=0(I.J)N3=NB-1 DO 7 M = 2 »N3 DO 6 J=1.NAD2(J.N2>=D(J,N2>-W(J.M+l)/THB CALL RECUR<B,F2.FI.D2.F3.il.NA.N2) DO 60 1=1»NA DO 60 J = 1♦ N2 ECM.I.J)=F3(I.J)FI{I.J)=F2(I.J)F 2( I * J )=F3(I .J)CONTINUED (1,N2)=-A2*TCB/A3 D(NA.N2)=-A4*TCD/A3 DO 57 1=1.NA DO 57 J = 1»N2 D2(I.J)=D(I.J)DO 58 1=1,NA DO 58 J = 1» N2
5253
8
6
607
57
ON
58 Fl< I,J)=F1(I,J)*(THB/A3)DO 56 J=1,NA
56 D2(J.N2)=D(J.N2)-W<J.N4)/<2.*A3>CALL RECUR (B2.F2.F1 ,D2,F3«11 »NA# N2)DO 59 J=1,NA
59 F3(J «N2 )=—F3(J•N2) + TCCC----- SYSTEM OF EQUATIONS SOLVED BY GAUSS-JORDAN ELIMINATION
CALL KJELL(F3»11.11.NA)C----- THIS SECTION PERFORMS THE FORWARD SUBSTITUTION
DO 75 1=1.NA 75 V(I * 1)=F3(I.1)
V(N2,1)=1.DO 64 M = 1»N 3 DO 77 1=1.NA DO 77 J = 1* 1 W3{I,J)=0.00 77 K=1,N2
77 W3(I.J)=W3(I.J)+E(M •I .K)*V(K.J)DO 2 1=1.NA
2 WD( I•M+1)=W3(I ,1)64 CONTINUE
DO 65 1=1.NA65 WD( I,1)=V(I.1 )
DO 35 1=1,NA DO 35 J=2,N4
35 W(I,J )=WD(I .J-l)CALL CLOX{II)12=11-111 1 = I IIF(TIME-TMAX) 73,74.74
73 IF(ICO.EG.IFREQ) GO TO 72 IC0=IC0+1 TIME=TIME+DELT GO TO 71
72 WRITE(6 » 99) TIME,12CALL MATOUTIWD.NA.NB.l1 .11,0.0,0.0)
'•a
IC0 = 1TIME=TIME+OELT GO TO 71
74 CONTINUE GO TO 30
333 STOPEND -
$IBFTC RECUR DECKSUBROUTINE RECUR<A .B.C,D.E.NO•NE•NF)
C A IS THE SQUARE COEFFICIENT MATRIXC B I S THE E(I) MATRIXC C IS THE E(I-l) MATRIXC D IS THE CONSTANT TERM MATRIXC E IS THE E (1+1) MATRIXC NO IS THE DIMENSION OF THE COEFF. MATRIXC NE IS THE NUMBER OF INTERIOR POINTS ALONG THE J AXISC NF IS (NE+l)
DIMENSION A (ND.ND).B(ND.ND)»C(ND.ND),D(ND.ND)* E {NDi ND),Alt 11, 11), A 12(11*11)CALL MULT(A *8 *A1*NE•NE•NF.ND*ND.ND*ND•11 * 11)CALL SUB(A 1.C.A2.NE.NF.ND.ND,ND.NO.ND.ND)CALL ADD(A2.D.E.NE.NF,ND.ND,ND.ND.ND.ND)RETURNEND
®IBFTC KJELL DECKSUBROUTINE KJELL(A ,I A .JA,M)DIMENSION A (IA . JA)«p (1 1 *11) .C (11 .11)
C----- KJELL SOLVES THE SYSTEM USING GAUSS-JORDAN ELIMINATIONN =M +1
19 Z=A(1.1)IF(Z-0. ) 11.6 »11
6 K=N-1DO 9 1=2.K ZK=A(I.1)IFCZK-O.>7.9.7
7 DO 8 J=1,N
^3CO
J
C(I.J)=ACI.J)A( I. JJaAU.J)
8 A C 1 « J )=CC I * J )GO TO 1 1
9 CONTINUE WRITE 16,25)
25 FORMAT (30X.I8HNO UNIQUE SOLUTION)GO TO 18
11 00 12 J=2,N DO 12 1=2,M
12 B (I— 1# J— 1)=A(I» J )—A (1•J)*A(I,l)/A(l,l)DO 13 J =2,N
13 B(M, J - 1 ) =A( I , J ) /A {1 • 1)N=N- 100 14 J = 1 ,N 00 14 '1 = 1, M
14 AC I,J)=B(I,J)IF (N-l)19,16,19
30 FORMAT (40X.F10.3)16 CONTINUE 18 RETURN
ENDSIBFTC SMULT DECK
SUBROUTINE SMULT (A,B*C,N1«N2« N3,Mi,M2,M3,M4,M5,M6) C COMPUTES CCN1.N3) = ACN1CN2) S B (N2(N3) A BEING SPARSE
DIMENSION ACMl,M2),3<M3,M4), CCNi5,M6)00 1 1=1,N1DO 1 J=1,N3
1 C(I,J ) = 0.0 DO 2 I = 1 » N1 DO 2 J=1,N2IFCAC1,J),EQ,0,0) GO TO 2 XI = AC I,J)DO 3 K=1,N3
3 C CI,K ) = CCI.K) + X1*0CJ.K)2 CONTINUE
END
O
80
APPENDIX D
THE GAUSS-JORDAN ELIMINATION METHOD
THE GAUSS-JORDAN ELIMINATION METH0t>
The system of s im ultaneous equa t ions r e s u l t i n g from the
a p p l i c a t i o n of the r e c u r s io n equa t ion i n th e development can
be w r i t t e n i n the form:
Xt +a, nx 0+ ----- . . . . +al n xnnb1ttl l Al +a12x2
a 21x l +a22x 2+
(a)
+a2nxn=b2 ^(1)
an l x l +an2x 2+ +annxn=bn (c)
Using the Gauss-Jordan e l im in a t io n method, we d iv id e equa t ion
1(a) by the c o e f f i c i e n t of th e f i r s t unknown in t h a t equa t ion
o b ta in in g equa t ion 2(a) below. *Ee then m u l t ip ly equa t ion
2 (a) by th e c o e f f i c i e n t o f th e f i r s t unknown i n each of the
remaining equa t ions of 1 and o b ta in as fo l lo w s :
x l +^a l 2 //al l ^ x2+------ ------- + a ln'/ a l l ^ xn=bl ' / a H ^
a 21x l t ^a 21a 12/ a u ) x 2+* • • * •<-^a 21a l n/ a l l ^ xn=a21bi / a l l ^( 2 )
an l x l + (an l a 12/ a H ) x 2+........ + <an i a ln / a n ) xn f an l b l / a l l ( c)i
Next we s u b t r a c t 2(b) from 1 (b ) , 2(c) from 1 ( c ) , e t c , and
l e t t i n g 2(a) become 3(c) we g e t :
81
S2
(a 2 2 - a2 i a i 2/ an ) x 2+ ------ + (a 2n"a 2 1 a ln / a l l ) x n=b2 ~a 2 1 bl //a l l
(3 )(a n 2 “an l a l 2 / a l l ) x 2 4- - - f |' (a nn“an l a l n / a l l ) xn:::V an l b l / a i l (b)
Ca l l - a n l ) x 1+ (a 1 2 - a 12 / a i i ) x 2+ ------* (a ln ~ a ln / a l l )x n=bl - bl / a l l ( c >
R ep ea tin g th e procedure w i l l reduce th e system o f e q u a tio n s
a d d i t io n a l ly , u n t i l e v e n tu a lly th e w hole system i s reduced
to a column v e c to r which c o n s t i t u t e s th e d e s ir e d s o lu t io n o f
th e s im u lta n eo u s e q u a tio n s .
C on sid erin g th e c o e f f i c i e n t m atrix A augmented w ith the
column v e c to r b , we can e x p r ess th e e lem en ts o f th e reduced
augmented c o e f f i c i e n t m atrix a s :
i s 2 , 3 , . . . . ,m
c i - l , ,i_i=a i j " (a l j a i l / a l l ^ J‘=2 >3, • • • • »na H^O
cm, j - l " a l j ^ a H 3 - 2 , 3 , » • • • , n
a H ^0
In th e s e e q u a tio n s:
i-yow number o f o ld m a tr ix A
j=column number o f o ld m a tr ix A
m^maximum row number
n=maximum column number
a=an elem ent o f o ld m a tr ix A
c=an elem ent o f new m a tr ix C
The G auss-Jordan method i s used to s o lv e NA s im u lta n e
ous e q u a tio n s in th e Booy m ethod. The method i s l i s t e d as
83
a su b ro u tin e ,!KJELL" on p .7 8 and th e f lo w c h a r t i s g iv en on
p .S 4 .
84-
STOP
85
APPENDIX E
THE CHOLESKI DECOMPOSITION METHOD
THE CHOLESKI DECOMPOSITION METHOD
A system o f n l in e a r ly in d ep en d e n t, s im u lta n eo u s equa
t io n s l i k e th e one r e s u lt in g from u s in g th e backward d i f f e r
ence method can be w r it te n in th e fo l lo w in g m atr ix n o ta t io n :
(1)
w here: [A] i s an (n x n) m atr ix
[b] i s an(n x 1) column m atrix
M i s an (n x 1) column m atrix
Assuming th a t th e c o e f f i c i e n t m atrix A i s sym m etric, we know
th a t we can f in d a m atrix L such th a t
[l] [l] T = A (2 )
where we d e f in e 1 ^ = 0 , j > i . S u b s t itu t in g (2 ) in to (1 ) g iv e s
= (3)
R ew ritin g (2 ) in elem ent form g iv e s :
i- l l 0 ^ l 121 . . . . . l ln
121 122 0 . . . 0 0 122 ** ** 12n
.M i 1n2 ----- 1nn 0 0 ___
SYM. a 22
P erform ing th e m u l t ip l ic a t io n g iv e s :
l ^ 1=a11_ l 11= i / ^ 1
1 2 1 1 l l =a 2 1 ------- - 12 1 =1 l i a 21
87
For d ia g o n a l c a se when i= j we g e t :
1.r i - i H
D e fin in g th e m a tr ix [dj * j x iT p a g iv e s
M Cm » [b 3
E q uation (4 ) can be w r it t e n a s :
", '~d-
(4 )
111 0 ..............0
121 122 0 . . . . 0
1 , _______ 1n l nn n
P erform ing th e m u l t ip l ic a t io n g iv e s :
l l l d 1? b1--------W 1! !
121dl - 122d2s:b2 -------- d2= (b 2- l 2 1 d l ) / 122
In g e n e r a l we g e t : i - 1
Backward s u b s t i t u t io n in t o L X - D y i e l d s :
x = l ~ b d n nn n
xi =1i ii [ d l ' f c - i likXlc]
The computer program c o n s i s t s o f two su b r o u t in e s , CHOL 1
and CHOL 2 * . The f i r s t perform s th e d eco m p o sitio n and com
p u tes w hereas CHOL 2 perform s th e forw ard and backward
*The su b r o u tin e s w ere ob ta in ed from Dr. H. C h r is t ia n s e n , C iv i l E n g in eerin g De p t . , B. Y. U.
s u b s t i t u t i o n s . The program i s optimized f o r symmetric band
m a t r i c e s . Since th e method r e q u i r e s a p o s i t i v e d e f i n i t e co
e f f i c i e n t m a t r ix , th e pentadifegonal m a tr ix has to be t r a n s
formed i n t o a symmetric, p o s i t i v e d e f i n i t e form b e fo re th e
Choleski method i s a p p l i e d . The c o e f f i c i e n t m a tr ix i s read
i n a r r a y form ta k in g advantage of th e f a c t t h a t symmetry and
band c o n d i t io n s e x i s t .
I f th e c o e f f i c i e n t m a t r ix can be w r i t t e n as fo l lo w s ;
a l l a 12 a 13 0
a 22 a23 a 24
0
0
th e c o e f f i c i e n t a r r a y would be read i n t h i s manner:
A(I)«
a l l
a 12
a 13
a 22
a 23
a 24
a nn
Besides the column m a t r ix on th e r i g h t which i s read in , the
program a l s o u t i l i z e s an a r r a y named KEY(I,1) whose elements
a r e th e band width per row. R e fe r r in g to the c o e f f i c i e n t
m a tr ix , th e KEY a r r a y would be:
KSY(I)=
3
6
9
12
For f u r t h e r in fo rm at io n and d i s c u s s io n on th e Choleski method
see r e f . (10) pp.ZW~.Z77.A program l i s t i n g of the backward d i f f e r e n c e method
u t i l i z i n g the Choleski decomposit ion method i s shown on pp.
92-100. The flow c h a r t of CHOL 1 i s shown on page 90 and the
flow c h a r t of CHOL 2 on page 91, the nomenclature of th e flow
c h a r t s corresponding to t h a t of th e computer programs.
90
91
S JOB
C
THS750 CHOLESKI K. GUNDERSEN 7DIMENSION W{100.1).NAME(13).D(100.1).8(100.1).A{100.100),C(1100.!) 1 . KEY(100.1)
3 0 J 1 = 0CALL C L O X ( J J )J 2 = J J — J 1 J 1 = J J NCE = 0 I T = 0
--- THIS SECTION READS AND WRITES INPUT DATALOGICAL ZEROREAD(5.10) (NAME(M),M=1,13)READ(5.11) NA.NB.IFREQ READ(5 »12) DA.DB.DELT.TMAX READ(5.12) AA ,AB.FKA.FKBREAD(5.12) TCA.TCB.TCC.TCD.HA.HB.HC.HO READ(5♦13) ZERO
10 FORMAT(13A6)12 FORMAT!8E10.0)11 FORMAT(3110)13 FORMAT(LI)
N=NA*NBI F ( . N O T . Z E R O ) GO TO 3 DO 1 1 = 1 , N
1 W( I ,1 )=0 .GO TO 4
3 R E A D ( 5 , 1 2 ) ( W{ I , 1 ) , I = 1 , N )4 WRI TE( 6 , 2 0 ) ( N A M E ( M ) , M = 1 . 1 3 )
2 0 FORMAT( 1 H 1 , 1 3 A 6 )WR I T E ( 6 , 2 1 ) N A . N B , D A , D B , D E L T
2 1 FORMAT ( 1 H 0 , 3 H N A = , I 3 , 1 1 X , 3 H N E - - , I 3 » 1 1 X . 3 H D A = 1 . 5 H D E L T = , E 1 0 . 3 )
. E 1 0 . 3 . 4 X , 3 H D B = . E 1 0 « 3 . 4 X
WRI TE( 6 » 2 2 ) A A . A B . F K A . F K B . T MA X WRI TE( 6 , 2 3 ) H A . H B . H C . H D . T C A . T C B . T C C . T C D
2 3 FORMAT( 1H , 3 HHA=E1 0 . 3 . 4 X . 3 H H B = . E 1 0 • 3 . 4 X , 3 HHC= . E 1 0 . 3 . 4 X . 3 H H D = , E 1 0 . 3 2 . 4 X , 4 H T C A = . F 7 . 1 , 4 X , 4 H T C 0 = » F 7 . 1 , 4 X . 4 HTCC= . F 7 . 1 , 4 X , 4 HTCD=, F 7 . 1 )
vOro
22 FORMAT(1H •3HAA= »El0,3,4X »3HAB=*Ei 0•3,4 X »3HKA=* E10•3*AX * 3HK8=, E 10 • 33,4X.5HTMAX=,E10.3)WRITE(6,24)
24 FORMAT!1H0,42HTEMPERATURES PRINTED IN FORMAT— (N) TEMP--)99 FORMAT!1H •I OX,5HTIME=,E12.5,16HI NT• TIME DIFF.=.112)
332 FORMAT!5X,E12.8)C----- THIS SECTION GENERATES NECCESSARY PARAMETERS
ICO = l TIME=0.0 ANA =NA BNB=NBY=DA/(ANA-l•)X=DB/(BNB-l•)THA=DELT*AA/(Y**2.)THB=DELT*AB/!X**2.>RHOC=FKA/AAAl=HA*DELT/(RHOC*X>A2=HB*DELT/(RHOC*Y)A3=HC*DELT/(RHOC*X)A4 =HD*D£LT/(RHOC*Y)N=NA*NB DO 9 1=1,N
C-----THIS SECTION GENERATES THE PENTA-DIAGONAL MATRIXDO 9 J = 1,N
9 A(I,J)=0.DO ICO 1=1,N
100 A ! I » I ) =- ( 2• *THA+2» *THB+1•)A (1 * 1)=— (THA+THB+A1+A2+0,5)A(N.N)=-(THA+THB+A3+A4+0.5)A(NB,NB)=-(THA+THB+A2+A3+0.5)N3=NB*(NA~1)+lA(N3»N3)=— (THA+THB+A1+A4+0,5)Nl=N—NB DO 110 1=1,Ni N 13 = I+N3
110 A(I *N 13)=THA
'Ou>
N12=NB+1 00 120 I=N12,N N 14 = I—NB
120 A (I,N14)=THA N2=N— 1 NN = 0DO 130 1=1*N2NN=NN+1IF(NN*EQ.NB) GO TO 14A(I.I+1)=THBGO TO 130
14 NN=0130 CONTINUE
KK = 0DO 15 I=2.N KK=KK+1IF(KK.EQ.NB) GO TO 16 A(I,I-1)=THB GO TO 15
16 KK=015 CONTINUE
N4=NB-1DO 210 I=2 »N4 N 5=I+NBA ( I• I )=-(2.*THA+2«*THB+l.+2.*A2)
210 A (I * N 5)- 2 •* THAM6=N3+1DO 220 I=N6,N2 N7=I—NBA (I•I)=— (2•* THA + 2* *THB+1•+2•*A4)
220 A(I,M7)=2.*THA N8=1+NB N9=N3—NBDO 230 1=N8•N9•NBA (I•I)=-(2.*THA+2«*THB+l.+2.*Al)
230 A ( I,1 + 1)=2•* THB
sO
N10=2*NB N 11=N-NBDO 240 I-N1 0 , N1 1 ,NB A ( I ,I)=-(2.*THA+2.*THB+l.+2.*A3)
240 A (I»1-1)=2.*THBN23=N6-NB NY =N 1 0— 1 NX=NB+2
75 CONTINUEDO 62 I=NX » NY DO 62 J = l,N
62 A (I,J)=2.*A<I.J)IF(NX.EQ.N23) GO TO 73 NX=NX+N13 NY=NY+NB GO TO 75
73 CONTINUEC----- THIS SECTION GENERATES THE CORRESPONDING COLUMN MATRIX
C----- THIS SECTION GENERATES THE KEY ARRAYK S UM—0DO 55 1=1,N1KEY!I,I)=N12+KSUM
DO 45 1=1.N 45 D(I,1)=0.
D ( 1.1)= + Al* TCA + A2*TCB DO 41 1=2,N4
41 D(I,1>=+2.*A2*TCBO (NO »1)=+A3*TCC+A2*TCB D (N »1)=+A3*TCC + A4*TCD D (N 3 * 1>=+Al*TCA+A4*TCD DO 42 I=N6 » N2
42 D(I, 1 )=+2.*A4*TCD DO 43 I=N8.N9.N8
43 D{ I , 1 )=+2.*Al*TCA DO 44 1=N10.N 11 * NB
44 D(I,1)=+2.*A3*TCC
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55 KSUM=KEY{ I » 1 1 I2=N1+11=1
56 KEY!12.1)=KSUM+N12— I I F( 12•EQ *N ) GO TO 57 KSUM=KEY(12.1)12=12+11=1+1 GO TO 56
57 CONTINUECALL MATOUTlD.N.l.100.1.0.0.0.0)CALL CLOX(JJ)J2=JJ— J 1WRITE(6» 99) TIME.J2 J1=JJCALL MATOUT(W.N,l.100.1.0.0.0.0)T IME = TIME+OELT
51 CONTINUE00 46 1=1.N
46 B (I•1)=0(I»l)+W(I.l)B{ 1 . 1 )=D(1.1) + W(1,1)/2.D (NO .1)=D(NO.1}+W<NB.l)/2*B (N 3.1 ) =0(N3.1)+W(N3.1 )/2.B (N * 1 ) =0 (N » 1 ) + W ( N • 1 ) /2 •NY=N10— 1 NX=NB+2
65 CONTINUEDO 61 I=NX•NY
61 B(I.1)=2.*B(I ,1 )IF(NX.EQ.N23) GO TO 63 NX=NX+NB NY=NY+NB GO TO 65
63 CONTINUEC----- THIS SECTION GENERATES THE COEFFICIENT ARRAY
N 19=N12
vOCT\
NR = 0 11 = 1DO 52 K = 1,NJ=KDO 53 1=11,N19 C ( I ♦ 1 ) =— A ( K # J )J=J + 1
53 CONTINUEIFlK.EQ.Nl.OR.K.GT.Nl) GO TO 59 I 1=N19+1 N 19 =N1 9+N12 GO TO 52
59 NR=NR+1 I 1=N19+1 N 19 =N19+N12—NR
52 CONTINUEC----- THIS SECTION SOLVES THE SYSTEM USING CHOLESKI METHOD
CALL CHOLKC. 1100,KEY, 100,NCE.N)CALL CH0L2(C,1100,B,100*KEY,100 »N)
C----- THIS SECTION SOLVES FOR THE TRUNCATION ERRORIT=IT+1 IF(IT.EQ.2) GO TO 330
TR = VJ (17,1)TR2 =B(17,1 )GO TO 331 •
330 TR3=B(17,1)TRUN1=0.5*(TR3-2,*TR2+TR)/DELTTRUN2 = ( 1./12.)*(W<19,1)-4.*W(18,1>+6.*W(17,1>-4.*W116,1>+W(15,I))/ 1(X**2.)TRUN3={1,/12.)*{W(31,1>-4.*W(24,1)+6.#W(17•1)-4.*W(10,1>+W(3,1))/t
2Y**2.)TRUNC=TRUN1— TRUN2— TRUN3 I T = 0
331 CONTINUEDO 101 1=1,N
101 W (I , 1 )=0(1,1)
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CALL C L O X ( J J )J 2 = J J - J I J 1 = J JI F { T I ME - T MA X ) 4 8 . 4 7 , 4 7
4 8 I F { I C O . E Q . I F R E Q > G O TO 4 9 I C O = I C O + l TI ME=TI ME+D£ LTGO TO 51
4 9 WRI TE( 6 , 9 9 ) T I M E . J 2 W R I T E ( 6 , 3 3 2 ) TRLNCCALL M A T O U T ( B , N , l , 1 0 0 , 1 , 0 » 0 . 0 , 0 )I C0 = 1TI ME=TI ME+DELT GO TO 51
4 7 CONTINUE GO TO 3 0
3 3 3 STOP END
S I B F T C CHOL1 DECKSUBROUTINE C H O U { A , NA, KEY, NKEY, NCE, N> DI MENSI ON A ( N A , 1 ) , K E Y ( N K E Y , 1 )
C----------- CHOLI PERFORMS THE CHOLESKI DECOMPOSI TI ONI F ( N C E . G T . O ) NCE =0 1 = 1DO 1 J = 1 , NI F ( A ( I . 1 ) . L E . 0 . 0 ) GO TO 2 A ( I , 1 ) = S Q R T ( A ( I , 1 ) )X 1 = A ( 1 , 1 )1 = 1 + 1M= K E Y ( J , 1 )I F ( I • GT• M) GO TO 1 DO 3 L = I , M
3 A { L , 1 ) =A ( L , 1 ) / X 1 I 1 = J + 1 1 2 = 1 1+M- 2I F ( J . N E . l ) 1 2 = 1. 2 - K E Y ( J - l , 1 )
HNC
HNC
HNC
HNCHNC
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0 0 4 L = I 1 , I 2I F ( A U , D . E O . O . O ) GO TO 4 X 1 = A < 1 , 1 )I 3 = K E Y £ L - 1 • 1 ) + l 0 0 5 1 4 = 1 , MA U 3 , l ) = A U 3 , l ) - X l * A U 4 , n
5 1 3 = 1 3 + 1 4 1 = 1 + 11 I =M+1
GO TO 62 NCE = J6 RETURN
END$ I 8 FTC CH0 L2 DECK
SUBROUTINE C H 0 L 2 ( A , N A , B , N B , KEY, NKEY. N ) O I MEN SI ON A ( N A , 1 ) , B ( N 8 , 1 ) , KEY ( NKEY, 1 )
C----------- CH0 L2 PERFORMS THE BACK S UBS TI TUTI ONJ = 1DO I 1 = 1 , NB( I , 1 ) = 6 ( I , 1 ) / A < J . l )1 1 = 1 + 1 I 2 = K E Y < 1 , 1 )I F ( I . N E . l ) 1 2 = 1 2 —KE Y( I —1 , 1 )1 2 = 1 2 + 1 1 - 2 J = J + 1I F ( I 1 . G T . I 2 ) GO TO 1 X 1 = B ( 1 , 1 )DO 2 J 1 = I 1 , 1 2 B ( J 1 , 1 ) =B ( J 1 » 1 ) — X1 * A ( J • 1 )
2 J = J + 1 1 CONTINUE
J =KEY( N , 1 )DO 3 L 4 = 1 , N I =N + 1 —L4B ( I , 1 ) = B \ I , 1 ) / A ( J , 1 )I F ( I . E Q . 1) GO TO 3
HNC
HNC
HNCHNCHNCHNCHNCHNCHNC
HNC
HNC
HNCHNCHNC
HNC
HNCHNC
HNCHNC
HNC
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J=J-1I 2 = 1“ XI1=KEY(12.1)-2+I IF ( I ,NE , 2 ) 11=1 l-KEY(I2-l ,1 )IF(I.GY.II) GO TO 3 M = 11DO 4 M4 = I,11B(I2»1)=Q(12,1)-A(J.1)*B(M,1 ) M=M—1
4 J=J-1 3 CONTINUE
RETURN END
HNCHNC
HNCHNCHNCHNCHNCHNCHNC
100
101
BIBLIOGRAPHY
102
REFERENCES CITED
1. Anderson, J . T. A Review of Dig i t a l - Comp u te r Hea t -T ra n s fe r Progr ams . ASME Paper no. 65-WA7hT-48, 1965*
2. Anderson, J . T . , B o t je , J . M. and K o e f fe l , W. K. D i g i t a lCompu te r S o lu t i on of Complex T ra n s ie n t Heat Trans f e r Problems. West V i r g i n i a U n iv e r s i ty B u l l e t i n , Engineering Expreiment S t a t i o n , Technica l B u l l e t i n No. 62.
3. Booy, M. L. "A Numerical S o lu t io n of P o i s s i o n ' s andL a p la c e ' s Equat ions With A p p l ica t io n s t o Slow Viscous Flow," Jour n a l of Ba s i c E n g in ee r in g , December 1966, pp. 725-733.
4. Bramble, James H. ( e d . ) Numer i c a l S o lu t io n of P a r t i a lD i f f e r en t i a l E qua t ions , pew York: Academic P re s s , 1966.
5. Carslaw, H. S. and J a e g e r , J . C. Conduction of Heat i nS o l i d s . Oxford: At the Clarendon P r e s s , 1959*
6. Gaumer, G i lb e r t R. " S t a b i l i t y of Three F i n i t e D i f fe re n ceMethods of Solving f o r T ra n s ie n t T em pera tu res ," ARS Journa l , pp. 1595-1597- (Oct. 1962)
7. Gay, B. and Cameron, P. T. The E f f i c i e n c y of Numer i c a lS o l u t i ons of th e Heat-Con d u c t io n Equ a t io n . AS?IE Paper no7~b7-WA/HT-17', 1957.
8. Hamming, R. Y«. Numerical Methods f o r S c i e n t i s t s andEngineer s . New York: McGraw-Hill Book Co., I n c . , 1962.
9. Hawgood, John. Numerical Methods in A lgo l . New York:McGraw-Hill Book Co., I n c . , 1955.
10. I r v in g , J . and Mull ineux, N. Mathematics i n Phy s i c s andE ng inee r ing . New York: Academic P re ss I n c . , 1959.
11. I saacso n , E. and K e l l e r , H. B. A nalys is of Nume r i c a lMethods. New York: John Wiley & Sons, I n c . , 1955.
12. James, M. E. e t . ' a l . Analog and D i g i t a l Computer Methodsi n Engineering A n a ly s i s . Scran ton : I n t e r n a t i o n a l TextbooE~Uompany, 1954.
13. Lee, John A. N. Numerical Analysis f o r Computers. NewYork: Reinhold P u b l ish in g Co., 1955.
103
14. R a ls to n , Anthony and W il f , H erbe r t S. MathematicalMethods f o r Di g i t a l Comput e r s . New York: John Wiley & Sons, I n c . , I960.
‘ 15. Richtmyer, R. D. Dif f e r e n c e Methods f o r I n i t i a l - ValueProblems. New Y o r k : ' i n t e r s c i e n c e P u b l i s h e r s , I n c . ,1 9 5 7 . '
16. S chne ider , P. J . Conduct i o n Heat Tr a n s f e r . Reading,• Mass.: Addison Wesley P u b l ish in g Company, I n c . , 1957.
17. Weeg, G. P. and Reed, G. B. I n t r o d u c t io n to Numer i c a lA n a ly s i s . Waltham, Mass.: B l a i s d e l l P u b l ish in g Co.,1 9 5 5 . r
18. Wendroff, Burton . Theor e t i c a l Numer i c a l A n a ly s i s . NewYork: Academic P re s s , 1 9 ^ .
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21. P ersona l communication w i th Dr. Henry C h r i s t i a n s e n ,C i v i l Engineering Department,. Brigham Young U n iv e r s i t y .
INVESTIGATION OF IMPLICIT METHODS
FOR SOLUTION OF THE FOURIER EQUATION
An A b s t rac t of a. Thesis
P resen ted to the
Department of Mechanical Engineering
Brigham Young U n iv e r s i ty
In P a r t i a l F u l f i l lm e n t
of the Requirements f o r the Degree
Master of Science
by
K j e l l S t e in a r Gundersen
May, 1963
ABSTRACT
The purpose cf t h i s t h e s i s was to compare s e v e r a l im p l i
c i t methods f o r s o l u t i o n of th e F o u r ie r conduc t ion equa t ion
w i th r e s p e c t to accuracy and computer t im e.
A FORTRAN program was w r i t t e n f o r each method so lv ing a
r e c t a n g u la r p l a t e problem with convec t ive boundary c o n d i t io n s ,
in c lu d in g o r th o t r o p i c p r o p e r t i e s , a r b i t r a r y g r i d , spacings in
th e two d i r e c t i o n s , a r b i t r a r y i n i t i a l t e m p e ra tu re s , and
a r b i t r a r y f l u i d tem pera tu res and hea t t r a n s f e r c o e f f i c i e n t s
a t th e fo u r b o u n d a r ie s .
The methods were compared and i t was shown t h a t the
Choleski decomposit ion method was f a s t e s t . a n d most a c c u ra te
i n most c a se s , whereas th e Booy method showed . l im i te d a p p l i
c a t i o n because of s t a b i l i t y problems. The Gauss e l im in a t io n
i s a c c u r a te , bu t slow, and th e Gauss-Ss ide l method i s depend
en t upon th e r eq u i r ed accuracy and the method of a c c e l e r a t i o n .
APPROVED: